control and optimization

The 5th International Conference on CONTROL AND OPTIMIZATION WITH INDUSTRIAL APPLICATIONS

BOOK OF ABSTRACTS

27-29 August, 2015 Baku, Azerbaijan

Editor in-Chief Aliev Fikret (Azerbaijan) Deputy Editor in-Chief Gasimov Yusif (Azerbaijan) Editorial Board Aida-zade Kamil (Azerbaijan) Bashirov Agamirze (Turkey) Lakestani Mehrdad (Iran) Larin Vladimir (Ukraine) Mahmudov Nazim (Turkey) Mutallimov Mutallim (Azerbaijan) Nasiboglu Efendi (Turkey) Poqorilyy Sergey (Ukraine) Executive Editor Latifa Agamalieva (Azerbaijan)

The 5th International Conference on

CONTROL AND OPTIMIZATION WITH INDUSTRIAL APPLICATIONS 27-29 August 2015

BOOK OF ABSTRACTS

BAKU - 2015

PREFACE Dear colleagues,

We are pleased to present you the Book of Abstracts of the 5th International Conference on Control and Optimization with Industrial Applications-COIA-2015, held in 27-29, August, 2015 in Baku, Azerbaijan. It was nice to continue this series of conferences founded in 2005 and supported by the Ministry of Communications and High Technologies (former Ministry of Communications and Information Technologies) of the Republic of Azerbaijan. The 2nd conference-COIA-2008 was held also in Baku. Considering the interest of the scientists and to attract new countries and institutions, it was decided to organize the 3rd one-COIA-2011 at Bilkent University, Ankara, Turkey, and COIA-2013 in Borovets, Bulgaria. Then we again came back to the country of the idea of this conference and were pleased to meet our friends and well known scientists on control and optimization here. We honored that more that 250 participants from about 20 countries attended this event. We hope all of them enjoyed being here, had a good platform to discuss their new ideas and trends, share new results. We hope to meet all of you in the next COIA that is planning to be held at St.John's University, New York, USA in 27-29, June, 2017.

CONFERENCE ORGANIZERS

4

The 5th International Conference on Control and Optimization with Industrial Applications, 27-29 August, 2015, Baku, Azerbaijan

CHAPTERS

INVITED TALKS.................................................................................

21-30

CONTROL AND OPTIMIZATION.................................................

31-178

INTELLIGENT AND FUZZY CONTROL.....................................

179-226

NUMERICAL METHODS.................................................................

227-282

MATHEMATICAL MODELING......................................................

283-376

APPLICATIONS...................................................................................

377-440

5


CONTENTS INVITED TALKS Aspects of quantum information M. Abdel-Aty...................................................................................................................

21

Multi-agent networked systems with adversarial elements T. Başar...........................................................................................................................

21

Convex optimization: from embedded real-time to large-scale distributed S. Boyd............................................................................................................................

22

Recent advances in global multi-objective optimization Y. Evtushenko, M. Posypkin........................................................................................

22

Natural gas to liquid transportation fuels and chemicals: process synthesis and global optimization framework Ch. A. Floudas................................................................................................................

25

Human-centric decision making: complex systems and multiagent paradigms J. Kacprzyk.....................................................................................................................

26

On a history of development of the optimal control theory in Azerbaijan M. Mardanov...................................................................................................................

27

To question of the motivation potentially-energy possibilities of the process of the development complecately structured field T.Sh. Salavatov, R.B. Mamadzadeh, A.V. Mamadov................................................... 28 Linear complementary dual codes, a survey P. Solé...............................................................................................................................

29

CONTROL AND OPTIMIZATION A not on the optimal control problem for rigid body motions N. Abazari, Y. Yayli.......................................................................................................

31

Charged balls method M.E. Abbasov..................................................................................................................

32

On a volterra-type optimal control problem with a delay N.G. Abdullayeva, K.B. Mansimov...............................................................................

33

6


Linearized principle of maximum in a problem of control of population dynamics A.I. Agamaliyeva, K.B. Mansimov.................................................................................

34

About optimal control problem for delayed switching systems Ch. Aghayeva...................................................................................................................

35

The problem of optimal control by the transient processes in oil pipelines of complex structure K.R. Aida-zade, Y.R. Ashrafova……………………………………………………… 36 Using PSO for optimal gain selection of integrator controllers in a two-area power system with governor backlash A. Akbarimajd, S. Fallahi............................................................................................

38

Algorithm for solving of the generalized Sylvestre equation F.A. Aliev, V.B. Larin.....................................................................................................

41

Model of Roesser for gas lift process in oil production N.A. Aliev, R.M. Tagiev.................................................................................................

43

Investigation of transient processes in trunk pipelines with intermediate pumping stations J.A. Asadova..................................................................................................................... 46 Asymptotic method for solution of certain optimization and control problems I.M. Askerov, N.A. Ismailov..........................................................................................

49

Kalman filtering for wide band noise driven systems A.E. Bashirov, K. Abuassba...........................................................................................

51

The use of particle swarm optimization for optimal fractional control realization Dj. Boudjehem, B. Badreddine......................................................................................

53

The problem of mayer for discrete and differential inclusions with initial boundary constraints G. Çiçek, E.N. Mahmudov……………………………………………………………

56

Sufficient conditions of optimality for third order polyhedral differential inclusions with boundary value constraints S. Demir, E. Mahmudov................................................................................................

58

BK-spaces generated by using the fractional difference operator S. Ercan, Ç.A. Bektaş.....................................................................................................

61

Respect to two spectra stability of the inverse problem for singular Sturm-Liouville operator A. Ercan, E. Panakhov.................................................................................................

63

7


On controllability and reversibility for some class of the 3D - linear modular dynamic systems F.G. Feyziev, G.H. Mammadova , F.N. Nabi-zade......................................................

65

Optimal contol under set-membership uncertainty (synthesis and realization of optimal closable feedbaks) R. Gabasov, F.M. Kirillova............................................................................................

67

On a linear optimal control problem of Roesser-type systems S.Sh. Gadirova.................................................................................................................

70

Necessary optimality conditions in a discrete control problem with a functional constrained right end of trajectory E.A. Garayeva, K.B. Mansimov.....................................................................................

72

On a minimization of one domain spectral functional Y.S. Gasimov, A. Aliyeva, N. Allahverdiyeva...............................................................

73

Algorithm to the solution of a shape optimization problem for the eigenvalues of Pauli operator Y.S. Gasimov, N.A. Allahverdiyeva, A.Aliyeva, L.I. Amirova..................................

75

Optimization of thickness of finite rectangular plate with circular holes and different boundary condition to minimize weight structure using eso method A. Ghannadiasl, M. Seyedhashemi Dijvejin.................................................................

78

Optimization of hole size and location to maximize the structural strength of concrete beams without additional reinforcement in opening region A. Ghannadiasl, S. Ebadi...............................................................................................

81

A feedforward backpropagation neural network method for the best estimation of outrage rate in medium voltage R. Ghasemi.......................................................................................................................

83

On a determination of the right hand side of the string oscillations equation in the mixed problem H.F. Guliyev, G.G. Ismailova........................................................................................

84

Optimal control problem for equations of flexural-torsional vibrations of the bar H.F. Guliyev, A.T. Ramazanova....................................................................................

86

An optimal control problem for hyperbolic equation with non-local boundary condition with controls at the coefficients H.F. Guliyev, H.T. Tagıyev, A.A. Mehdiyev................................................................. 89 On a boundary control problem for a thin plate oscillations equation H.F. Guliyev, Kh.I. Seyfullayeva...................................................................................

8

91


On numerical solution to an optimal control problem with constraint on the state for a parabolic equation S.I. Guseynov, S.R. Kerimova......................................................................................... 94 Investigation of the optimal control problem for linear impulse systems K.K. Hasanov, Kh.T. Huseynova...................................................................................

97

Investigation of the optimal control problem for heating rod process using boundary control method of moments K.K. Hasanov, L.K. Hasanova......................................................................................

100

On a numerical algoritm for the solution of the inverse problem with respect to potential N.Sh. Huseynova, M.M. Mutallimov, M.Sh. Orucova................................................

103

On a discrete-continuous control problem with functional constrained right end of trajectory G.A. Huseynzadeh...........................................................................................................

106

Optimal control problems with mixed control-state constraints: optimal exploitation of renewable resources M.H. Imanov....................................................................................................................

107

Optimal control problem with controls in coefficients of quasilinear elliptic equation A.D. Iskenderov, R.K. Tagiyev......................................................................................

108

On a linear problem of control of rosser-type hybrid system A.Ya. Jabbarova..............................................................................................................

111

Open optimal problems in voting theory M.M. Konstantinov, P.H. Petkov...................................................................................

112

Conjugate operator method and its application A.L. Karchevsky..............................................................................................................

113

On weak subgradients in nonconvex optimization and optimality conditions R. Kasimbeyli..................................................................................................................

115

Construction of control for one-dimensional heat equation with a delay D.Y.Khusainov, E.I.Azizbayov, I.A.Dzhalladova.......................................................

118

On a strengthening of the discrete maximum principle M.J. Mardanov, T.K. Melikov......................................................................................

121

4D optimal control problem for a Volterra-hyperbolic integro-differential equation of Bianchi type with non-classical Goursat conditions I.G. Mamedov................................................................................................................... 123 9


The optimum control of fluacting system A.C. Mammadov, B.M. Yusifov, I.Z. Aliyev................................................................

126

Approximate solution of a mixed problem, and the optimal control problem for systems with distributed parameters in Hilbert space A.C. Mamedov, B.M. Yusifov, H.H. Alieva.................................................................

128

Finite-approximate controllability of evolution equations N.I. Mahmudov...............................................................................................................

131

On one gaslift problem described by the system of delay differential equations M.M. Mutallimov, U.Z. Imanova...................................................................................

132

Necessary optimality condition in one discrete two-parameter system K.B. Mansimov, S.T. Aliyeva, Zh.B. Ahmadova, A.I. Agamaliyeva...........................

134

On optimality of quasisingular controls in stochastic control problem K.B. Mansimov, R.O.Mastaliyev....................................................................................

135

Necessary optimality condition in a problem of optimal control of one continuous system K.B. Mansimov, Sh.M. Rasulova.................................................................................... 137 Queeuing model with instantaneous and delayed feedback A. Melikov, A. Rustamov..............................................................................................

138

Feedback optimal boundary control of the process of heating the material taking into account the impact of the external environment R.S. Mammadov............................................................................................................... 141 Definition of the suboptimist solution and its finding in the bull programming problem with interval data K.Sh. Mammadov, A.H. Mammadova........................................................................... 142 Optimal dosing strategies against susceptible and resistant bacteria M. Imran..........................................................................................................................

145

Computation of derivative tensors using hypercomplex-steps and application to optimization problems H.M. Nasir.......................................................................................................................

145

Analogue of the discrete maximum principle and investigation of singular controls in a discrete two-parameter control problem М.М. Nasiyyati................................................................................................................

149

10


Heuristic grid method for traveling Salesman problem F. Nuriyeva, G. Kizilates................................................................................................

150

Design of low order controllers for unstable infinite dimensional plants H. Ozbay...........................................................................................................................

153

Asymptotic behaviour of eigenvalues of hydrogen atom equation E. Panakhov, I. Ulusoy..................................................................................................

154

Development of the method for solving combinatorial optimization problems and its application for data compression S.D. Pogorilyy, A.V. Potebnia........................................................................................

155

Quadratic transformations with applications to optimization B. Polyak...........................................................................................................................

158

A meta-heuristic algorithm for the two-dimensional strip packing problem F. Sabaz, H. Kutucu.......................................................................................................

158

On an extremal problem for Goursat-Darboux type inclusion in infinite domain M.A. Sadygov, J.J. Mamedova, H.S. Akhundov..........................................................

160

An extreme problem for a Volterra type integral inclusion M.A. Sadygov...................................................................................................................

163

High order optimality conditions for p-regular inequality constrained optimization problem E. Szczepanik, A. Tret'yakov........................................................................................

168

Placement of optimal capacitor by PSO algorithm F. M. Shahir, P. Mohammadalizadeh, M. Farsadi, F.N. Heris...................................

169

Optimal control of the mobile source for process of intra sheeted burning in oil production R.A. Teymurov................................................................................................................

169

Necessary conditions of optimality of the generalized controls in the system described by the Dirichlet problem М. А. Yagubov, A.A. Yagubov.......................................................................................

172

Necessary conditions of optimality of the singular with respect to components controls in the Goursat-Darboux systems Sh.Sh. Yusubov................................................................................................................

175

INTELLIGENT AND FUZZY CONTROL Fuzzy model and algorithm for solving of E-shop income maximization F.A. Aliev, E.R. Shafizadeh, G.R. Hasanova, J. Zeynalov........................................... 11

179


Analysis of the effectiveness of the methods of recognition of authorship of texts in the Azerbaijani language K.R. Aida-zade, S.G. Talibov.........................................................................................

183

Presentation and analysis of fuzzy rules productions using a modified fuzzy Petri nets M.A. Ahmedov, V.A. Mustafaev, Sh.S. Huseynzade....................................................

184

Fuzzy initial value problems for the second order differential equations Ö. Akin.............................................................................................................................

186

An indicator operator algorithm for solving a second order fuzzy initial value problem Ö. Akin, T. Khaniyev, F. Gokpinar, B. Turksen, S. Bayeg.......................................... 190 The modeling the complex systems of the oil production: intelligent and fuzzy control by L.Zadeh's fuzzy sets theory M.I. Aliyev, I.M. Aliyev, E.A. Isaeva, A.M. Aliyeva, V.M. Baxishov.......................... 193 A new approach to numerical solution of linear fuzzy Fredholm integral equations using iterative method based on fuzzy Bernstein polynomials R. Ezzati, Sh. Ziari, S.M. Sadatrasoul.........................................................................

196

Solution method for a fuzzy delay diferential equation A.G. Fatullayev, N.A. Gasilov, Ş.E. Amrahov............................................................

197

Solving a non-homogeneous linear system of interval differential equations N.A. Gasilov, Ş.E. Amrahov, A.G. Fatullayev............................................................

197

Theoretical and applied aspects of fuzzy structured systems with fuzzy logic of decision-making K.S. Imanov.....................................................................................................................

198

Modeling of fuzzy hierarchical structures described by graphs of partial ordering with the highest vertex K.S. Imanov.....................................................................................................................

202

Feature selection method for recognition of hand-printed characters E. Ismayilov.....................................................................................................................

208

Formulating andobtaining t-best approximation in a fuzzy normed spaceusing some optimization methods K. Ivaz, A. Beiranvand..................................................................................................

210

The existence theorem for fuzzy differential equation M.M. Mutallimov, A.A. Murtuzayeva, B.M. Gasimov, A.Kh. Abdullayev...............

211

12


Indicators of sustainable development strategy Azerbaijan Republic for the investigation of fuzzy methods S.M. Salimov, A.M. Maharramov................................................................................

214

Generalized fuzzy entropy optimization methods with application on wind speed data A. Shamilov, S. Senturk, N. Yilmaz...............................................................................

217

Economic and mathematic model in a virtual business by applying a fuzzy instrument R.Y. Shykhlinskaya, E.R. Shafizadeh, Т.F. Murtuzaliyev…………………………... 221 Fuzzy model of profit maximization in online store R.Y. Shikhlinskaya, N.D. Hajiyev, F.A. Mirzayev....................................................... 223 Chaos genetic algorithm based on fuzzy logic M.J. Varnamkhasti..........................................................................................................

225

NUMERICAL METHODS Numerical solution to the problem of regulating heat supply with feedback K.R. Aida-zade, V.M. Abdullaev..................................................................................

227

Feedback control under different types and forms of observations K.R. Aida-zade, S.Z. Guliyev………………………………………………………….

230

The sweep algorithm for solving the system of partial hyperbolic equations

describing the motion in oil production F.A. Aliev, N.A. Aliev, K.K. Gasanov, A.P.Guliev........................................................ 233 Study of the method of solving the problem for a system of partial differential equations describing the process gas lift N.А. Aliyev, V.Yu. Babanly, А.М. Aliyev.....................................................................

234

A method to solving some identification problem F.A. Aliev, N.A. Ismailov, Y.S. Gasimov, A.A. Namazov, M.F. Rajabov..................

237

Algorithm for finding the coefficient of hydraulic resistance by a small parameter method F.A. Aliev, N.A. Ismailov, A.A. Namazov, K.G. Gasimova.........................................

240

Determination the hydraulic resistance coefficient in gas-lift process by the straight line method F.A. Aliev, N.S. Mukhtarova, N.A. Safarova, M.F. Rajabov......................................

242

On convergence rate of sequence of linear integral operators A.J. Aliyeva, J.M. Aliyev.................................................................................................

245

13


Wavelet bases of hermite cubic spline for linearly constrained quadratic optimal control problems E. Ashpazzadeh, M. Lakestani, R. Mohammadzadeh.................................................. 249 Numerical calculation of processes, described by block-diagonal system with nonseparated initial-boundary conditions between blocks Y.R. Ashrafova................................................................................................................. 252 Hybrid suboptimal control with saturations for the chaotic systems A. Bakkouche, N. Mansouri...........................................................................................

253

2×2 systems of nonlinear integral equations and its approximate solution Z.K. Eshkuvatov, H.H. Hameed, N.M.A. Nik Long, B. M. Taib................................

257

Numerical solution to the system of equations of parabolic type with non-linear boundary conditions G.G. Gasimov...................................................................................................................

257

On one dependence in the two-product model of the economic dynamics S.I. Hamidov....................................................................................................................

259

Eigenfaces vs. fisherfaces vs. SVM and wavelet: recognition using class specific linear projection S. Irandoust-Pakchin.......................................................................................................

262

Uncertainty of Brownian motion and stochastic integral processes M. Khodabin................................................................................................................

262

On an application of hybrid methods G.Yu.Mehdiyeva, V.D.Aliyeva, M.N.Imanova, T.M.Askerov....................................

263

Some recommendation for definition of symmetry of the multistep methods G.Yu. Mehdiyeva, V.R. Ibrahimov, V.D. Aliyeva, A.M. Quliyeva.............................

266

On a factorization of matrix pencils S.S. Mirzoyev...................................................................................................................

269

Optimal control of nonlinear quadratic systems via iterative technique and hybrid functions approximation based on biorthogonal multiwavelets R. Mohammadzadeh, M. Lakestani, E. Ashpazzadeh.................................................

271

An estimate for the curvature of an order-convex set A.B. Ramazanov..............................................................................................................

274

Approximation algorithm to the solution of the optimal stabilization problem for discrete periodic output systems N.A. Safarova, N.I. Velieva, Sh.A. Faracova..............................................................

276

14


Approximate solution of the synthesis problem with measurement errors: discrete case N.I. Velieva, L.F. Agamalieva, K.G. Gasimova............................................................

279

MATHEMATICAL MODELING Definition of pressure field in reservoir deformable under vibrowave methods of oil reservoir stimulation E.M. Abbasov, N.A. Agayeva......................................................................................... 283 Differentiation of a signal based on the operation of integration A.P. Afanasyev, S.V. Emelyanov....................................................................................

285

Boundary value problems for real order differential equations A. Ahmadkhanlu.............................................................................................................

286

On an approach to numerical solution of coefficient-inverse problems for a parabolic equation with nonlocal conditions K.R. Aida-zade, A.B. Rahimov………………………………………………………..

286

Iterative processes and infinite triangle matrices A.M. Akhmedov...............................................................................................................

288

Application of the contour integral method to the solution of the problem for parabolic equation N.A. Aliyev, O.H. Asadova, N.Q. Mamedova...............................................................

290

Basics of mathematical modeling of the gas lift in the well-rerservoir system F.A. Аliev, M.A. Jamalbayov, M.F.Guliyev.................................................................

293

On a Fredholm property of the Dirichlet problem for the Helmholtz equation on the union of two rectangles N.A. Aliev, M.M. Mutallimov, R.T. Zulfugarova, M.A. Namazov, E.H.Mamedhasanov......................................................................................................... 295 Weighted consensus metrics for evaluating scientific journals R. Alguliyev, R. Aliguliyev, N. Ismayilova....................................................................

299

New trends of competition policy of the state in the industry G. Aliyeva........................................................................................................................

301

Investigation of a boundary value problem for a second order ordinary differential equation O.H. Asadova, A.Kh. Abbasova, H.K. Kazimov..........................................................

302

15


On a regularization of some integral equations R.M. Babayev..................................................................................................................

305

Numerical study of the process of displacement of oil by steam based on the global pressure approach D. Baigereyev...................................................................................................................

307

A new approach for five-axis motion of two-parameter families of spheres by using spacelike curves in Minkowski space S. Baş, V. Asil, T. Korpinar............................................................................................

310

On nonlinear wave dynamics in saturated porous media N.V. Bayramova, M.M. Tagiyev, M.M. Mutallimov, E.H. Mamedhasanov.............

311

Measuring the efficiencies of in-house manufacturing and outsourcing of a garment manufacturing firm Ch. Kao.............................................................................................................................

314

Calculating technical efficiency scores using DEA S. Ebadi, A. Ghannadiasl...............................................................................................

316

Modeling of hydrodynamic connection between aquifers Kh.M. Gamzaev, Y.V. Volkova....................................................................................

320

On a model of diagnosing pipelines paraffinization G.G. Gasimov, S.O. Huseynzade..................................................................................

322

Numerical method of determination of the resistance coefficient as a function of fluid velocity and of point of the pipeline section under unsteady flow regime S.Z. Guliyev....................................................................................................................

324

Warehouse management system V. Hasanaliyev, I. Cavadov, K. Aliyev..........................................................................

327

Interpenetrated environment motion in the pipeline by considering the temperature changes A.B. Hasanov, U.R. Babayeva, Sh.A. Hasanova, K.A. Mirzeyeva............................

330

On a class of degenerated quazilinear elliptic equations with non-standart growth condition S.T. Huseynov................................................................................................................

333

Operational – medium-term forecast of electrical energy consumption based on the regional modeling approach A.F. Huseynov, N.A. Yusifbeyli, V.Kh. Nasibov..........................................................

335

Boundary value problems for irrational order differential equations M. Jahanshahi.................................................................................................................

339

16


Solution of one problem for the parabolic type linear loaded differential equation Z.F. Khankishiev.............................................................................................................. 339 Complex quantitative and basis of complex quantitative E.S. Mahmudov, A.A. Haqverdiyev..............................................................................

342

Mathematical modeling in the molecular structure design of materials by using Hartree-Fock-Roothaan theory B.A. Mamedov, E. Çopuroğlu........................................................................................

346

The dependence of the vector Meson’s electromagnetic form factor on the factorization scale Y.V. Mamedova...............................................................................................................

347

Option of sensors for controlling flexible manufacture system J.F. Mammadov, G.N. Atayev, A.H. Huseynov, Z.A. Sadikhov.................................. 350 An approach for exploring new customer requirements in quality function deployment M. Molani, I. Mahdavi, B. Shirazi................................................................................

353

A new method on Fredholm property of a boundary value problem with nonlocal boundary conditions for a three-dimensional elliptic equation Y.Y. Mustafayeva...........................................................................................................

354

Finding routes with minimal number of transfers for public transportation E. Nasiboğlu, M.E. Berberler, Z.N. Odabas Berberler……………………………..

356

A mathematical model for the bin packing problem with group constraints and fuzzy relations R. Nasiboglu, B.T. Tezel, E. Nasibov............................................................................

359

The band collocation problem in telecommunication networks U. Nuriyev, H. Kutucu, M. Kurt, A. Gursoy..............................................................

362

Features of SAS Enterprise Guide for probabilistic modeling system, macroeconomic analysis and forecasting T. Prosyankina-Zharova, O. Terentiev, P. Bidyuk, A. Gasanov……………………

365

Impulsive twopoint boundary value problems for nonlinear qk -difference equations Y.A. Sharifov..................................................................................................................

368

Development of the two-dimensional axisymmetric model of gas lift N.М. Temirbekov, A.K. Turarov..................................................................................

370

17


Model matching h2 output feedback controller design using linear matrix inequality optimization method for obtaining negative stiffness on 4 Pole U-type hybrid electromagnet B.C. Yalçın, K. Erkan....................................................................................................

374

Numerical solution of integro-differential equation systems M. Zarebnia.....................................................................................................................

376

APPLICATIONS The comparison of B-convex, B-1-convex and convex functions G. Adilov, I. Yesilce........................................................................................................

377

Draft labview (laboratory virtual IDE devices) in Azerbaijan Technical University N.B. Akhmedov, A. Humbatova, S. Ismayilov, N. Ramazanova.................................

377

Evaluation of financial stability in industrial enterprises R.M. Aliyev......................................................................................................................

380

A search for repeated polypeptide chain patterns: organization and use of prior structural knowledge V. Amikishiyev, T. Mehdiyev, J. Shabiyev, G.N. Murshudov....................................

382

The equations of heat conductivity for polytrophic gas E.M. Akhundova, A.A. Akhundov.................................................................................

385

Estimation of financial risks in conditions of uncertainty P.I. Bidyuk, O.M. Trofymchuk, A. S. Gasanov, S.H. Abdullayev..............................

388

Characterization of defects using inverse analysis with emat signal and electromagnetism-like mechanism H. Bougheda, T. Hacib, H. Acikgoz, M. Chelabi, Y. Le Bihan...................................

391

Regarding hyperbolic prototype solutions for some nonlinear fractional partial differential equations H. Bulut, H.M. Baskonus................................................................................................

394

An application of the new function method to nonlinear partial differential equations H. Bulut, T. Akturk, Y. Gurefe....................................................................................

396

Application of new investigation methods of deep-pumping wells with viscous - plastic oil J.R. Damirova.................................................................................................................

398

18


Building dynamic homographs of large proper names A.Fatholahzadeh.............................................................................................................

400

Secret sharing schemes and syndrome decoding S. Çalkavur......................................................................................................................

400

Train and check data samples optimal volumes selection method A.S. Gasanov, N. A. Murga, S.H. Abdullayev...............................................................

402

Development of startup-ecosystem in Azerbaijan A. Jafarov.........................................................................................................................

405

Morphometric study of the horizontal dismemberment density of mountain geomorphosystems in the greater caucasus (within Azerbaijan) to optmize resource management M.M. Mehbaliyev............................................................................................................. 408 Mınımızatıon prıncıple of eıgenvalues and rayleıgh quotıent of a boundary value-transmıssıon problem O.Sh. Mukhtarov, K. Aydemir.....................................................................................

412

On the use of computer networks L.M. Nusretli....................................................................................................................

412

The central limit theorem for the family of the first moments of reaching the level of a random walk, described the first-order AR(1) autoregression process F.H. Rahimov, S.A. Aliev, A.D. Farhadova..................................................................

414

Integral limit theorems for the first reaching the level of the random walk described by the first-order AR1 autoregression process F.H. Rahimov, T.E. Hashimov, A.D. Farhadova.........................................................

417

K∞ – robust control of object is with time delay on state R. Rustamov....................................................................................................................

419

Interpolation approach to search hidden results in GPR data R. Samet, E. Çelik, E. Şengönül, S. Tural, M. Özkan.................................................

422

Assessment of scientific productivity on higher education in Azerbaijan: cross-universities analysis S.A. Shabanov.................................................................................................................

425

Survivable network design problems in distribution systems F.A. Sharifov...................................................................................................................

429

19


Relationship between composite piezoelectric properties and crystal chemical parameters of piezofiller and polymer matrix F.N. Tatardar, M.A. Kurbanov, Z.A.Dadashev...........................................................

432

Multifunctional mobile robot M. Tatur, M. Zhartybayeva, K. Iskakov, T. Babayev, A. Pashayev, E. Sabziev.......

432

Design of a service oriented architecture for efficient resource allocation in mass customization R. Vatankhah, A.Vatankhah..........................................................................................

435

Fuzzy-based analysis of social networks L.A. Zadeh, A.M. Abbasov, Sh. N. Shahbazova...........................................................

437

Nonlinear predictive control teleoperation for EAMA: East Articulated Maintenance Arm J. Wu, H. Wu, Y. Song, Y. Cheng, T. Zhang, Zh. Yang……………………………..

439

20


INVITED TALKS

ASPECTS OF QUANTUM INFORMATION Mahmoud Abdel-Aty1 2 1

Zewail City of Science and Technology, Egypt 2 Faculty of Science, Sohag University, Egypt e-mail: [email protected]

In this communication we discuss different aspects of Bioinformatics models and its application quantum information and quantum computer. We focus on the dynamics of charge qubits coupled to a nanomechanical resonator under influence of both a phonon bath in contact with the resonator and irreversible decay of the qubits. Even in the presence of enviroment, the inherent entanglement is found to be rather robust. Due to this fact, together with control of system parameters, the system may therefore be especially suited for quantum computer. Our findings also shed light on the evolution of open quantum many-body systems.

MULTI-AGENT NETWORKED SYSTEMS WITH ADVERSARIAL ELEMENTS Tamer Başar1 1

University of Illinois, Urbana, Illinois, USA e-mail: [email protected]

The recent emergence of multi-agent networks in general, and cyber-physical systems in particular, has brought about several non-traditional and non-standard requirements on strategic decision-making, thus challenging the governing assumptions of traditional control and game theory. Some of these requirements stem from factors such as: (i) limitations on memory, (ii) limitations on computation and communication capabilities, (iii) heterogeneity of decision makers (machines versus humans), (iv) heterogeneity and sporadic failure of channels that connect the information sources (sensors) to decision units (strategic agents), (v) limitations on the frequency of exchanges between different decision units and the actions taken by the agents, (vi) operation being conducted in a hostile environment where disturbances are controlled by adversarial agents, (vii) lack of cooperation among multiple decision units, (viii) lack of a common objective shared by multiple control stations, and (ix) presence of multiple layers in the 21


topologies of the underlying networks. These all lead to substantial degradation in performance and loss in efficiency if appropriate mechanisms are not in place. The talk will identify the underlying challenges, particularly those that are brought about by the adversarial nature of the environment, and also dwell on the research opportunities this broader framework creates for communication, control and game theory. In this context, issues of network resilience, reliability and security will be discussed, with some specific applications in networks with static and dynamic (mobile) nodes, with adversary-inflicted topological changes.

CONVEX OPTIMIZATION: FROM EMBEDDED REAL-TIME TO LARGE-SCALE DISTRIBUTED Stephen Boyd1 1

Information Systems Laboratory, Department of Electrical Engineering, Stanford University, USA e-mail: [email protected]

Convex optimization has emerged as useful tool for applications that include data analysis and model fitting, resource allocation, engineering design, network design and optimization, finance, and control and signal processing. After an overview, the talk will focus on two extremes: real-time embedded convex optimization, and distributed convex optimization. Code generation can be used to generate extremely efficient and reliable solvers for small problems, that can execute in milliseconds or microseconds, and are ideal for embedding in realtime systems.

At the other extreme, we describe methods for large-scale distributed

optimization, which coordinate many solvers to solve enormous problems. RECENT ADVANCES IN GLOBAL MULTI-OBJECTIVE OPTIMIZATION Yuri Evtushenko1, Mikhail Posypkin2 1

2

Dorodnicyn Computing Centre RAS, Russia Institute for Information Transmission Problems, Dorodnicyn Computing Centre RAS, Russia e-mail: [email protected] Many numerical methods for solving multiobjective optimization problems have been

proposed so far[1]. Most of them are heuristic, i.e. they don’t guarantee the optimality of the found solutions. In this paper we extend the non-uniform space covering technique [2] for the global multiobjective optimization. This method constructs the finite set of feasible points and 22


proves its global  -optimality. The space-covering algorithm is guaranteed to converge to a global  -Pareto set within a specified tolerance in a finite number of steps. A multiobjective optimization problem is stated as follows: F ( x)  min, x  X ,

where F : R n  R m is a continuous mapping and X  R n is a compact set. Function F (x) is a vector objective comprising m scalar objectives f (1) ( x),..., f ( m) ( x) . For a point y  R m define





SW ( y)  z  R m : z  y and NE ( y)  z  R m : z  y. A Pareto frontier P(Y ) for a set Y  R m

is defined as follows P(Y )  y  Y : SW ( y)  Y  y. The goal of multiobjective optimization is to find P(Y ) , where Y  F (X ) and such X *  X that F ( X * )  P(Y ) .

Except for special cases where the Pareto set is finite or representable by a finite collection of faces of polyhedron it is in general very difficult to determine the entire Pareto set. Therefore the suitable approximation concept is needed. For a positive   0 define an  -Pareto frontier Y as an arbitrary set satisfying the following properties: 1. Y  Y , 2. P(Y )  Y , 3. Y  NE (Y    em ) . Notice that for   0 an  -Pareto frontier is generally not unique and 0-Pareto frontier coincides with Pareto frontier. An  -Pareto frontier can be constructed deterministically. Paper [3] describes the algorithm that generates a finite approximation and a proof of its  -optimality. Notions of Pareto frontier and  -Pareto frontier can be generalized to a case of an arbitrary combination of criteria maximization and minimization. Let  m be a set of mdimensional vectors with components equal to -1 or 1. For a vector    m notation y2   y1 means that the following m inequalities are simultaneously valid: ( y2(i )  y1(i ) )(i )  0 for i  1,..., m .

23


For y  R m and    m define sets D ( y)  {z  R m : z   y} and D* ( y)  {z  R m : y   z} . The

 -frontier P (Y ) is defined as follows: P (Y )  { y  Y : Y  D (Y )  y} . The union of  frontiers is called the effective frontier of the set Y : Peff (Y ) 

 P (Y ) .

 m

Proposition 1. The effective frontier of a compact strictly convex set coincides with its boundary. The set H (Y ) 

 D ( P (Y )) is called the effective hull of a set Y .  *

 m

Proposition 2. Given an arbitrary compact set Y , it holds that Y  H (Y )  Conv(Y ) , where Conv(Y ) is a convex hull of a set Y . Generally speaking, the effective hull is “closer” to the set Y than its convex hull is and, in this sense, provides a more accurate description of this set. Proposition 2 implies that, for a compact convex set, its effective hull and convex hull coincide with the set itself. We can define  -approximation of the  -frontier in the same way as  -Pareto set: 1. Y  Y , 2. P (Y )  Y , 3. Y  D* (Y   ) . The union of  -approximations for all possible    m is called the  -effective frontier of a set

Y . The intersection H  

D (Y   ) is called the   *

 -effective hull of a set Y . The figure

 m

below depicts the  -effective frontier (white points),  -effective hull (shadowed gray) of as set

Y. The  -Pareto hull can be utilized in other problems where one needs to describe an image of some function, see [3] for examples. Suppose Y  F (X ) , where where F : R n  R m is a continuous mapping and

X  R n is a

compact set. Then we can run algorithm for constructing  -Pareto set

2 n times for all

possible combinations of criteria minimization and maximization given by elements of  m

24


thereby obtaining  -Pareto frontier. From this frontier approximations we can easily derive the

 -Pareto hull. Moreover we can construct the convex set encapsulating Y based on the   following observation Y  Conv  Y    .     m 

References 1. Lotov A.V., Bushenkov V.A., Kamenev G. K., Interactive Decision Maps: Approximation and Visualization of Pareto Frontier, Kluwer Academic, Boston, 2004. 2. Evtushenko Yu.G., USSR Comput. Math. Math. Phys. Vol.11, No.6, 1971, pp.38–54. 3. Evtushenko Yu.G., Posypkin M.A., A deterministic algorithm for global multi-objective optimization, Optimization Methods and Software, Vol.29, No.5, 2014, pp.1005-1019.

NATURAL GAS TO LIQUID TRANSPORTATION FUELS AND CHEMICALS: PROCESS SYNTHESIS AND GLOBAL OPTIMIZATION FRAMEWORK Christodoulos A. Floudas1 1

Texas A&M Energy Institute, Erle Nye '59 Chair Professor for Engineering Excellence, Texas A&M University, USA e-mail: [email protected]

Heavy dependence on petroleum and high greenhouse gas (GHG) emissions from the production, distribution, and consumption of hydrocarbon fuels pose serious challenges for the United States (US) transportation sector. Depletion of domestic petroleum sources combined with a volatile global oil market prompt the need to discover alternative fuel-producing technologies that utilize domestically abundant sources. One of the primary aims in the discovery of novel energy processes is to explore the conversion of natural gas to liquid transportation fuels and high value chemicals. The first part of this presentation will outline the needs and introduce natural gas to liquids and chemicals process alternatives which employ the reverse water-gasshift reaction and a plethora of process alternatives through a superstructure-based optimization framework. The second part of the presentation will address important decisions at the process design and process synthesis level. A thermochemical based process superstructure, its mixed-integer nonlinear optimization (MINLP) model, and systematic approaches for its global optimization will be discussed. Simultaneous heat, power and water integration takes place at the process synthesis stage. 25


Computational studies will illustrate the potential of the proposed approaches for the joint production of fuels and chemicals.

HUMAN-CENTRIC DECISION MAKING: COMPLEX SYSTEMS AND MULTIAGENT PARADIGMS Janusz Kacprzyk1 1

Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland e-mail: [email protected]

We are concerned with broadly perceived distributed information and decision systems, networks, etc. which can involve technical devices (e.g. robots, computer systems), human beings (individuals, groups and maybe even organizations), software agents, etc. They constitute a (possibly) synergistic combination of technology, people and organization aimed at facilitating the communication, cooperation, collaboration, coordination, etc. They should possibly contribute to a more effectively and efficiently functioning to attain some common/shared goal, with mutual benefits for the participating parties. In this talk we focus on multiagent systems which involve, as a key component, a human being, or artificial entity that exhibits some human behavior inspired characteristics. They can be considered as individuals acting just by following their intentions or goals, without paying an explicit attention to those of their fellow agents, and as a group of individuals who have to explicitly pay attention to the intentions, goals or choices of their fellow agents which may be different, even conflicting. We are basically concerned with the human (or human inspired) agents dealt with from an economic perspective, we practically equated with decision making. More specifically, we follow the perspective of the agent-based computational economics viewed as boiling down to the study of economic processes modeled as dynamic systems of interacting economic agents. We consider some aspects of economic (decision making) models that can be employed to make the agents (individuals and groups of individuals) involved in the multiagent systems better follow how the human beings behave and operate, notably in relation to some sophisticated types of rationality, emotions, and other feelings. We show some new paradigms in decision making type models, mostly of the decision analytic type and game theoretic type, and present some examples of experiments, mainly in behavioral economics and neuroeconomics, which clearly suggest a discrepancy between 26


solutions adopted by humans and obtained by using directly the traditional decision analytic and game theoretic models that are in principle based on a greedy utility maximization. We basically indicate that in many real cases the human being is in general not a deliberate, hence a relatively slow, decision maker driven by a greedy and selfish utility maximization, which is a point of departure to virtually all traditional formal models, but is rather an emotional, fast decision maker who is often willing to faster arrive at a decision, even if it is “worse”, and – what is maybe more important – whose behavior is often motivated by a willingness to be fair to others, expecting a reciprocal reaction. We also consider some systemic aspects related to the functioning of such human centric systems: (1) coordination that boils down to how to making different constituents of the systems work together for fulfilling desired overall goals, (2) collaboration that boils down to a joint functioning of the constituents with each other to attain shared goals, and (3) cooperation that boils down to a joint acting of the constituents for their common/mutual benefit. We consider them also from an information theoretic, cognitive, psychological, etc. points of view. We also mention information and knowledge sharing as a crucial element, and mention relations to human affects, judgments and biases, etc. We conclude with a brief account of importance of such a complex systems and economic multiagent systems approach for solving real world human centric problems.

ON A HISTORY OF DEVELOPMENT OF THE OPTIMAL CONTROL THEORY IN AZERBAIJAN Misir Mardanov1 1

Institute Mathematics and Mechanics, National Academy of Sciences, Baku, Azerbaijan e-mail: [email protected]

In the history of civilization, especially in recent years, in the period of rapid development in science and technology, the problem of choosing the best, have caused great interest. In the XII-XIX centuries, this type of problems has been solved by the application of the methods of classic analysis and variational calculus. Since the beginning of the XX century, the expansion of production in all areas, and a gradual reduction of limited natural resources, energy, made necessary the optimal and more efficient use of the resources materials and staff time. Since some of these problems could not be solved with the help of the classical methods of 27


calculus of variations it became necessary to develop new areas and methods of mathematics. The mathematical theory for the solution of these type problems was developed in the 50s of the last century and was called as mathematical methods of the optimal control. The most important step in this direction is associated with the name of Pontryaginin’s "maximum principle". It should be noted the essential impact of the Soviet mathematicians to the above-mentioned areas of modern mathematics. Investigation of the extremal problems in Azerbaijan was firstly met in the works of academician Zahid Khalilov in the beginning of 60s. But the establishment of the scientific school on the mathematical methods of the optimal control is associated with the name of Professor Goshgar Ahmadov. In the early 60s with his leadership in the Department of Differential Equations of the Baku State University, started a scientific seminar devoted to the optimal control. In 1972 the first in the Soviet Union scientific conference in this field was held in Baku. As a result of all these efforts the scientific schools on control and optimization in Azerbaijan were founded and a generation of researchers began to work in this field. At present, more than 20 doctors, 40 PhDs are working in the academic and educational institutions in our country and abroad. In the report detail information will be presented about the establishment of the optimal control theory in Azerbaijan, the significant results obtained by the Azerbaijani researchers since 50s of the past century.

TO QUESTION OF THE MOTIVATION POTENTIALLY-ENERGY POSSIBILITIES OF THE PROCESS OF THE DEVELOPMENT COMPLECATELY STRUCTURED FIELD T.Sh. Salavatov1, R.B. Mamadzadeh1, A.V. Mamadov1 1

Azerbaijan State Oil Academy, Petroleum and Reservoir Engineering Department, Baku, Azerbaijan e-mail: [email protected]

During development and operation of oil deposits often it is necessary to collide with unforeseen conditions and with impossibility of full formalization of decision making in a choice and realization of the necessary geological- technical measures. Oil deposit together with wells and all communications represents complex dynamic system for designing, analysis and management which needs new methods based on principles

28


and large systems theory methods. It is also known that in accordance with the principle of integrity, according to which the large system cannot uniformly be described exactly and for its analysis in different levels are required various methods and models, the traditional determined approach to the description of development processes of fields and production is necessary, but not sufficient and significantly limits the management opportunities. In nowadays the cybernetic methods of technological parameters analysis of development process enable under the current trade information without realization of expensive research operations to make authentic and early diagnosing of transients development caused by change of system status, definition of hydrodynamic flows directions change, account of which is necessary in decision making on regulation of well operation, choice of objects for realization of different kind of geological - technical measures. LINEAR COMPLEMENTARY DUAL CODES: A SURVEY Patrick Solé1 1

Telecom, Paristech, CNRS, Paris, France e-mail: [email protected]

Linear complementary codes (LCD) are linear codes that intersect with their dual trivially. This concept was introduced by Massey, following an Information Theoretic motivation. It was rediscovered more recently by Carlet and Guilley [1] from Boolean masking considerations, of interest in embarked cryptography. The two main results so far in the theory of LCD codes are their asymptotic goodness (Sendrier [6]), and the characterization of the cyclic subclass (Massey [4]). In the present work, we consider the more general subclass of quasicyclic complementary dual codes (QCCD). We use the duality driven CRT decomposition championed in Ling and Solé [2, 3] to derive a criterion for a QC code to be LCD. Since that decomposition was useful to study self-dual quasi-cyclic codes [7] it is natural to consider it gain for studying LCD codes. We also generalize LCD concept [6] to higher fields and rings, and give constructions from rings to _elds using a Gray map. On a more combinatorial vein we give a linear programming upper bound on the size of a LCD code of given length and minimum distance. (joint works with Steven Dougherty, Cem Gueneri, Jon-Lark Kim and Buket Ozakaya) References 1. Carlet C., Guilley S., Complementary dual codes for counter-measures to side-channel attacks, 29


Proceedings of the 4th ICMCTA Meeting, Palmela, Portugal, 2014. 2. Ling S., Solé P., On the algebraic structure of quasi-cyclic codes I:finite fields", IEEE Trans. Inform. Theory, Vol.47, 2001, pp.2751-2760. 3. Ling S., Solé P., On the algebraic structure of quasi-cyclic codes III: generator theory, IEEE Trans. Inform. Theory, Vol.51, 2005, pp.2692-2700. 4. Massey J.L., Reversible codes, Inform. and Control, Vol.7, 1964, pp.369-380. 5. Massey J.L., Linear codes with complementary duals, Discrete Math., Vol.106-107, 1992, pp.337-342. 6. Sendrier N., Linear codes with complementary duals meet the Gilbert-Varshamov bound, Discrete Math., Vol.285, 2004, pp.345-347. 7. Yang X., Massey J.L., The condition for a cyclic code to have a complementary dual, Discrete Math., Vol.126, 1994, pp.391-393.

30


CONTROL AND OPTIMIZATION

A NOT ON THE OPTIMAL CONTROL PROBLEM FOR RIGID BODY MOTIONS Nemat Abazari1, Yusuf Yayli2 1

Department of Mathematics, Faculty of Mathematical Sciences University of Mohaghegh Ardabili, Ardabil, Iran 2 Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey e-mail: [email protected]

In this research the smooth trajectories are computed in the some Lie groups as a motion planning problem by using a Frenet frame to the rigid body system to optimize the elastic energy function which is spent to track any curve in semi-Riemannian 3-manifolds [1,2]. A method is proposed to solve motion planning problem that minimize the integral of the square norm of Darboux vector of a curve in semi-Riemannian 3-manifolds. This method uses the coordinate free maximum principle of optimal control and results in the theory of Hamiltonian systems [3,4]. Keywords: semi-Riemannian, Frenet frame, optimal control, Hamiltonian systems, rigid body motions. AMS Subject Classification: 49L99, 49Q99, 49K35.

References 1. Jurdjevic V., Monroy-Perez F., Variational problems on Lie groups and their homogeneous spaces: elastic curves, tops and constrained geodesic problems in nonlinear geometric control theory, World Scientific, Singapore, 2002. 2. Jurdjevic V., Geometric Control Theory, Advanced Studies in Mathematics, Vol 52. Cambridge University Press, Cambridge,1997. 3. Biggs J., Holderbaum W., Planning rigid body motions using elastic curves, Math. Control Signals Syst. 20: 2008, pp.351-367. 4. Lopez R., Differential geometry of curves and surfaces in Lorentz-Minkowski space, University of Granada, 2008.

31


CHARGED BALLS METHOD* M.E. Abbasov1 1

Saint-Petersburg State University, Petersburg, Russia e-mail: [email protected]

Mechanical analogies in some cases make it possible to build efficient algorithms for solving mathematical programming problems. A well-known heavy ball method [1], [2] allows to solve unconstrained optimization problems. The indisputable advantage of such methods is their ideologic transparency and the confidence in convergence, which follows from the laws of mechanics. Besides that, selectable parameters which can affect the rate of convergence arise in this type of algorithms. In this work we will consider the problem  x  a  min  x X, 





where a is an arbitrary point form R n , X  x  R n | f ( x)  0 and f : R n  R is continuously differentiable convex function. To solve this problem a new algorithm based on mechanical principles is proposed. Keywords: convex analysis, minimal distance, projection. AMS Subject Classification: 90C25, 65K05.

References 1. Dorofeev P. A., Generalized Heavy Ball Methods, Unpublished manuscript, VINITI , No. 5779– 85, 1985 [in Russian]. 2. Zavriev S.K., Kostyuk, F.V. Heavy-ball method in nonconvex optimization problems, Computational Mathematics and Modeling, Vol. 4, No.4, 1993, pp.336-341.

*

The work is supported by the Saint-Petersburg State University under Grant No 9.38.205.2014

32


ON A VOLTERRA-TYPE OPTIMAL CONTROL PROBLEM WITH A DELAY N.G. Abdullayeva1, K.B. Mansimov1 1

Institute of Control Systems of ANAS, Baku, Azerbaijan e-mail: [email protected], [email protected]

The report is devoted to derivation of necessary optimality conditions in an optimal control problem described by Volterra type integral equations with a delay. Let it be required to minimize the functional

S u    xt1  ,

(1)

at restraints ut , x U  R r , t  T  t0 ,t1  ,

(2)

t

xt    f t , s, xs , xhs , u s  ds , t  T  t0 ,t1  ,

(3)

t0

xt   t  ,

t  ht0 , t0   Et0 .

(4)

Here f t , s, x, y, u  is the given n -dimensional vector-function continuous in totality of variables together with f x t , s, x, y, u  , f y t , s, x, y, u  ,  x  is the given scalar function satisfying the Lipschits condition and having the derivatives in any direction, t 0 , t1 are given,

t  is the given n - dimensional continuous initial vector-function, ht  ht   t  is the given continuously-differentiable function, moreover ht   0 . Under the made assumptions, in the considered paper different first order necessary optimality conditions of type 1, 2 were obtained. Keywords: Volterra-type equation, time delay system, nonsmooth control problem, necessary optimality conditions. AMS Subject Classification: 49K15.

References 1. Demyanov V.F., Rubinov A.M. Fundamentals of non-smooth analysis and quasi-differential calculus. М. Nauka, 1990, 432 p. 2. Demyanov V.F., Vinogradova T.K., Nikulina V.N., etc. Nonsmooth problems of theory of optimization and control, LGU publ. 1982, 324 p.

33


LINEARIZED PRINCIPLE OF MAXIMUM IN A PROBLEM OF CONTROL OF POPULATION DYNAMICS A.I. Agamaliyeva1, K.B. Mansimov2 1

Baku State University, Baku, Azerbaijan Institute of Control Systems of ANAS, Baku, Azerbaijan e-mail: [email protected], [email protected]

2

In the report we study a problem of optimal control of population dynamics. Let it be required to minimize the quality functional x1

S u     z T1 , x , z T2 , x , ..., z Tk , x dx

(1)

x0

at restraints ut , x   U  R r ,

t, x  D  t0 , t1  x0 , x1  ,

zt t , x   f t , x, zt , x , yt , x , ut , x  , t , x   D  t0 , t1  x0 , x1  , zt0 , x   ax  ,

x  x0 , x1  ,

(2) (3) (4)

x1

yt , x    g t , x, s, z t , s , u t , s ds .

(4)

x0

Here Ti  t0 ,t1  , i  1, k

g t, x, s, z, u 

t0  T1  T2  ...  Tk  t1 

are the given points, f t , x, z, y, u 

is the given n -dimensional vector-function continuous in totality of variables

together with partial derivatives with respect to z, y, u 

z, u  ,  a1 , a2 ,..., ak 

is the given

continuously differentiable scalar function, t 0 , t1 , x0 , x1 are given, ax  is the given measurable and bounded vector-function, u t , x  is r - dimensional measurable and bounded vectorfunctions of control actions, U is the given nonempty, bounded and convex set of optimal control problem of type (1)-(5) and describes an optimization model of population dynamics taking info account competition for dwelling resources between individuals with different adaptive characteristics 1, 2. By means of the increments method, a necessary optimality condition is established in the form of the linearized condition of maximum. Keywords: dynamics population, linearized principle maximum, necessary optimality conditions, convex control domain. AMS Subject Classification: 49K20. 34


References 1. Bukina A.V., Bukin Yu.S. Investigation of the model of population dynamics by the methods of optimal control problem, Izvestia Irkutskogo Gosudarstvennogo Universiteta. Seriya Matematiki, 2010, No.3, pp.59-66. 2. Bukina A.V., Identification of speciation model by the methods of optimal control theory, Zhurnal JFU, Ser. math. i fiz., 2008, No.3, pp.231-235.

ABOUT OPTIMAL CONTROL PROBLEM FOR DELAYED SWITCHING SYSTEMS Charkaz Aghayeva1,2 1

Faculty of Engineering,Anadolu University, Turkey Institute of Control Systems of ANAS, Azerbaijan e-mail: [email protected]

2

Uncertainty and time delay are associated with many real phenomena, and often they are sources of complicated dynamics. Systems with uncertainties have provided a lot of interest for problems of nuclear fission, communication systems,self-oscillating systems and etc., where the influences of random disturbances cannot be ignored. Many real stochastic process cannot be considered as Markov process, because their future behavior obviously depends not only on their present, but also on their previous states. The differential equations with time delay can be used to model processes with a memory, when the behaviour of the system depends on values of the process of the past.

This research

concerns stochastic optimal control problem of switching systems with delay. The constraint defined by the functional equality on the right end. Consider the following stochastic control system with delay:





dxtl  g l xtl , xtlh , u tl , t dt  f l xt , xt h , t dwtl t  t l 1 , t l  xtl 1   l 1 t , t  t l  h, t l , l  0,1,..., r  1,



xtll1   l 1 xtll , t l



u tl U l  u l ,  L2F

, t

(2)

l  0,..., r  1, xt10  x 0 ,

l 1 , t l ; R

m

| u t,U l

where U l , l  1,...,r are non-empty bounded sets. Let

35

l

 Rm

 l , l  1,...,r

continuous functions  l  , l  1,...,r : tl 1  h, tl 1   Nl  Ol and

(1)

h0.

(3)

 be the set of

(4) piecewise


The problem is concluded to find the control

u1, u 2 ,...,u r

and the switching law t1 , t 2 ,...,t r

which minimize the cost functional: tl  r  J (u )  E   l xtll  p l xtl , u tl , t dt   l 1  tl 1  

    



(5)

which is determined on the decisions of the system (1)- (3), which are generated by all admissible controls U  U 1 U 2  ...U r at conditions:





Eq r xtrr , t r  0.

(6)

To establish the necessary condition of optimality in form of maximum principle Ekeland’s Variational Principle is used. Keywords: differential equation with delay, stochastic switching systems, optimal control problem, necessary condition of optimality. AMS Subject Classification: 93E20.

THE PROBLEM OF OPTIMAL CONTROL BY THE TRANSIENT PROCESSES IN OIL PIPELINES OF COMPLEX STRUCTURE† K.R. Aida-zade1,2, Y.R. Ashrafova2 1

Azerbaijan State Oil Academy, Baku, Azerbaijan Institute of Control System of ANAS, Baku, Azerbaijan e-mail: [email protected], [email protected] 2

We consider the problem of optimal control of the regimes of transient processes in the pipelines of complex structure, differs from many previous cases examined for individual linear sections of main pipelines [1-3]. To be specific, we consider the pipeline containing five linear segments, the structure of which is shown in Figure 1, the arrows in the figure indicate the expected formal (not actual ) direction of fluid flow. Linearized system of differential equations for unsteady isothermal laminar flow of dripping liquid with constant density  in a linear pipe k of length †

This work was supported by the Science Development Foundation under the President of the Republic of Azerbaijan – Grant № EİF/GAM-2-2013-2(8)-25/06/1.

36


lk and diameter d k of oil pipeline network can be written in the following form [3]:

 P k ( x, t ) Q k ( x, t )   2ak Q k ( x, t ),   x t  k k - P ( x, t )  c 2 Q ( x, t ) , x  (0, l ), t  [0, T ], k  1,...,5, k  t x 

(1)

where Pk ( x, t ), Qk ( x, t ) – are the flow rate and pressure of flow of the segment k of the pipe network; c – is the sound velocity in oil;

2a ( k , s ) 

32 is the coefficient of dissipation, (d ( k , s ) ) 2

  const is the kinematic coefficient of viscosity. There are pumping stations at the nodes of pipeline which determine the given pumping regimes Q1 (0, t )  u1 (t ), Q3 (0, t )  u3 (t ),

Q4 (0, t )  u4 (t ), Q5 (0, t )  u5 (t ).

(6)

The following junction conditions are satisfied at internal nodes of the network:

P1 (l1 , t )  P3 (l3 , t )  P 2 (l2 , t ),

Q1 (l1 , t )  Q 2 (l2 , t )  Q3 (l3 , t )  0,

P 4 (l4 , t )  P5 (l5 , t )  P 2 (l2 , t ),

Q 4 (l4 , t )  Q5 (l5 , t )  Q 2 (l2 , t )  0.

(3)

We’ll suppose that an initial state of the process at t  0 is unknown, but the functions Q k ( x,0)  Q0k ( x), P k ( x,0)  P0k ( x), k  K  {1,3,4,5}, x [0, l k ],

(3)

which defining an initial state of the process belong to some admissible set of functions : D  {Q0k ( x), P0k ( x) : x  [0, l k ], k  K} and all the conditions of existence and uniqueness of the

solution to the corresponding initial condition are satisfied The problem consists in finding the smallest transient time T and the values of boundary controls u(t )  (u1 (t ), u3 (t ), u 4 (t ), u5 (t )), t  (0, T ], where the functional

  l

R J (u, T )  T  [Q k ( x, T ; u )  QTk ( x)]2  mesD D kK 0



(4)

 [ P k ( x, T ; u )  PTk ( x)]2 dxd (Q0k )d ( P0k )

gets its minimal value. The functional (4) defines the estimate of the average value of the deviation the state at time t  T from the given desired state (QTk ( x), PTk ( x)) , k  K  {1,3,4,5} of the process for all admissible initial conditions (Q0k ( x), P0k ( x))  D, k  K ;  (Q0k ,  ( P0k )  are the given distributed functions of initial conditions on the set of D ; R  the penalty coefficient, the value of which tends to   . In a study of boundary value and optimal control 37


problems plays an important role the time interval [t 0 , T ] during which the state of the process no longer depends on the initial conditions at t  0 [2],[3]. We use the first order iteration methods of optimization for numerical solution to the problem of optimal control, based on application of obtained formulas for the gradient of the target functional with respect to control functions. The results of carried out numerical experiments of the solution to the problem of optimal control and analyze will be given at the report. Keywords: optimal control, transient processes, oil pipelines network, fluid flow, partial differential equations. AMS Subject Classification: 76M99, 35L04, 65Y04, 49J20, 35Q35.

References 1. Aida-zade K.R., Asadova J.A. Study of Transients in Oil Pipelines, Automation and Remote Control, 2011, Vol.72, No.12, pp.2563-2577. 2. Aida-zade K. R., Ashrafova E. R. Localization of the points of leakage in an oil main pipeline under non-stationary conditions, Journal of Engineering Physics and Thermophysics, Vol.85, No.5, 2012, Springer Science+Business Media New York, pp.1148-1156. 3. Charnyi I.A., Unsteady Motion of Real Fluid in Pipes Nedra, Moscow, 1975,.199p. (in Russian).

USING PSO FOR OPTIMAL GAIN SELECTION OF INTEGRATOR CONTROLLERS IN A TWO-AREA POWER SYSTEM WITH GOVERNOR BACKLASH Adel Akbarimajd1, Samira Fallahi1 1

Faculty of Electrical Engineering, University of Mohaghegh Ardabili, Iran e-mail: [email protected], [email protected]

1. Introduction. In interconnected power systems, local load changes and abnormal circumstances such as generation outage would cause mismatch in frequency and planned inter area power exchange. These disturbances should be compensated through a Load-Frequency Control (LFC). Load frequency control is preferred to be implemented in decentralized form and it is usually established based on a signal called Area Control Error (ACE) which is a linear combination of interchange and frequency errors [1, 2]. Although different decentralized

38


controllers including nonlinear [3] and intelligent [4] controllers have been introduced for LFC, conventional integral controllers are still popular because of their low cost and simplicity. In many real systems there is a governor deadband that imposes a nonlinear effect to the system which cannot be neglected in many cases. This nonlinear effect tends to produce steady oscillations in frequency and tie-line transient response. In this paper we employ a particle swarm optimization method for tuning of parameters of integral controllers in a two-area power system with governor backlash. 2. Studied system. The understudy power system is the system studied in [5] composed of two areas connected via a power line called tie line. Each area is fed by its power source and the power flows between areas though AC tie-line. As the areas are interconnected, load changes in an area causes frequency changes in both areas and changes inter-area power flow. Block diagram of the two area power system is shown in Fig. 1 wherein nonlinear governors, reheat turbines and generators. 3. PSO algorithm for optimal tunings of integrator gains. In PSO Particle Swarm Optimization (PSO), solutions of a problem are considered as particles with a position and velocity. Particles move in the environment to find optimal solution. PSO

Fig. 1. Block diagram of understudy two-area interconnected power system extracted from [5].

captures the concept of social intelligence and uses interaction among particles to solve the problem. Each particle, at each iteration, keeps the position in which it has the best result. Then it shares the information about the best position with the other particles. Position and velocity of particles is updated as: t vidt 1  w.vidt  c1 . 1 .( pidt  xidt )  c2 . 2 .( p gd  xidt )

(1)

xidt 1  xidt  vidt 1 ,

(2)

39


where vidt and xidt are component d in dimension d of the ith particle velocity and position in iteration t, c1 and c2 are constant weight factors, pi is best position achieved so long by particle i, pg is best position found by the neighbors of particle i,  1 , 2 are random factors in the [0,1] interval and w is inertia weight. Using PSO optimum values of integrator gains K1 and K2 (shown in Fig. 1) is obtained. To this end we defined an objective function as: ∞

∞

∞

∞

∞

∞

𝐽 = ∫0 𝑡|𝑑𝑓1|𝑑𝑡 + ∫0 𝑡|𝑑𝑓2|𝑑𝑡 + 10 ∗ ∫0 𝑡|𝑑𝑝|𝑑𝑡 + ∫0 𝑡|𝐴1|𝑑𝑡 + ∫0 𝑡|𝐴2|𝑑𝑡 + ∫0 𝑡|𝑝𝑐1|𝑑𝑡 + ∞

∫0 𝑡|𝑝𝑐2|𝑑𝑡

(3)

where 𝑑𝑓1 is frequency change in area 1, 𝑑𝑓2 is frequency change in area 2, 𝑑𝑝 is power change in tieline, 𝐴1 is amplitude is sinusoidal input of backlash in area 1, 𝐴2 is amplitude is sinusoidal input of backlash in area 2, 𝑝𝑐1 is control signal of area 1 and 𝑝𝑐2 is control signal of area 2. 4. Simulation results. After optimization of integrator gains by PSO, we simulated the system in to scenarios. In the first scenario 1% load change in area 1 is considered and in the second scenario 1% load change in area 2 is considered. The results are compared with those of controllers designed in [5]. Simulation results of frequency changes are shown in Figures 2 and 3. It is obvious that the controllers designed by PSO have better results in terms of transient time and sustaining oscillations. DF2

DF1 0.03

0.04 GDB+PSO GDB

0.02

GDB+PSO GDB

0.02 0.01

0 0

-0.02 -0.01

-0.04 -0.06

-0.02

0

50

100

-0.03

150

0

50

Time (seconds)

100

150

Time (seconds)

Fig. 2. Frequency change in two areas in scenario 1: 1% load change in area 1 DF1

DF2

0.03

0.04 GDB+PSO GDB

0.02

GDB+PSO GDB

0.02

0.01

0 0

-0.02 -0.01

-0.04

-0.02 -0.03

-0.06 0

50

100

150

0

50

100

150

Time (seconds)

Time (seconds)

Fig. 2. Frequency change in two areas in scenario 2: 1% load change in area 2

Keywords: particle swarm optimization, two-area power system, backlash nonlinearity, integral control. AMS Subject Classification: 93, 49.

40


References 1. Khodabakhshian A., Hooshmand R., A new PID controller design for automatic generation control of hydro power systems. Int J Electr Power Energy Syst., 2010, Vol.32, No.5, pp.375–82. 2. Alrifai M.T., Hassan M.F., Zribi M., Decentralized load frequency controller for a multi-area interconnected power system, Int J Electr Power Energy Syst., 2011, Vol.33, No.2, pp.198–209. 3. Mi Y., Fu Y., Wang C., Wang P., Decentralized sliding mode load frequency control for multiarea power systems, Power Systems, IEEE Transactions on, Vol.28, No.4, 2013, pp.4301-4309. 4. Mathur H.D., Manjunath H.V., Extended fuzzy logic based integral controller for three area power system with generation rate constraint, ICIT 2006 IEEE international conference on industrial Technology, 2006, pp.917–21. 5. Tsay, Tain-Sou., Load–frequency control of interconnected power system with governor backlash nonlinearities, International Journal of Electrical Power & Energy Systems, Vol.33, No.9, 2011, pp.1542-1549.

ALGORITHM FOR SOLVING OF THE GENERALIZED SYLVESTRE EQUATION F.A. Aliev1, V.B. Larin2 1

Institute of Applied Mathematics, Baku State University, Baku, Azerbaijan 2 Institute of Mechanics, Academy of Sciences of Ukraine, Kiev, Ukraine e-mail: [email protected], [email protected]

An algorithm of constructing of the solution of the generalized Sylvester equation, which is based on the transformation of Cayley and the relation of Bass, is considered. It is noted that the use of the Cayley transform allows to essentially simplify the computational procedure of the algorithm. It is shown that the algorithm can be realized by using the procedures of symbolic calculations of the package MATLAB. 1. Introduction. An algorithm of the solution of the generalized Sylvester equation EX  AXB  C ,

(1)

is considered, in which as in [1] is used the Bass relation. However in contrast to [1], here the Cayley transform [2] is applied, which allows to significantly simplify the computational procedure and to use (if necessary) the solution (1) of the procedures for symbolic computation of the package MATLAB for construction with high accuracy. 2. Bass relation [1,3]. Let us rewrite the equation (1) in the following form

41


I  I  M 1    F1   B , X  X 

 B M1    C

0 , E 

I F1   0

(2) 0 . A

It is necessary to transform (2) to the form I  I  M p    F p   (B) , X  X 

(3)

here (B) – some polynomial of matrix B . For suggesting (2) to the form (3) can be used an algorithm [1]. Obviously, if (B)  0 (for example, (B) – is the characteristic polynomial of the matrix B ), then the relation (3) is transforming to the following linear equation relatively X : I  M p    0. X 

(4)

3. Cayley transformation [2]. Let the matrices M1 , F1 are defining by (2). We will define the matrix Z as follows: Z  (M 1  F1 ) 1 (M 1  F1 ) .

(5)

The value of the parameter  is defined by the condition of invertibility of the matrices

M1  F1

and

B  I .

Taking

into

account

that

Z  I  2 (M 1  F1 ) 1 F1 ,

I  I  Z  I  2(M 1  F1 ) 1 M 1 we can rewrite (2) as follows:  ( Z  I )    ( Z  I )   B . X  X 

Let transform this relation to the form. I 

I 

I 

I 

I 

I 

Z        Z   B    B , Z  ( B  I )   ( B  I ) , X  X  X  X  X  X  I  I  Z      R , R  ( B  I )(B  I ) 1 . X  X 

(6)

The relation (6), in which the matrix Z is defining by (5), can be considered as the Cayley transformation of the relation (2). Note that the structure of (6) coincides with the structure of (2), which simplifies the procedure for determining of the matrix M p in (4), however, in contrast to (2), in (6) is not assumed the invertibility of the matrix F1 . Thus, can be stated that the Cayley transformation allows to simplify the procedure of defining of the matrix M p in (4).

42


4. The procedures of the symbol calculations. Let consider more detail the possibility of the realization of considered algorithm by procedures of symbol computations of the package I 

MATLAB. As follows from (4), the matrix   defines the null space of the matrix M p . Thus, X  if to rewrite the relation (4) in the form M p N  0 , where the matrix N defines the null space of the matrix M p , then for finding N can be used the procedure null.m of package MATLAB.  N1   , the matrix N1 is quadratic, we can write the N 2 

Dividing the matrix N into blocks: N  

following relation for the solution (1): X  N 2  N11 . Thus, for the realization of the described algorithm is required the procedures of the inversion of the matrix, finding its characteristic polynomial and finding of its null space. All these calculation operations allow the realization by the use of procedures of symbolic calculations of the package MATLAB. Keywords: generalized Sylvester matrix equation, Bass relation, Cayley transformation. AMS Subject Classification: 15B99, 11C08, 11C20.

References 1. Larin V. B. Solution of Matrix Equations in Problems of Mechanics and Control, Int. Appl. Mech.,Vol.45, No.8, 2009, pp.847–872. 2. Aliev F.A., Bordyug B.A., Larin V.B. Calculation of the optimal stationary regulator, Techn. Cybernetics, No.2, 1985, pp.143–151. 3. Kwakernaak H., Sivan R., Linear Optimal Control Systems, Wiley, Interscience, New York, 1972, 650 p.

MODEL OF ROESSER FOR GAS LIFT PROCESS IN OIL PRODUCTION N.A. Aliev1, R.M. Tagiev2 1

2

Baku State University, Baku, Azerbaijan Institute of Mathematics and Mechanics ANAS, Baku, Azerbaijan e-mail: [email protected]

As it is known, gas-lift method is one of the main stages of the oil production for, which designed the different mathematical models of 1, are designed to describe the motion in gas-lift 43


process. Further, with their assistance different tasks are set, such as, to produces the maximum amount of oil with the minimum gas supply 2, the determination of the coefficient of hydraulic resistance and formation parameters of the gas and liquid mixture (GLM) 1 and etc. However, all these tasks have been solved for the lumped differential equations derived from the distributed systems of differential equations of hyperbolic type, describing the spatial movement of the gas lift process assistance of different versions of the averaging method. Based on the last it can be stated that, mainly, these researches are approximate and are more relevant researches of the initial case, where 3 are presented their solutions, with the assistance of special infinite series that provide approximate solutions with any accuracy. Consider the following system of hyperbolic equations, describing the motion in gas-lift process . P( x, t ) Q( x, t )  F   2aQ( x, t ),   x t   F P( x, t )  c Q( x, t ) , x  (0, l ), t  (;),  t x 

(1)

with the boundary conditions

P(0, t )  P0 (t ),  Q(0, t )  Q0 (t ),

t   ;,

(2)

where F, a, c, l, and T are positive constants, а and given P0 (t ) and Q0 (t ) real values of smooth functions, and unknowns to be determined. For the applicability of the model of Roesser in the system (1) is necessary to make the following change P( x, t )  R( x, t )  Q( x, t ) ,

(3)

noting that by using solutions of problems (1) - (2) the initial values of the unknowns and are determined from the view 4,5 (see eq. (3)), where it is presented as a series of unknown coefficients, which are determined not from the system of differential and algebraic equations. Here, R( x, t ) a new unknown function, replacing P( x, t ) . Using (3) in the system (1) we find:

 R Q  x   x    Q   R   t t

1  c Q R  2a     Q, F  F x t  F c Q . F x 44

(4)


Taking designations: Q( x, t )  W ( x, t ), x R( x, t )   ( x, t ). t

(5)

from (4) we obtain :

 R( x, t )  c 1 2a   x   F 2  1W ( x, t )  F  ( x, t )  F Q( x, t ),     Q( x, t )    ( x, t )  c W ( x, t ). F  t

(6)

Thus, as the system (5), the system (6) has the form which is applicable the Roesser scheme. As R( x, t ) and P( x, t ) and Q( x, t ) are studied with the by the difference scheme. Next, is considered an example where P0 (t ) and Q0 (t ) are constants. In this work are presented the Roesser model, which is a discrete analog of the system of hyperbolic equations, described the motion in gas-lift process in oil production. Showing that in the transition to the discrete case there is a perturbation, which are determined from the relevant systems of linear algebraic equations. Note that, this model can be successfully used in the determination of the coefficient of hydraulic resistance in the lift and coefficient GLM in the joint end of the annular space and beginning of the lift, the determination of the optimal mode, etc..

Keywords: hyperbolic equation, gas lift, difference equations, model of Roesser. AMS Subject Classification: 39A10, 39A12

References 1. Aliev, F.A., Ismailov, N.A. Inverse problem to determine the hydraulic resistance coefficient in the gas lift process. Appl.Comput. Math., Vol.12, No.3, 2013, pp.306-313. 2. Aliev, F.A.. Minimax solution to the problem of choosing the optimal modes of gas lift, Reports of ANAS., 201, No.1. pp.27-36. 3. Guliev A.P., Tagiev R.M., Gasimova K.G., Computational algorithm to solution of boundary problem of the system of hyperbolic type in gas lift process. Proceedings of the Institute of Applied Mathematics, Vol.3, No.1, 2014, pp.105-111. 4. Roesser R., A discrete state space model for linear image processing. IEEE Trans. Autom. Contr., vol.AC-20, No.1, 1975.

45


5. Aliev N.A., Aliev, F.A., A.P. Guliyev, Ilyasov M.Kh., Method series to solving a boundary value problem for the system of hyperbolic equations, arising in the oil production. Proceedings of the Institute of Applied Mathematics, Vol.2, No.2, 2013, pp.113-136.

INVESTIGATION OF TRANSIENT PROCESSES IN TRUNK PIPELINES WITH INTERMEDIATE PUMPING STATIONS J.A. Asadova1 1

Institute of Control Systems of ANAS, Baku, Azerbaijan е-mail: [email protected]

In the work, we propose the numerical approach to solution to the problem of optimal control of transient processes in trunk pipelines with intermediate pumping stations, as well as derive principal formulas and give algorithms of its numerical solution based on first order optimization procedures. We derive the corresponding analytical formulas for the components of the gradient of the criterion functional with respect to the control actions on the class of piecewise continuous functions. We give the results of numerical experiments obtained by solving several model problems. For the purpose of decreasing the financial charges when transporting hydrocarbon feedstock, related to the difficulty of keeping accurate record of the quantity of transported feedstock under transient processes, as well as for the purpose of increasing efficiency of exploiting oil pumping station facilities and in order to avoid emergency situations on the pipeline, it is necessary to investigate the problem of optimal control of transient processes on the pipeline for carrying out them in the shortest time while at the same time meeting technological constraints on the oil delivery process. The present work is dedicated to address such issues. We consider the process of unsteady fluid motion in the pipeline with intermediate n  1 pumping stations. This process in the linearized formulation is described by the following system of hyperbolic equations in dimensionless variables [3]:  p n1     ui (t ) ( x  xi )   2b  ,   x i1 t   p   ,  x  t

b

46

al , 0  x  1, 0  xi  1 , t  0 . c

(1)


Here a is the dissipation factor; l  the pipeline length; c  the velocity of sound in liquids; b  the dimensionless frictional coefficient; p  p( x, t ) and    ( x, t )  the average values of the pressure and velocity in the section, respectively;  (x)  the Dirac delta-function; xi  (0, l ) and ui (t ), i  1,...,n  1  the locations of discharge heads and their strength, produced at the

intermediate pumping stations. There are boundary, initial, and final conditions ( x, t )  0  const, p( x, t )  p0 ( x), 0  x  1, ( x, t )  T  const,

p( x, t )  pT ( x),

(0, t )  un (t ), (1, t )  un1 (t ) ,

0  x  1,

t 0 ,

(2)

t  T,

(3)

t [0,T ] ,

(4)

where T is the transient-process time, after which a new steady-state regime (3) will progress. In the considered problem, the control is fulfilled by changing the pressure generated by the intermediate pumping stations and by changing the flow rate (velocity) or the pressure at the ends of the pipeline. It is required to find the functions u(t )  (u1 (t ),...,un1 (t )) , under which the transient-process time T is minimal. The criterion functional is as follows:

  r ( x, t )   

T  DT l

J (u, T )  T 

2

1

T

T



 r2  p( x, t )  pT ( x) dxdt  min . 2

(5)

0

When controlling the change of the pressure or flow rate on the pumping stations, it is necessary to meet certain technical constraints on the values of these control parameters given the Q  H characteristics of the pumps [3]: psti ( ( xi , t ))  ui (t )  psti ( ( xi , t )) , i  1,..., n  1 ,

u n  un (t )  un ,

u n1  un1 (t )  un1 ,

t 0.

t  0,

(6) (7)

Here psti ( ( xi , t )) , psti ( ( xi , t )) , i  1,...,n  1 , u i , ui , i  n, n  1 , are given lower and upper bounds of the admissible values of the pressure (discharge heads) or velocities (flow rates), respectively, which the pumping stations can work with. It is necessary to meet certain technological constraints on the values of the pressure, which can be considered as phase constrains: p  p( x, t )  p,

x  (0,1),

t  0, T  ,

(8)

proceeding from the requirements of endurance of the pipeline and of providing normal operating conditions of the facilities [3]. For numerical solution to the problem of optimal control of transient processes in the pipeline, we propose to use iterative first order optimization procedures, which are based on the 47


application of analytical formulas for the gradient of the criterion functional with respect to the control parameters. Formulas for the components of the gradient of the criterion functional with respect to the strength of discharge heads produced by the intermediate pumping stations. Using the method of variation of the optimizable functions [2], we prove that the functional (5) in the optimal control problem (1), (2), (4)-(8) is differentiable with respect to ui (t ) , i  1,...,n  1 , in the space L2 [0, T  DT ] , and its gradient is determined by the formula:

gradu J (u, T )   1 ( xi , t ) , i

i  1,...,n  1 ,

t [0, T  DT ] .

(9)

Here the pair ( 1 ( x, t ) , 2 ( x, t ) ) is solution to the corresponding adjoint boundary-value problem. Formulas for the components of the gradient of the criterion functional with respect to the flow rate fed by the pumping stations at the ends of the pipeline. gradu J (u, T )   2 (0, t ) ,

t [0, T  DT ] ,

(10)

gradu J (u, T )   2 (1, t ) ,

t [0, T  DT ] .

(11)

n

n 1

Formulas for the components of the gradient of the criterion functional with respect to the point of time T 1

gradT J (u, T )  1   r1 ( ( x, T  DT )   ( x, T )  2T )( ( x, T  DT )   ( x, T ))dx  0

1

  r2 ( p( x, T  DT )  p( x, T )  2 pT ( x))(p( x, T  DT )  p( x, T ))dx

(12)

0

Using the formulas obtained in the work, we have conducted numerical analysis of dependence of the optimal transition-process time from its parameters, the number and locations of the pumping stations, the range of admissible values of the control actions, and the values of the initial and final steady-state regimes ([1]). These investigations generalize the results of the works ([1], [4]), obtained for boundary controls, in cases when there are intermediate pumping stations considered as control actions. Keywords: trunk pipeline, intermediate pumping stations, optimal transient period of process, steadystate regimes, control actions, range of admissible controls. AMS Subject Classification: 49J15, 49J35.

48


References 1. Aida-zade K.R., Asadova J.A., Study of transients in oil pipelines, Automation and Remote Control, Vol.72, No.12, 2011, pp.2563-2577.` 2. Vasilyev F.P., Optimization Methods, Moscow, Factorial Press, 2002, 824 p. (in Russian) 3. Guseynzade M.A., Yufin V.A., Unsteady Flow of Oil and Gas in Trunk Pipelines, Moscow, Nedra, 1981, 232 p. (in Russian) 4. Ilyin V.A., Two-endpoint boundary control of vibrations described by a finite-energy generalized solution of the wave equation, Diff. Equations, Vol.36, No.11, 2000, pp.1652-1659.

ASYMPTOTIC METHOD FOR SOLUTION OF CERTAIN OPTIMIZATION AND CONTROL PROBLEMS I.M. Askerov1, N.A. Ismailov2 1

2

Lenkaran State University, Lenkaran, Azerbaijan Institute of Applied Mathematics, Baku State University, Baku, Azerbaijan e-mail: [email protected], [email protected]

In the paper the procedure of Lagrangian factors and the principle of Lagrangian for the Lagrangian problem [2] are considered. Such problem has been studied in a large number of works [3]. But since the considered problem is in the specific form, it allows to study the corresponding problem in the different aspect. Let suppose that,  2  1  0 are fixed numbers, f 1 : R n   1 ,  2   R n , f 2 : Rn  1,  2   Rn are continuous functions and the vectors u 0 , u1 , ..., u m are boundary controllers from R n . The set of n -dimensional continuously differentiable vector functions, respectively, on the intervals

0, l 

and l1 , 2l  are denoted by C 1 (0, l , R n ) and C 1 (l1 , 2l , R n ) , where 0  l  l1  2l . If

y  ( y1 ,, yn ), z  ( z1 ,, z n )  R n , then we can denote that yz  y1 z1    yn z n . In this

expression, the left-hand vector is considered as row-vector and the right-hand vector is considered as column-vector. Q1 ( x) and Q2 ( x)

n  n -dimensional symmetric, continues

~ matrices with respect to x , but Q is n  n -dimensional symmetric matrix. In this paper, for

every fixed number    1 ,  2  , when u 0 , u1 , ..., um  R n minimization of the functional l

1 ~ J (u 0 , u1 ,..., u m ,  )  z (2l ) Q z (2l )   y( x) Q1 ( x) y( x)dx  2 0

49


2l

m

l1

i 0

  z ( x) Q2 ( x) z ( x)dx   i u i  i u i

(1)

and of the problem y ( x)  f 1 ( y( x),  )

(2)

y(0)  u0   u1  ...   mum

(3)

z( x)  f 2 ( z ( x),  )

(4)

z(l1 )  y(l ,  )   ( y(l ,  ))

(5)

has been considered in the set of solutions y()  C 1 (0, l , R n ), z ()  C 1 (l1 , 2l , R n ) .The algorithm of solution of this problem is given and this algorithm is applied to gas lift process [1, 4], where  is n  n dimensional matrix,  : R n  R n is continuously differentiable vector function and  i , i  1,..., m are given numbers or n  n - dimensional symmetric matrices. Keywords: optimal control, asymptotic method, gaslift process. AMS Subject Classification: 49J15, 49M25, 49N10.

References 1. Aliev F.A., Ismailov N.A.,Temirbekova L.N. Methods of solving the choice of extremal modes for the gas-lift process, Appl. Comput. Math., Vol.11, No.3, 2012, pp.348-357. 2. Alekseev V.M., Tixomirov V.M., Fomin S.V., Optimal Control, M.:Nauka, 1979, 430 p. 3. Asepkov L.T., Optimal control system with intermediate conditions, Applied Math. Mech, Vol.45, No.2, 1981, pp.215–222. 4. Mutallimov M.M., Askerov I.M., Ismailov N.A., Rajabov M.F., An asymptotical method to construction a digital optimal regime for the gas-lift process, Appl. Comput. Math., Vol.9, No.1, 2010, p. 77-84.

50


KALMAN FILTERING FOR WIDE BAND NOISE DRIVEN SYSTEMS Agamirza E. Bashirov1,2, Kanda Abuassba1 1

Eastern Mediterranean University, Gazimagusa, North Cyprus, via Mersin 10, Turkey 2 Institute of Cybernetics, ANAS, B. Vahabzade Str. 9, Az1141, Baku, Azerbaijan e-mail: [email protected], [email protected]

Filtering theory is heavily based on the white noise model of real noises. The most significant result in filtering theory, the Kalman filter, that originates from the pioneering work of Kalman [1] and found great applications in space engineering, telecommunication, etc., has been discovered for partially observable linear systems with a white noise disturbance. At the same time engineers observed that the real noises are not white indeed. At most, they are approximately white and, in general, far from being white. In this regard Fleming and Rishel [2] wrote that the real noises are wide band and white noises are the ideal case of wide band noises. When the parameters of white and wide band noises are sufficiently close to each other, white noises take place of wide band noises to make mathematical models simpler. Respectively, for more adequate estimation and control results, a mathematical method of handling and working with wide band noises is required. A white noise is a generalized derivative of a Wiener process. This derivative does not exist in the ordinary sense. Therefore, it can be suspected that the real noises are just “uncompleted derivatives” of Wiener processes because in some cases the white noise model of real noises produces more or less adequate results. If 𝑤 is a Wiener process, then its 𝜑𝑡 =

“uncompleted derivative” is equal to

𝑤𝑡−𝜀 −𝑤𝑡 𝜀

𝑡

= ∫𝑡−𝜀

𝑑𝑤𝑠 𝜀

, 𝑡 ≥ 0, where 𝜀 is a small

positive number. Here 𝜑𝑡 behaves as a wide band noise with the covariance function Λ𝜎 = 𝐼(𝜀−𝜎) 𝜀2

, 0 ≤ 𝜎 ≤ 𝜀, where 𝐼 is an identity matrix of respective dimensions. Generally, a wide band noise is defined as a random process 𝜑 such that its

autocovariance function Λ𝑡,𝜎 is a nonzero function on some interval [0, 𝜀] and vanishes outside. Based on the preceding discussion, the integral representation 𝑡

𝜑𝑡 = ∫max(0,𝑡−𝜀) Φ𝑡,𝑠−𝑡 𝑑𝑤𝑡 , 𝑡 ≥ 0, where Φ is a continuous in the first variable and square integrable in the second variable matrixvalued function, 𝜀 > 0, and 𝑤 is a vector-valued Wiener process, becomes a realistic representation for wide band noises. We call the function Φ as the relaxing (damping) function

51


of the wide band noise 𝜑. This representation was intensively used in Bashirov [3] to solve different control and filtering problems for wide band noise driven systems. The main difficulty of working with wide band noises is that they are observed only by autocovariance function. At the same time, there are infinitely many distinct wide band noises, which have the same autocovariance function. Therefore, it is important to obtain filtering and control results in terms of autocovariance function rather than relaxing function. Such results are called invariant results. This problem was raised in Bashirov [4] as an unsolved problem in mathematical systems and control theory. Below we present an invariant Kalman filtering result for a linear signal system disturbed by a wide band noise and an observation system by a white noise. Consider the system {

𝑥 ′ 𝑡 = 𝐴𝑥𝑡 + 𝜑𝑡 , 𝑥0 = 𝜉, 𝑑𝑧𝑡 = 𝐶𝑥𝑡 𝑑𝑡 + 𝑑𝑣𝑡 , 𝑧0 = 0,

where 𝑥 and 𝑧 are vector-valued signal and observation processes, 𝐴 and 𝐶 are matrices, 𝜑 is a wide band noise, represented in the preceding integral form, 𝜉 is a Gaussian random variable with zero mean, 𝑤 and 𝑣 are vector-valued Wiener processes, and 𝜉 , 𝑤 and , 𝑣 are independent. Note that the signal system is given in terms of derivative while the observation system in terms of differential. By this, we stress on the fact that unlike white noises, which are generalized derivatives of Wiener processes and do not exist in the ordinary sense, wide band noises are well-defined random processes. We additionally assume that the relaxing function Φ of the wide band noise 𝜑 is not known. Instead, its autocovariance function Λ is known. Under these conditions the best estimate (in the mean square sense) process 𝑥̂ for the preceding system is uniquely determined as a solution of the equation 𝑑𝑥̂𝑡 = (𝐴𝑥̂𝑡 + 𝜓𝑡,0 )𝑑𝑡 + 𝑃𝑡 𝐶 ∗ (𝑑𝑧𝑡 − 𝐶𝑥̂𝑡 𝑑𝑡), 𝑥̂𝑡 = 0, 𝑡 > 0, where 𝐶 ∗ is the transpose of the matrix 𝐶 and 𝜑 is a unique solution of the partial differential 𝜕

𝜕

∗ equation (𝜕𝑡 + 𝜕𝜃) 𝜓𝑡,𝜃 𝑑𝑡 = 𝑄𝑡,𝜃 𝐶 ∗ (𝑑𝑧𝑡 − 𝐶𝑥̂𝑡 𝑑𝑡), 𝜓0,𝜃 = 𝜓𝑡,−𝜀 = 0, −𝜀 ≤ 𝜃 ≤ 0, 𝑡 > 0.

Additionally, the functions 𝑃, 𝑄, and 𝑅 are uniquely determined as the solutions of the deterministic ordinary and partial differential equations 𝑑 ∗ 𝑃 = 𝐴𝑃𝑡 + 𝑃𝑡 𝐴∗ + 𝑄𝑡,0 + 𝑄𝑡,0 − 𝑃𝑡 𝐶 ∗ 𝐶𝑃𝑡 , 𝑑𝑡 𝑡 𝜕 𝜕 ( + ) 𝑄𝑡,𝜃 = 𝐴𝑄𝑡,𝜃 + Λ∗𝑡,−𝜃 + 𝑅𝑡,0,𝜃 − 𝑃𝑡 𝐶 ∗ 𝐶𝑃𝑡 , 𝜕𝑡 𝜕𝜃 and 52

𝑃0 = 𝑐𝑜𝑣 𝜉,

𝑡 > 0,

𝑄0,𝜃 = 𝑄𝑡,−𝜀 = 0, −𝜀 ≤ 𝜃 ≤ 0, 𝑡 > 0,


𝜕

𝜕

𝜕

∗ (𝜕𝑡 + 𝜕𝜃 + 𝜕𝜏) 𝑅𝑡,𝜃,𝜏 = −𝑄𝑡,𝜃 𝐶 ∗ 𝐶𝑃𝑡 , 𝑄𝑡,𝜏 , 𝑅0,𝜃,𝜏 = 𝑅𝑡,−𝜀,𝜏 = 𝑅𝑡,𝜃,−𝜀 = 0, −𝜀 ≤ 𝜃, 𝜏 ≤ 0, 𝑡 > 0.

Moreover, the mean square error of estimation is equal to 𝑒𝑡 = 𝑃𝑡 . An important feature of these equations is that they don’t contain unknown relaxing function Φ, but contain known autocovariance function Λ. Keywords: Kalman filtering, white noise, wide band noise. AMS Subject Classification: 93E11.

References 1. Kalman R.E., A new approach to linear filtering and prediction problems, Transactions ASME, Ser. D (J. Basic Engineering), Vol. 82, 1960, pp. 35-45. 2. Fleming W.M. and Rishel R.W., Deterministic and Stochastic Optimal Control, Springer-Verlag, 1975, 222 p. 3. Bashirov A.E., Partially Observable Linear Systems under Dependent Noises, Systems & Control: Foundations & Applications, Birkhauser, 2003, 334 p. 4. Bashirov A.E., Problem 2.1. On error of estimation and minimum of cost for wide band noise driven systems, in: Eds: V.D. Blondel and A. Megretski, Unsolved problems in mathematical systems and control theory, Princeton University Press, 2004, pp. 67-73.

THE USE OF PARTICLE SWARM OPTIMIZATION FOR OPTIMAL FRACTIONAL CONTROL REALIZATION Djalil Boudjehem1, Boudjehem Badreddine2 1

Advanced Control Laboratory, Department of Electronics and Telecommunication, University of Guelma, Guelma, Algeria 2 Advanced Control Laboratory e-mail: [email protected]

In this paper we propose the use of a particle swarm optimization for the realization of a fractional controller, that permits to ensure the desired specifications. The controller parameters are obtained by solving an optimization problem, which is formulated via diffusive representation of the fractional operator. The diffusive representation use, helps obtaining a direct usable realization of the controller. Illustrative numerical example and simulations in the time domain are given and commented. The non standard operator use, as for example fractional operators, appeared strongly in different disciplines. The theoretical and convenient interest of these operators is established 53


henceforth well. The fractional operators are widely used in control systems to construct fractional regulators such as fractional PI, PID ...etc. [1-5]. The uncertain dynamic control system requires the employment of control laws that are able to ensure a good compromise between performances and Robustness. An alternative is the use of the fractional control where the concept of the robustness is based on the property of the invariance of the fractional differential equation. The fundamental property of this control is to preserve as much as possible, and on all the domain of uncertainty, the dynamic features imposed by the control of the nominal system, up to time scaling (or to frequency scaling). The fractional control is achieved by diffusive representation.[6-9] We apply this concept to control a DC motor where its transfer function is uncertain. The uncertainty here, is carried at the mechanical load and the current loop constant time. The non strict invariance consist of minimizing an adequate cost function. The optimal controller is then achieved by the use of the particle swarm optimization algorithm (PSO), which is one of the metaheuristic algorithms widely applied in the last decade [10-12]. Its use helps in most cases to improve the quality of the obtained results. As an application, we choose a DC motor. The open loop uncertain transfer function of this motor can be written as: H  ( s) =

1 . J s(1  Tbc s)

The uncertainty is carried at the moment of inertia J "motor+load", the time constant

Tbc of the current loop or on the two at the same time.

(a)

(b)

Figure. Step responses of the velocity for different values of

54

J and Tbc

0


J and Tbc are uncertain where:  = [ J min , J max ]x[Tbcmin , Tbcmax ] . 0 is the nominal parameter that contains the nominal values of J and Tbc noted by J 0 and Tbc0 respectively. Figures 1, 2 ( a,b) show the step responses of the velocity for different values of J , and

Tbc using optimal fractional controller. the used fractional controller helps to get step responses with iso-overshoots (iso-damping). Keywords: fractional systems, diffusive symbol, diffusive representation, modeling, minimization, particle swarm optimization, iso-damping. AMS Subject Classification: 26A33, 34A08, 49M25, 49K99.

References 1. Boudjehem D., Boudjehem B., Boukaache A., Reducing dimension in global optimization. International Journal of Computational Methods, Vol.8, No.03, 2011, pp.535–544. 2. Boudjehem D.,Boudjehem B., A fractional model for robust fractional order Smith predictor, Nonlinear Dynamics, Vol.73, No.3, 2013, pp.1557–1563. 3. Boudjehem B., Boudjehem D., Fractional order controller design for desired response, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, Vol.227, No.12, 2013, pp.243–251. 4. Casenave, C., Montseny, G., Identification and state realization of non-rational convolution models by means of diffusive representation, Control Theory and Applications, IET, Vol.5, No.07, 2011, pp.934–942. 5. Jesus, I. S., Machado, J. T., Fractional control of heat diffusion systems. Nonlinear Dynamics, Vol.54, No.3, 2008, pp.263–282. 6. Laudebat, L., Bidan, P., Montseny, G., Modeling and Optimal Identification of Pseudodifferential Electrical Dynamics by Means of Diffusive Representation", IEEE Transactions on Circuits and Systems I, Vol.51, No.9, 2004, pp.1801–1813. 7. Liu, J., Ren, X., Ma, H., A new PSO algorithm with random C/D switchings. Applied Mathematics and Computation, Vol.218, No.19, 2012, pp.9579–9593. 8. Montseny, G. Diffusive representation of pseudo-differential time-operators. LAAS, 1998. 9. Montseny, G., Simple Approach to Approximation and Dynamical Realization of Pseudodifferential Time Operators such as Fractional ones, IEEE Transactions on Circuits and Systems II, Vol.51, No.11, 2004, pp.613–618. 10. Montseny, G., Diffusive wave-absorbing control: Example of the boundary stabilization of a thin flexible beam. Journal of Vibration and Control, C, Vol.18, No.11, 2012, pp.1708–1721. 55


11. Poli, R., Kennedy, J., Blackwell, T., Particle swarm optimization. Swarm intelligence, Vol.1, No.1, 2007, pp.33–57. 12. Maurice C., Kennedy, J., The particle swarm-explosion, stability, and convergence in a multidimensional complex space, Evolutionary Computation, IEEE Transactions, Vol.6, No.1, 2002, pp.58-73.

THE PROBLEM OF MAYER FOR DISCRETE AND DIFFERENTIAL INCLUSIONS WITH INITIAL BOUNDARY CONSTRAINTS Gulseren Çiçek1, Elimhan N. Mahmudov2,3 1

Istanbul University, Faculty of Science, Dep. of Math., Vezneciler, Istanbul, Turkey Industrial Engineering Dep. Faculty of Management, Istanbul Technical University, Maçka Istanbul, Turkey 3 Azerbaijan National Academy of Sciences, Institute of Control Systems, Baku, Azerbaijan e-mail: [email protected], [email protected] 2

This paper is devoted to derive optimality conditions for the Mayer problem for second order differential inclusions :

(PC)

minimize  ( x(1), x(1))

(1)

x(t )  F ( x(t ), x(t )), a.e. t  0,1 ,

(2)

x(0)  M , x(0)  N ,

where F :

:

2n



2n

 P(

n

n

) is a set-valued mapping ( P(

is proper single valued function, M , N 

n

) is a family of subsets of

(3) n

),

are subsets. The problem is to find an

arc x(t ) of Mayer problem (1) – (3) for the second order differential inclusions satisfying (2) almost everywhere (a.e.) on  0,1 and the boundary conditions at t  0 (3) that minimizes the Mayer functional   x(1), x(1)  . We mark this problem with (PC). Here, a feasible trajectory x() is taken to be an absolutely continuous function on a time interval [0,1] together with the

first order derivatives for which x()  L1n 0,1 . Discrete and continuous time problems with higher order ordinary and partial differential inclusions have wide applications in the field of mathematical economics and in problems of control dynamical system optimization and differential games [1,3,4,6]. In particular, the problems including the second order discrete and discrete-approximate inclusions are studied by Mahmudov [5]. Notice that a lot of investigations on the second order differential inclusions 56


(SODIs) usually are devoted to existence and viability problems [2]. In the classical Mayer problem for the SODIs with initial boundary constraints make the examined optimal control problem quite complicated. For construction of optimality conditions we begin with the second order discrete problem and then using first and second order difference operators and an auxiliary multifunction, we approximate the convex problem (PC) by the discrete approximation problem. Generally, there are some difficulties in constructing adjoint inclusions and transversality conditions at the endpoints t  0 and t  1, respectively. We achieve by the approximation and formulation of the equivalence theorems. Let us formulate the main theorem for problem (PC) in the convex case. Theorem. For the optimality of the trajectory x(t ) in the convex problem (PC) it is sufficient that there exists a pair of functions  x *(t ), v *(t ) simultaneously not equal to zero, satisfying the second order Euler-Lagrange differential inclusion

 d 2 x *(t ) dv *(t )   , v *(t )   F *  x* (t );  x(t ), x(t ), x(t )   ,a.e. t   0,1 , (i)  2 dt  dt  the transversality conditions at the endpoints t  0 and t  1 (ii) (iii)

dx *(1)   ,  x *(1)  0  x(1), x(1)  ,  v *(1)  dt  

 dx * (0)   x* (0)  K N*  ,  dt 

v* (0) 

dx * (0)  K M* ( x(0)) , dt

respectively and the condition ensuring that the locally adjoint mapping F * is nonempty at a given point (iv)

d 2 x(t )  F  x(t ), x(t ); x *(t )  , a.e. t   0,1 dt 2





where F ( x, u; v* )  v  F ( x, u) : v, v*  H ( x, u, v* ) is the argmaximum set for a multivalued mapping

F.

Here we assume x *(t ) , t [0,1] , to be absolutely continuous function together with the first order derivative and and

d 2 x *() ()  L1n  0,1 . Besides v* (t ), t  0,1 is absolutely continuous 2 dt

dv* ()  L1n  0,1 . dt 57


Keywords: differential inclusion, Mayer problem, Euler-Lagrange, adjoint multivalued approximation, dual cone, second order transversality. AMS Subject Classification: 49k 20, 49k24, 49J52, 49M25, 90C31.

References 1. Aubin J. P., Frankowska H., Set-Valued Analysis, Birkhäuser, Boston, 1990. 2. Cernea A., On the existence of viable solutions for a class of second order differentialinclusions, Discuss. Math. Differ. Incl., Vol.22, No.1, 2002, pp.6778. 3. Mahmudov E.N., Approximation and optimization of higher order discrete and differential inclusions, Nonlinear Differential Equations and Applications ,Vol.21, 2014, pp.1-26. 4. Mahmudov E.N., Approximation and Optimization of Discrete and Differential Inclusions, Elsevier, 2011. 5. Mahmudov E.N., Optimization of Second Order Discrete Approximation Inclusions, Numerical Functional Analysis and Optimization, DOI: 10.1080/01630563.2015.1014048 6. Mordukhovich B.S. Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol.330 and 331, Springer, 2006.

SUFFICIENT CONDITIONS OF OPTIMALITY FOR THIRD ORDER POLYHEDRAL DIFFERENTIAL INCLUSIONS WITH BOUNDARY VALUE CONSTRAINTS Sevilay Demir1, Elimhan Mahmudov2, 3 1

2

Istanbul University, Faculty of Science, Dep. of Mathematics, Istanbul, Turkey Industrial Engineering Department Faculty of Management, Istanbul Technical University, Turkey 3 Azerbaijan National Academy of Sciences, Institute of Control Systems, Azerbaijan e-mail: [email protected], [email protected]

We study a third order polyhedral optimization problem (𝑃𝐶 ) given by third order polyhedral differential inclusions (PDIs): 1

minimize 𝐽[𝑥(∙)] = ∫0 𝑔(𝑥(𝑡), 𝑡)𝑑𝑡 (PC)

subject to

𝑥 ′′′ (𝑡) ∈ 𝐹 (𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡)) , 𝑎. 𝑒. 𝑡 ∈ [0,1],

(1) (2)

𝑥(0) ∈ 𝑀0 , 𝑥 ′ (0) ∈ 𝑀1 , 𝑥 ′′ (0) ∈ 𝑀2 , 𝑥(1) ∈ 𝑁0 , 𝑥 ′ (1) ∈ 𝑁1 , 𝑥 ′′ (1) ∈ 𝑁2 .

(3)

Here 𝐹: ℝ3𝑛 → 𝑃(ℝ𝑛 ) is a polyhedral set-valued mapping, 𝐹(𝑥, 𝑣1 , 𝑣2 ) = {𝑣3 : 𝑃0 𝑥 + 𝑃1 𝑣1 + 𝑃2 𝑣2 − 𝑄𝑣3 ≤ 𝑑}. 𝑃0 , 𝑃1 , 𝑃2 and Q are 𝑚 x 𝑛 dimensional matrices, 𝑑 is a 𝑚-dimensional

58


column-vector. Additionally, 𝑀𝑖 and 𝑁𝑖 , i=1,2,3 are polyhedral set and 𝑔(∙, 𝑡): ℝ𝑛 → ℝ is a polyhedral function. We have to find a solution 𝑥̃(𝑡) of the problem (1)-(3) for the third order differential inclusions satisfying (2) almost everywhere (a.e.) on [0,1] and the initial and endpoint conditions (3) that minimizes the functional 𝐽[𝑥(∙)]. [1-3] Theorem 1. In order for trajectory 𝑥̃(𝑡), 𝑡 ∈ [0,1]

lying interior to 𝑑𝑜𝑚𝐹 to be an

optimal solution of the third order PDIs of problem (PC), it is sufficient that there exists an absolutely continuous function

𝑥 ∗ (𝑡) satisfying the following third order Euler-Lagrange

differential inclusion almost everywhere (𝑎)

−

𝑑 3 𝑥 ∗ (𝑡) 𝑑𝜆(𝑡) 𝑑 2 𝜆(𝑡) ∗ ∗ ∗ ∈ 𝑃 𝜆(𝑡) − 𝑃 + 𝑃 + 𝜕𝑔(𝑥̃(𝑡), 𝑡), 0 1 2 𝑑𝑡 3 𝑑𝑡 𝑑𝑡 2 𝑥 ∗ (𝑡) = 𝑄 ∗ 𝜆(𝑡)

(𝑏) < 𝑃0 𝑥̃(𝑡) + 𝑃1

𝑑𝑥̃(𝑡) 𝑑 2 𝑥̃(𝑡) 𝑑 3 𝑥̃(𝑡) + 𝑃2 − 𝑄 − 𝑑 , 𝜆(𝑡) > = 0, 𝑑𝑡 𝑑𝑡 2 𝑑𝑡 3

𝑎. 𝑒. 𝑡 ∈ [0,1],

𝑎. 𝑒. 𝑡 ∈ [0,1],

and the transversality conditions 𝑑2 𝑥 ∗ (0) 𝑑𝜆(0) ∗ + 𝑃 − 𝑃1∗ 𝜆(0) ∈ 𝐾𝑀∗ 0 (𝑥̃(0)), 2 2 𝑑𝑡 𝑑𝑡 ∗ (0) 𝑑𝑥 − − 𝑃2∗ 𝜆(0) ∈ 𝐾𝑀∗ 1 (𝑥̃′(0)), 𝑑𝑡

(𝑐)

𝑥 ∗ (0) ∈ 𝐾𝑀∗ 2 (𝑥̃′′(0)), (𝑑)

− 𝑑𝑥 ∗ (1) 𝑑𝑡

𝑑 2 𝑥 ∗ (1) 𝑑𝜆(1) − 𝑃2∗ + 𝑃1∗ 𝜆(1) ∈ 𝐾𝑁∗0 (𝑥̃(1)), 2 𝑑𝑡 𝑑𝑡 + 𝑃2∗ 𝜆(1) ∈ 𝐾𝑁∗1 (𝑥̃′(1)),

−𝑥 ∗ (1) ∈ 𝐾𝑁∗2 (𝑥̃′′(1)).

Corollary 1. Let us consider the Bolza problem with cost functional 1

𝐽1 [𝑥(∙)] = ∫0 𝑔(𝑥(𝑡), 𝑡)𝑑𝑡 + 𝜑0 (𝑥(1), 𝑥 ′ (1), 𝑥 ′′ (1)) and differential inclusion (2) for boundary value conditions (3), where 𝑔, 𝜑0 are polyhedral functions. Then for optimality of the trajectory 𝑥̃(𝑡) in the Bolza problem the transversality conditions at the endpoints t = 0 and t = 1 should be as follows: (𝑒)

𝑑2 𝑥 ∗ (0) 𝑑𝜆(0) + 𝑃2∗ − 𝑃1∗ 𝜆(0) ∈ 𝐾𝑀∗ 0 (𝑥̃(0)), 2 𝑑𝑡 𝑑𝑡 𝑑𝑥 ∗ (0) − − 𝑃2∗ 𝜆(0) ∈ 𝐾𝑀∗ 1 (𝑥̃′(0)), 𝑑𝑡 𝑥 ∗ (0) ∈ 𝐾𝑀∗ 2 (𝑥̃′′(0)),

59


(𝑓)

− 𝑃2∗

𝑑𝜆(1) 𝑑𝑡

+ 𝑃1∗ 𝜆(1) ∈ 𝐾𝑁∗0 (𝑥̃(1)) , 𝑃2∗ 𝜆(1) ∈ 𝐾𝑁∗1 (𝑥̃′(1)),

−𝑥 ∗ (1) ∈ 𝐾𝑁∗2 (𝑥̃ ′′ (1)) ,

𝑑 2 𝑥 ∗ (1) 𝑑𝑥 ∗ (1) ∗ ( ,− , 𝑥 (1)) ∈ 𝜕𝜑0 (𝑥̃(1), 𝑥̃ ′ (1), 𝑥̃ ′′ (1)). 𝑑𝑡 2 𝑑𝑡 Corollary 2.

Let us consider the Bolza problem with cost functional 𝐽1 [𝑥(∙)] and

differential inclusion (2) with initial conditions 𝑥(0) = 𝛼0 , 𝑥 ′ (0) = 𝛼1 , 𝑥 ′′ (0) = 𝛼2 where 𝛼0 , 𝛼1, 𝛼2 are fixed vectors. Then for optimality of the trajectory 𝑥̃(𝑡) in the Bolza problem the transversality conditions will be given by the relation: 𝑑 2 𝑥 ∗ (1) 𝑑𝜆(1) 𝑑𝑥 ∗ (1) ∗ ∗ ( + 𝑃2 − 𝑃1 𝜆(1), − − 𝑃2∗ 𝜆(1) , 𝑥 ∗ (1)) ∈ 𝜕𝜑0 (𝑥̃(1), 𝑥̃ ′ (1), 𝑥̃ ′′ (1)). 2 𝑑𝑡 𝑑𝑡 𝑑𝑡 Now an application of these results is demonstrated by solving the variational problem of a functional with a single function, but containing its second order derivations [4,5]. Accordingly, boundary value problem is this: 1

minimize 𝐽2 [𝑥(∙)] = ∫0 𝐿(𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥′′(𝑡), 𝑡)𝑑𝑡 (PL)

subject to

𝑥 ′′′ (𝑡) ∈ 𝐹 (𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡)) , 𝑎. 𝑒. 𝑡 ∈ [0,1],

𝑥(0) = 𝛼0 , 𝑥 ′ (0) = 𝛼1 , 𝑥 ′′ (0) = 𝛼2 ,

𝑥(1) = 𝛽0 , 𝑥 ′ (1) = 𝛽1 , 𝑥 ′′ (1) = 𝛽2

where 𝐹 is a polyhedral set-valued mapping, the Lagrangian L is a real-valued function with the continuous first derivatives and 𝑥(∙) ∈ 𝐶 2 ([0,1]). Let us denote 𝜕𝐿 𝜕𝐿(𝑥̃(𝑡), ̃𝑥 ′ (𝑡), 𝑥̃ ′′ (𝑡) ) = , 𝑘 = 0,1,2, 𝜕𝑥 (𝑘) 𝜕𝑥 (𝑘) and require that 𝐿(𝑥, 𝑣1 , 𝑣2 ) − 𝐿 (𝑥̃(𝑡), ̃𝑥 ′ (𝑡), 𝑥̃ ′′ (𝑡)) ≥ < +
+ < ′ , 𝑣1 − 𝑥̃′ (𝑡) > 𝜕𝑥 𝜕𝑥

𝜕𝐿 , 𝑣 − 𝑥̃ ′′ (𝑡) > , ∀ (𝑥, 𝑣1 , 𝑣2 ) ∈ ℝ3𝑛 . 𝜕𝑥 ′′ 2

Then by Theorem 4.3.[1] for optimality of 𝑥̃(𝑡) the Euler-Lagrange inclusion is valid: (−

𝑑3 𝑥 ∗ (𝑡) 𝑑𝜓2∗ (𝑡) 𝜕𝐿 + + , 𝑑𝑡 3 𝑑𝑡 𝜕𝑥

𝜓2∗ (𝑡) +

𝑑𝜓1∗ (𝑡) 𝜕𝐿 𝜕𝐿 + ′ , 𝜓1∗ (𝑡) + ) 𝑑𝑡 𝜕𝑥 𝜕𝑥′′

∈ 𝐹 ∗ (𝑥 ∗ (𝑡); (𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡), 𝑥 ′′′ (𝑡)) , 𝑡).

60

(4)


Suppose now that in the problem (𝑃𝐿 ), 𝑑𝑜𝑚 𝐹(∙, 𝑡) ≡ ℝ3𝑛 . Then, F *   0,0  so that

x*  v*  0 . It means that x *( k ) (t )  0, t 0,1, k=0,1,2,3. Therefore, it follows from (4) that 𝑑𝜓2∗ (𝑡) 𝜕𝐿 + = 0, 𝑑𝑡 𝜕𝑥

𝜓2∗ (𝑡)

𝑑𝜓1∗ (𝑡) 𝜕𝐿 + + ′ = 0, 𝑑𝑡 𝜕𝑥

𝜓1∗ (𝑡) +

𝜕𝐿 = 0. 𝜕𝑥′′

(5)

Now, beginning from the last relation by sequentially substitution in (5), we derive the EulerPoisson equation for problem (𝑃𝐿 ): 𝜕𝐿 𝑑 𝜕𝐿 𝑑2 𝜕𝐿 − ( ′ ) + 2 ( ′′ ) = 0. 𝜕𝑥 𝑑𝑡 𝜕𝑥 𝑑𝑡 𝜕𝑥

(6)

It is well known that in calculus of variations for optimality of trajectory x(t) the Euler-Poisson equation (6) is necessary condition. Note that for the problem of calculus of variations (𝑃𝐿 ) the transversality condition of Theorem1 is superfluous. Keywords: polyhedral, third order, differential inclusions, transversality. AMS Subject Classification: 49K20, 49K24, 49J52, 49M25, 90C31.

References 1. Aubin J.-P., Cellina A., Differential Inclusions, Springer-Verlag, 1984, Grundlehren der Math., Wiss. 2. Clarke F.H., Optimization and Nonsmooth Analysis, 1983, Wiley. 3. Mahmudov E., Approximation and Optimization of Discrete and Differential Inclusions, 2011, Elsevier. 4. Mahmudov E., Approximation and optimization of higher oerder discrete and differential inclusions, NoDEA, 21, 2014, pp.1-26. 5. Mordukhovich B.S., Variational Analysis and Generalized Differentiation, Vol.I and II, Springer, 2006, Springer-Verlag Berlin Heidelberg.

BK-SPACES GENERATED BY USING THE FRACTIONAL DIFFERENCE OPERATOR Sinan Ercan1, Çiğdem A. Bektaş1 Firat University, Department of Mathematics, Elazıg e-mail: [email protected]

1

The difference operator of fractional order and its applications is studied by P. Baliarsingh in [8,9].   p  denote the Gamma function of a real number 61

p and


p 0, 1, 2, 3,... . By the definition, it can be expressed as an improper integral as 

  p    et t p1dt . 0

For a proper fraction  , a fractional difference operator   : w  w and its inverse defined in [8] as follows: 





 xk     1 i 0

i

    1  1    i  xk i ,    xk     1 xk i . i !   i  1 i ! 1    i  i 0

In particular, we have



1 2



1 2 

1 1 1 5 7 xk  xk  xk 1  xk 2  xk 3  xk 4  xk 5  ... 2 8 16 128 256

1 3 5 35 63 xk  xk  xk 1  xk 2  xk 3  xk 4  xk 5  ... 2 8 16 128 256

for   1 2 . He introduced in [9] the BK-spaces given as follow,







 j 0









  ,  , u    x   xk   w :   u j  x j     k

where    , c, c0 . u   un  is a sequence satisfying certain conditions and for a proper 

fraction  ,   xk     1 i 0

i

   1 xk i . He examined some topological properties of i !   i  1

these spaces and he also computed their dual spaces. In this paper, using the definition of difference operator of fractional order [1-3] and using definitions which are given in [4-6] by M. Mursaleen and A.K. Noman [5-7], we introduce

 

the BK-spaces of non-absolute type c0  

 

c 



 

 

and c   . We also prove that c0  

and

spaces are linearly isomorphic to the spaces c0 and c , respectively. Lastly, we

   , c     and we compute the   ,   and  

determine the Schauder basis of the c 



0

 

duals of these spaces.  Keywords: difference operator   ,   ,   and   duals.

AMS Subject Classification: 46A45, 46B20.

62


References 1. Ganie A.H., Sheikh N.A., On some new sequence spaces of non-absolute type and matrix transformations, Journal of Egyptian Math. Society, 2013, 21, pp.108-114. 2. Wilansky A., Summability Through Functional Analysis, in: North-Holland Mathematics Studies, Elsevier Science Publishers, Amsterdam, New York, Oxford, 1984. 3. Asma Ç., Çolak R., On the Köthe-Toeplitz duals of some generalized sets of difference sequences, Demonstratio Math., 33, 2000, pp.797-803. 4. Başar F., Summability Theory and Its Applications, Bentham Science Publishers, ISBN: 978-160805-252-3, 2011. 5. Mursaleen M., Noman A.K., On the spaces of  -convergent and bounded sequences, Thai J. Math., 2, 2010, pp.311-329. 6. Mursaleen M., Noman A.K., On some new difference sequence spaces of non-absolute type, Math.Comput. Mod., 52, 2010, pp.603-617. 7. Mursaleen M., Noman A.K., On some new sequence spaces of non-absolute type related to the spaces

p

and



II, Mathematical Communications, 16, 2011, pp.383-398.

8. Baliarsingh P., Dutta S., A unifying approach to the difference operators and their applications, Bol. Soc. Paran. Mat., (3s), Vol.33, No.1, 2015, pp.49–57. 9. Baliarsingh P., Some new difference sequence spaces of fractional order and their dual spaces Appl. Math. Comput, Vol.219, No.18, 2013, pp.9737-9742. 10. Ercan S., Bektaş Ç.A., On some sequence spaces of non–absolute type, Kragujevac Journal of Mathematics, Vol.38, No.1, 2014, pp.195-202.

RESPECT TO TWO SPECTRA STABILITY OF THE INVERSE PROBLEM FOR SINGULAR STURM-LIOUVILLE OPERATOR Ahu Ercan1, Etibar Panakhov1 Department of Mathematics, Fırat University, Elazig, Turkey e-mail: [email protected], [email protected]

1

In this study we consider stability inverse spectral problems associated with the singular Sturm-Liouville operator

L:

l  l  1  d2    q  x   2 dx  x2 

(1)

where q  L2  0,1 be real valued and l be an nonnegative integer. A meaningful question about the stability of the inverse spectral problems is following: 63


How much can two boundary value problems differ from eachother if their spectral functions differ a little on a given interval of values of the spectral parameter  . Let 1,0  1,1  ...  1,n  ... be the spectrum of the problem

l  l  1   L1 y   y   q1  x    y  y 0  x 1 x2   y  0   0 y 1  0

(2) (3)

Let us define the new problem corresponding to q  0 L0 y   y 

l  l  1 x2

y  y

(4)

with the Dirichlet boundary conditions (3). Let 0,n  be the spectrum of L0 with (3) boundary conditions. We can find a function q2  x  such that the sequences 2,n  is the spectrum of the problem

l  l  1   L2 y   y   q2  x    y  y x2   y  0   0 y 1  0

0  x 1

(5) (3)

In this study when the eigenvalues  j ,m  ,  j  1, 2  of boundary value problems (2) –(3) and (5) –(3) coincide numbers of N  1 for m  1, 2,..., N  1 , singular Sturm-Liouville operator (1) needs to examine how it can be different. Marchenko and Maslow deal with similar problem in the case of the spectral functions  j    of two boundary value problems coincide on given interval [1,2]. To benefit the result of this study , firstly when the eigenvalue  j ,m  ,  j  1, 2  of boundary value problems (2) –(3) and (5) –(3) coincide numbers of N  1 for m  1, 2,..., N  1 we must evaluate the difference of the spectral functions of these problems. The main theorem in this study is following: Theorem 1. If the eigenvalues  j ,m  ,  j  1, 2  of boundary value problems (2) –(3) and (5) –(3) coincide numbers of N  1 for m  1, 2,..., N  1 , then the spectral functions

 j ,n    satisfy inequality for m  N  1 , n 

N 2

64


 1 l 3  8 A  2 

max 1  N n 2

1,n  2,n

N N   1 l 3  l 1 l 2 1  2 8A  2   1 l 3  N 1  3  4 A  2   N 4 N 2  N N  N N      e 2  l 1 l 1  2  1 l 3 N 1   3  4 A   2  2  N 4N  N N    1 l 3  8 A  2 

N N   1 l 3  l 1 l 2 1  2 8A  2   1 l 3  N 1  2 3  4 A  2   N 4N  N N  N N N     Var 1,n      2,n     1,n   e 2 N     2   l 1 l 1  2  1 l 3 2 N 1   3  4 A   2  2  N 4N  N N  

1

where A   q2  t   q1  t dt  a2,k  a1,k 0

Keywords: inverse spectral problem, stability. AMS Subject Classification: 34A55, 34B24, 34D20.

References 1. Hryniv R ., Sacks P., Numerical Solution of the Inverse Spectral Problem for Bessel Operators, Journal of Comp. and Applied Math., 235, 2010, pp.120-136. 2. Marchenko, V.A., Maslov, K.V., Stability of the Problem of Recovering the Sturm-Liouville Operator from the Spectral Function,Mathematics of the USSR Sbornik, Vol.81, No.123, 1970, pp.475-502.

ON CONTROLLABILITY AND REVERSIBILITY FOR SOME CLASS OF THE 3D - LINEAR MODULAR DYNAMIC SYSTEMS F.G. Feyziev1, G.H. Mammadova2 , F.N. Nabi-zade3 1, 3

2

Sumgait State University, Azerbaijan, Sumgait Institute of Applied Mathematics, Baku State University, Azerbaijan, Baku e-mail: [email protected]

In the work the controllability and reversibility for the 3D - linear modular dynamic systems (3D-LMDS) are considered. This system is described by the following equations: x[n  1, c] 

∑A[  ] x[n, c   ]  ∑B[q] u[n, c  q], ∈P

GF ( p),

(1)

q∈Q

y[n, c]  ∑C[r ] x[n, c  r ],

GF ( p) .

r∈R

In (1),(2): c  C0 - the cellular space and n  T - the times space:

65

(2)


C0  {c  (c1 , c2 ) | c1 , c2 ∈{...,-1,0,1,...}}, T  {n | n  0,1,...} ; x[n, c] - the states, u[n, c] - the inputs and y[n, c] - the outputs of LMDS, where x[n, c]  ( x1[n, c],..., xm [n, c])T  [GF ( p)]m , u[n, c] = (u1[n, c],..., um[n, c])T ∈[GF(p)]m ,

y[n, c] = ( y1[n, c],..., ym[n, c])T ∈[GF(p)]m ; P , Q and R are the characteristic sets called the templates of the neighborhood of the “state at

states”, “state at inputs”, “output at states” correspondently; here A[  ] (  ∈ P) , B[q] (q ∈ Q) and C[r ] (r ∈ R) are m m dimensional matrices. In the work the concept of N -controllability and N -reversibility for the 3D-LMDS is given. In the work two cases are considered: а) For each i {1,...,n} when changing ( 1 ,...,  i ) in P i and q in Q 1 ,...,  i , q  and 1,...,  i , q  does not exist to satisfy 1 ≠1 ,...,  i ≠ i, q ≠q

and 1  ...   i  q  1  ...   i  q . b) For each i  {1,..., n} when changing ( 1 ,...,  i ) in P i and q in Q

1 ,...,  i , q  and 1,...,  i , q  exist to satisfy 1  1 ,...,  i   i , q   q  and

1  ...   i  q  1  ...   i  q .

In the case а) the following designations are introduced: A  ( A[ 1 ]....A[  P ]) ,





B  ( B[q1 ]....B[q Q ]) , V  ( B A  B ... A( N -1)  B) , U  col C, C  A,...,C  A( N -1) , where sign  means the element-wise multiplication of block matrices,

Ai (i  0,1,...,n) is i - th degree of

the block matrix A with respect to the  transaction. Theorem 1. In order to

LMDS described by the (1), (2) be fully N - controllable it is

necessary and sufficient fulfillment of the condition: rang V= m . Theorem 2. In order to

LMDS described by the (1), (2) be fully N - reversibility it is

necessary and sufficient fulfillment of the condition: rang U= m . In the work the criteria of N - controllable and N - reversibility in the case b) is given also. Keywords: N -controllability, N -reversibility, modular dynamic systems, cellular space. AMS Subject Classification: 12E20, 93B05, 15A69.

66


OPTIMAL CONTOL UNDER SET-MEMBERSHIP UNCERTAINTY (SYNTHESIS AND REALIZATION OF OPTIMAL CLOSABLE FEEDBAKS) R. Gabasov1, F.M. Kirillova2 1

2

Belarusian State University, Minsk, Belarus Institute of Mathematics, National Academy of Sciences of Belarus, Minsk, Belarus e-mail: [email protected]

The concept of feedback (which is one of the greatest discovery of mankind) has been in use for more than 4000 years. It forms the foundation for one of the most effective technique of controlling dynamical systems under uncertainty which is characteristic of almost all real time control systems. A main feature of the classical approach to the synthesis of optimal feedbacks was that they were designed on the base of deterministic models, even though they were intended for controlling systems under uncertainty. The known methods of constructing implementations of optimal feedbacks (maximum principle, dynamic programming) were oriented to classical control principle – closed-loop control principle and demand that an optimal feedback to be constructed before the process starts. On this way even in simple cases extreme computational difficulties arise named by R. Bellman as the curse of dimensionality. Computer technology progress has led to wide use of principle of real-time control when current values of control actions are being formed in the course of control process without explicit formulas obtained before the beginning of the process. At optimal state feedbacks the systems are closed at every instant of control processes and every state of the system is supposed to be known at this instant of time ( measured exactly).To obtain values of a control action the information on current states of the system are only used. A trajectory of a closedloop system outgoing from an admissible position ( , x) , x  X ,  T , is the same as the optimal trajectory corresponding to the optimal program for the position ( , x) . In the report we use only discrete control actions. At present the use of discrete control actions are more natural. Optimal discrete loops allow to avoid analytical difficulties which are not connected with real processes and to provide the quality of control systems which is close to extreme possible one. Classical optimal discrete disclosable feedbacks. In the class of discrete control actions we consider an optimal control problem : c ' x(t * )  max ; x  A(t ) x  B(t )u  M (t )w(t ) , x(t* )  x0 , x(t * )  X * ; u(t ) U , t  T , T  [t* , t * ] 67

(1)


where w(t ), t  T , is an unknown piecewise-continuous disturbances with values from bounded set W  {w 

nx

: w*  w  w*} , X *  {x  R n : g*  Hx  g * },U  {u  R r :| u | L} .

Tu  {t* , t*  h, , t *  h}, h  (t* , t * ) / N , N is a given integer. A function u () is called a discrete

one (with quantization period h) if u(t )  u(s), t  [s, s  h[, s  Tu . Let a control action u () and perturbation w()  (w(t ), t T ) generate a trajectory of (1): x(t | t* , x0 , u(), w()) , t  T , X t* (u)  {x 

admissible

nx

: x  x(t * | t* , x0 , u(), w()), w(t ) W , t  T } .

An

control action u () is said to be a (guaranteeing) program if X t* (u )  X * .The

quality criterion of a program is estimated by the functional J (u)  min c ' x, x  X t * (u) , is called a guaranteed value of the cost function of problem (1).The optimal (guaranteeing) program u 0 () is defined as J (u 0 ())  max J (u) . It steers system (1) to the terminal set X * at time t * with guarantee and provides the maximum of guaranteed value to the cost function J (u ) .The function u 0 () is a program solution to problem (1).

Let at discrete moments   Tu of a control process exact values of states of the system are known (measured exactly). To define a positional solution to problem (1) we consider a family of problems: c ' x(t * )  max; x  A(t ) x  B(t )u  M (t ) w(t ); x( )  z , x(t * )  X * ;

(2)

u (t ) U ; w(t ) W , t  T   [ , t * ],

depending on a scalar   Tu and a nx - vector z . Denote: u 0 (t |  , z), t  T  , stands for an optimal program of problem (2) for the position ( , z ) ; X  stands for the set of all states z 

nx

for which

there exist program solutions to problem (2) . A function u 0 ( , z)  u 0 ( |  , z), z  X ,  Tu , is said to be an optimal disclosable (discrete) state feedback (a positional solution to problem (1)), its construction is called the synthesis of an optimal system in the class of disclosable state feedbacks. Discrete closable state feedbacks. From the set Tu we choose a subset Tз  {t j  Tu , j  1, p} , t*  t1  t2  ...  t p  t * , of closing instants and define a program u () using the fact

that states of the system will be known at the instants t  Tcl Before the beginning of the control

68


process sets of closures X clp , X clp 1 ,, X cl1 to instants t  X cl are constructed. Every X clp includes such system states z 

nx

at the time instant t p which are guaranteed can be transferred to the

terminal set X * at the time instant t * , i.e. there exists admissible control action u (t p , t * ) such that the inclusion:

X tp* (u (t p , t * ) | z ) = {x 

n

: x  x(t * | t p , z, u (t p , t * ) w(t )), w(t ) W , t [t p , t * ]}  X * ,

holds. For the set X clp 1 one can say that it is a set of all states z 

nx

(3)

of the system at the time

instant t p 1 which an admissible control u (t p 1 , t p ) is guaranteed to transfer to the set X clp at the time instant t p , etc. As a result, we construct the set X cl0 and the admissible control action

u (t* , t1 ) . If sets X clp , X clp 1 ,, X cl1 are not empty and x0  X cl0 , then a compose of fragments, on which

inclusions

(3)

hold,

gives

a

program

(with

expanded

guarantee)

u ()  (u (t* , t1 ),, u (t p , t * )) to problem (1). Choose

X * {x 

n

a

number

  min c ' x, x  X * ,

substitute

the

set

X*

by

X * 

: c ' x   } and construct sets X clp , X clp 1 , X 1з  , following the above rules. It is

clear that maximum  0 at which u  (t* , t1 ) , u  (t j , t j 1 ) j  1, p  1 ; u  (t p , t * ) exist is equal to 0

0

0

maximum guaranteed value of the cost function of problem (1). A totality

u 0 () 

(u (t* , t1 ),, u (t p , t * )) is said to be an optimal program to the initial position (t* , x0 ) . The optimal program u 0 (t |  , z), t  T  , to problem (2) for the position ( , z ) is defined by above rules. Let X

be a set of all states z 

nx

for which optimal programs (with expanded guarantee) to

problem (2) exist. A function u 0 ( , z)  u 0 ( |  , z), z  X ,   Tu ,

(4)

Is said to be an optimal closable (discrete) state feedback, its construction is said to be the synthesis оf optimal control systems in the class of closable state feedbacks. For simplicity, the word "discrete" will be omitted below. If Tcl  Tu then function (4) is said to be a closed-loop state feedback. Thus, the optimal closable state feedback takes intermediate place between optimal disclosable state feedback and optimal close-loop state feedback.

69


While solving problem (1) practically, the sets X clp , X clp 1 ,, X cl1 are approximated by special polyhedrons which allow us to apply the effective dual methods of linear programming [1]. The presentation deals with the latest results on constructing in real time realizations of optimal closable state (and output) feedbacks obtained in Minsk. The preposterior analysis of a control system at the beginning of the process and at its course are introduced. When solving the problem for linear indeterminate systems under the condition of incomplete and inexact information about states and disturbances, the original problem is decoupled into three: identification, observation and control problems for deterministic systems. To accelerate computations at real-time constructing realizations of optimal closable feedbacks speed-up procedures are justified. The results are based on the dynamical variant of the adaptive method of linear programming [1] and the correcting procedure of current programs [2]. Keywords: control, feedbacks, real - time control. AMS Subject Classification: 49N05, 93C05.

References 1. Gabasov R., Kirillova F.M., Prischepova S.V., Adaptive Method of

Linear Programming,

pp.160-202, In Optimal Feedback Control, 1995, Springer- Verlag, London Ltd., 202 p. 2. Gabasov R., Kirillova F.M., Robust Optimal Control on Imperfect Measurement of Dynamic Systems States., Appl. Comput. Math., Vol.8, No.1, 2009, pp.54-69.

ON A LINEAR OPTIMAL CONTROL PROBLEM OF ROESSER-TYPE SYSTEMS S.Sh. Gadirova1 1

Institute of Control Systems of ANAS, Baku, Azerbaijan e-mail: [email protected]

In the paper, a problem on minimization of a linear functional determined on the solutions of Roesser-type linear system is studied. Necessary and sufficient optimality condition in the form of the discrete maximum principle in proved. Assume that the controlled process in described by the following system of two parameters difference equations

zt  1, x  At , x zt , x  Bt , x yt , x  f t , x, ut , x, 70

(1)


yt , x  1  Ct , x zt , x  Dt , x yt , x  g t , x, ut , x, with boundary condition zt 0 , x   ax,

yt , x0   bt .

(2)

Here At , x  , Bt , x  , C t , x  , Dt , x  are given discrete matrix functions of appropriable dimensions, zt , x , yt , x  is the n  m - dimensional state vector, ax  and bt  are the given

n and m - dimensional vector-functions, t 0 , t1 , x0 , x1 are given, and the difference x1  x0 is a natural number, f t , x, u  , g t , x, u  are the given n and m -dimensional vector-functions, respectably, u t , x  is the r - dimensional vector of control action with the values from the given nonempty and bounded set U  R r , i.е.

ut , x  U ,

t, x  T  X , T  t0 , t0  1..., t1  1, X  x0 , x0  1..., x1  1 .

(3)

Suds control functions are called admissible controls. On the solutions of boundary value problem (1)-(2) generated by all admissible controls, determine the functional S u  

x1 1

t1 1

x  x0

t t0

 cx  zt1 , x    d t yt, x1  .

(4)

Here cx  and d t  are the given n and m dimensional discrete functions. Consider a problem on the minimum of functional (4) at restraints (1)-(3). We call the admissible control delivering minimum to functional (4) at restraints (1)-(3) an optimal control, and the appropriate process ut , x , zt , x , yt , x  an optimal process. Applying the increments method 1-4 a necessary and sufficient optimality condition in the form of the discrete minimum principle was proved. Keywords: linear control problem, Roesser-type systems, necessary and sufficient condition, two parameter system. AMS Subject Classification: 49K10, 49K20.

References 1. Gabasov R., Kirillova F.М. Maximum principle in theory of optimal control, Minsk, Nauka i Tekhnika, 1974, 272 p. 2. Mansimov К.B., Mardavov М.J. Quality theory of optimal control of Goursat-Darboux systems, Baku, Elm, 2010, 360 p. 71


3. Gabasov R., Kirillova F.М. Optimization of Linear Systems, Minsk, BSU publ., 1973, 256 p. 4. Mansimov К.B. Discrete Systems, Baku University Press, 2013, 151 p.

NECESSARY OPTIMALITY CONDITIONS IN A DISCRETE CONTROL PROBLEM WITH A FUNCTIONAL CONSTRAINED RIGHT END OF TRAJECTORY E.A. Garayeva1, K.B. Mansimov1 1


Let the controlled process be described by the system of equations

zt  1, x   f t , x, zt , x , ut  , t  t 0 , t 0  1, ..., t1  1 ,

x  x0 , x0  1, ..., x1

zt 0 , x   yx  ,

(1) (2)

where yx  is determined from the equation

yx  1  g x, yx , vx  , x  x0 , x0  1,..., x1  1 , yx0   y0 .

(3)

Here f t , x, z, u  g x, y, v  is the given n -dimensional vector-function continuous in totality of variables together with f z t , x, z, u  g y x, y, v  ; y 0 , t 0 , t1 , x0 , x1 are given, and the difference t1  t0 , x1  x0 are natural numbers, the control functions u t  , vx  satisfy the constraints ut U  R r , t  t 0 , t 0  1, ..., t1  1,

vx V  R q , x  x0 , x0  1, ..., x1  1,

(4)

where U and V are the given nonempty and bounded sets. Let 1  y  , Gi x, z  , i  0, p be the given scalar functions continuous in totality of variables together with 1  y  y , Gi x, z  z , i  0, p . If the solution

zt, x, yx

of the problem (1)-(3) corresponding to the control

ut , vx , satisfies the relations x1 1

Si u   i  yx1    Gi x, z t1 , x   0 , x  x0

then such control is called an admissible control. The problem is in minimization of the functional

72

i  1, p ,


x1 1

S u   0  yx1    Gi x, z t1 , x 

(5)

x  x0

determined on the solutions of problem (1)-(3) generated by all possible admissible controls. In the paper we obtain the analogue of the discrete maximum condition 1-3. The case of convex domain of control is studied separately. Keywords: discrete control problem, maximum principle, necessary optimality conditions, functional constrained. AMS Subject Classification: 49K20.

References 1. Gabasov R., Kirillova F.M., Optimization Methods, Minsk, BSU publ., 1980, 400 p. 2. Propoi A.I., Elements of theory of optimal discrete processes, Мoscow, Nauka, 1973, 256 p. 3. Mansimov K.B., Discrete systems, Baku University Press, 2013, 151 p.

ON A MINIMIZATION OF ONE DOMAIN SPECTRAL FUNCTIONAL Y.S. Gasimov1, A. Aliyeva2, N. Allahverdiyeva2 1

Institute of Applied Mathematics Baku State University, Baku, Azerbaijan 2 Sumgait State University, Sumgait, Azerbaijan e-mail: [email protected]

We consider minimization problem with respect to domain for the eigenvalue of the biharmonic operator. Consider the following eigenvalue problem 2 u  u , x  D ,

u  0,

u  0, n

x  SD ,

(1) (2)

where  stands for the Laplace operator, D is convex bounded domain from Euclidian space

E n , S D - its boundary. Note that the problem (1), (2) describes the clamped plate under across vibrations and eigenvalues of this problem indeed are eigenfrequencies and eigenfunctions - eigenvibrations of this plate with domain D [1]. Let





K  D  E n : D  K0 , SD  C 2 ,

where D is a closure of D , K 0 is some subset of convex bounded domains from E n . 73


The problem is: to find a domain D  K that is a solution of the problem

1 D  min, where 1 D  is the first eigenvalue

S D  2 ,

(3)

of the problem (1)-(2) in the domain D (indeed the first

eigenfrequency of the plate with domain D ), S D is an area of S D . We use the theorem below. Theorem [4]. Let D  K gives minimum to the functional  j D  . Then for any D K is valid the inequality







(u j ( x)) 2 PD (n( x))  P

(n( x)) ds  0 D

SD

,

where u j x  is an eigenfunction corresponding to the eigenvalue 

(4)

j of the problem (1)-(2) in

the domain D  , PD x   max x, l , x  R n is a support function of the domain D , max is taken lD



over all eigenfunctions u j in the case of multiplicity of  j . For our case the condition (4) takes the form   u1 x  PD nx ds 

  u1 x  PD nx ds,

2

S

2

S

D

(5)

D

Let us show that unit ball B satisfies this condition. It is known that on the surface S B of

B is true u1 x   const, x  S B . Considering this in (5) we get

 P nxds   P nxds. D

B

SB

 P nxds   dx  S D

SB

In [2] is proved that

SB

D

.

(6)

SD

From the other hand [2]

 P nxds 2 .

(7)

B

SB

Since

S D  2 in the considered problem from (6) and (7) we obtain that unit ball

satisfies to the condition (5). As is shown in [3,5] in some cases  j D  is a quasi-convex with respect to D functional. In such cases (5) is also a sufficient condition.

74


Keywords: biharmonic operator, shape optimization, convex domain, support function, necessary condition. AMS Subject Classification: 49QО45, 49К50.

References 1. Guld S., Variational Methods for Eigenvalue problems, University of Toronto Press, 1966, 328 p. 2. Demyanov, V.F. Rubinov A.M., Basises of Non-smooth Analysis and Quazi-differential Calculas, Moscow, Nauka, 1990 (in Russian). 3. Gasimov Y.S. Some shape optimization problems for eigenvalues. Journal of Physics A: Mathematical and Theoretical, Vol. 41, No.5, 2008, pp.521-529. 4. Gasimov Y.S., Niftiyev A.A. On a dependence of the eigenfrequency of the plate under across vibrations on the domain, Problems of Durability and Plasticity, 2002, No.64, pp.91-95. 5. Gasimov Y.S. On a convex dependence of the eigenvalues of Schrödinger operator on the domain, News of Baku University, No.2, 2006, pp.29-33. 6. Boulkhemair A., Chakib A., Nachaoui A., Uniform trace theorem and application to shape optimization, Appl. Comput. Math., Vol.7, No.2, 2008, pp.192-205.

ALGORITHM TO THE SOLUTION OF A SHAPE OPTIMIZATION PROBLEM FOR THE EIGENVALUES OF PAULI OPERATOR Y.S. Gasimov1, N.A. Allahverdiyeva2, A.Aliyeva2, L.I. Amirova1 1

Institute of Applied Mathematics Baku State University, Baku, Azerbaijan 2 Sumgait State University, Sumgait, Azerbaijan e-mail: [email protected]

We study the eigenvalues of Pauli operator in the variable operator definition domain. It is known that Pauli operator describes the motion of a particle with spin (in differ from Schrödinger operator) in a magnetic field and is a generalization of the Schrödinger operator in a mathematical and quantum-physical meaning [1]. Various methods have developed to investigate such problems [2-5], and we use the methodology offered in [3, 4]. As follows from the basic postulates of quantum physics, eigenvalues  n of Pauli operator describe the total energy of a quantum system (in our case an electron with spin in a magnetic field) in a state  n , where  n is an eigenfunction corresponding to the eigenvalue  n [1]. Let

 be the set of all convex bounded closed domains from R 2 with smooth

boundaries and 75






K  D   , D  0 , S D  C 2 ,

where  0 is some convex subset of  , D is a closure and S D - boundary of the domain D . Consider the problem k D  min , D  K .

(1)

Here k is the k -th eigenvalue of the following spectral problem P   ,

  0,

xD,

(2)

x  SD ,

(3)

where P is Pauli operator generated by the expression P  Pa, v  J  B .  1 0

1

0

 ,     , P  a, v    i  a 2  V , Here J   0 1 0  1          , ,  x y 

B

a  a1 , a2   R 2 -vector potential,

(4)

V -smooth

enough

function,

B -magnetic field generated by a , i.e.

  a 2  a1 . If to consider these entire definitions one can obtain the following explicit x y

form of Pauli operator in two dimensional case     2 0   i  a   a2  a1  V  x y  P    2  i  a   a2  a1  V  0  x y       2 0     2ia1  a2   2ia2  a1   a  V  x y        2 0    2ia1  a2   2ia2  a1   a  V   x y   .

(5)

 1  Taking     , where 1 , 2  L2 D from (2) and (5) we get 2  1   2ia 2  a1  1  a 21  V1   1 x y .  2  2 2   2  2ia1  a 2   2ia 2  a1   a  2  V 2   2 x y  1  2ia1  a 2 

Denote P1    2ia1  a 2 

   2ia 2  a1   a 2  V , x y

76

(6)


P2    2ia1  a 2 

   2ia 2  a1   a 2  V . x y

Thus within some conditions on a we can consider eigenvalues of Pauli operator positive and numerated in increasing order considering their multiplicity 0  1  2  ... . Replacing  i  ex y i ,

i  1,2 ,  ,   R after some transformations (6) we get

1    1  V1     a 2 1 , 4  

(7)

1     2  V 2     a 2  2 . 4  

(8)

1 Taking     a 2 from (5), (7), (8) we can rewrite the problem (2), (3) in the form 4 ~ P   , x  D

 0 , ~

  V  0

where P  

x  SD

,

(9)

 .    V  0

The following theorem is proved. Theorem. In order to D   K provide minimum to the functional (1) subject to (2), (3) it is necessary the fulfillment of the condition max  k

 S





 k x  PD nx   PD nx  ds  0 , 2

D

for any D  K . Here  k x  is an eigenfunction corresponding to k  k ( D  ) in D  , max is taken over all eigenfuctions  k corresponded to  k in the case of its multiplicity, PD x   maxx, l , x  R 2 is a support function of the domain D , s -is a boundary element. lD

On the base of this result the following algorithm is offered for the solution of the problem (9). Algorithm. Step 1. Chose the initial domain D0  K and assume that Di  K , i  1,2,... are already known. Step 2. Solve the problem (9) in Di and find eigenfunction  i (x) . Step 3. Solve the variational problem  i  max ,

77


where  i 

 (i ) Pnx ds and find convex positively-homogeneous function Pi x  .



2

S Di

Step 4. Find the auxiliary domain D m as a subdifferential of the function Pi x  in the point 0. Step 5. Find the following domain from the relation Di 1  1   i Di   i D i , where 0   i  1 .

Step 6. Check up the exactness criteria. If it is not satisfied take Di  Di 1 and go to Step 2. Otherwise stop the iteration. Keywords: Pauli operator, extremal problem, necessary condition, variable domain, support function. AMS Subject Classification: 49QО45, 49К50.

References 1. Cycon H.L., Froese R., Kirsch W., Simon B., Schrodinger Operators with Applications in Quantum Physics and Global Geometry, Moscow, Mir, 1990, 406p. 2. Demyanov V.F., Rubinov A.M., Basis of Non-Smooth Analysis and Quasidifferential Calculus, Moscow, Nauka, 1990. 3. Gasimov Y.S. On a shape design problem for one spectral functional, Journal of Inverse and IllPosed problems, Vol.21, No.5, 2013, pp.629-637. 4.

Gasimov Y.S. Some shape optimization problems for eigenvalues. Journal of Physics A.: Mathematical and Theoretical, Vol. 41, No.5, 2008, pp.521-529.

5. Boulkhemair A., Chakib A., Nachaoui A., Uniform trace theorem and application to shape optimization, Appl. Comput. Math., Vol.7, No.2, 2008, pp.192-205.

OPTIMIZATION OF THICKNESS OF FINITE RECTANGULAR PLATE WITH CIRCULAR HOLES AND DIFFERENT BOUNDARY CONDITION TO MINIMIZE WEIGHT STRUCTURE USING ESO METHOD Amin Ghannadiasl1, Mirmohammad Seyedhashemi Dijvejin2 1

2

Faculty of Engineering, University of Mohaghegh Ardabili, Ardabil, Iran Department of Civil Engineering, Germi Branch, Islamic Azad University, Germi, Iran e-mail: [email protected]

This paper investigates the optimization of the thickness of rectangular plate with circular opening and different boundary condition to minimize weight by evolutionary structural optimization (ESO) method. The evolutionary structural optimization (ESO) method is one of 78


the most favorite techniques for topology optimization. The software ANSYS is used to handle the finite element analysis. The inequality constraints are the maximum Von Misses stress required on this rectangular plate with a circular opening. The effect of different boundary conditions is assessed. Finally, some numerical examples are presented to illustrate the optimization of the thickness on the weight of the plate structure with circular opening using ESO method. Introduction. In the last decades, structural optimization has become one of the most important topics of engineering applications. The Structural design optimization has been an interesting area of research in the field of engineering design for its ability to short the design cycle and to increase product quality [1]. Xie and Steven present a new method of structural optimization, the Evolutionary Structural Optimization method [2]. The principles of the ESO method are based on the concept of gradually removing unnecessary or inefficient material from a structure to attain an optimal design. The ESO method is easy to implement and link with existing finite element analysis software packages (e.g. Abaqus, Ansys, Nastran). The initial steps of development of the this method are employed in verifying the classical single load problems to exhibit its applicability by Xie and Steven [3-7], Hinton et al [8]. Problem statement. Supports are utilized to hold the plate structure firmly and pull up it from deflecting excessively. The coordinates of simple support positions are assigned as design variables in support position optimization problems. Therefore, the support position optimization problem can be defined: Minimize Max  i

i  1,...,m 

Subject to a j  a j  a j ad  f ai  where

(1)

( j  1,...,n ) (2)

 i and m are the absolute value of the ith nodal deflection of the structural FE model and

the total number of nodal deflections of interest, respectively. aj shows the design variable, representing the coordinate of the jth independent support position of n simple supports, and ad is a dependent support coordinate. a j and a j define the upper and lower bounds of the support positions, respectively. The maximal deflection may often replace its position from one point to another during the design optimization because the maximal and absolute values used in the objective function in equation (1) does not refer to the same point deflection. Also, more nodal

79


deflections need to be taken into account in the equation (1), which implies that m should take a larger number [9]. Objectives and scope of study. The purpose of this study is the optimization of the thickness of rectangular plate with circular opening and different boundary condition to minimize weight by evolutionary structural optimization (ESO) method. This paper focuses on the effect of the boundary condition on the weight of plate structure. The scope of this study is to simulate plate with circular holes via ANSYS software package to obtain the weight of the plate. Results. The optimum design of a rectangular plate structure with circular opening using Evolutionary Structure Optimization method is presented in this paper. The thickness of the plate sets it into the best value with easy procedure in this method. The ESO method is a very reliable and easily applicable method of structural optimization. Numerical examples are exhibited to illustrate the efficiency and simplicity of the ESO method. Keywords: rectangular plate, circular hole, ESO method, ANSYS. AMS Subject Classification: 49Q12, 74K20, 74P05.

References 1. Jia H., Jiang C., Du L., Liu B., Jiang C., Evolutionary Enhanced Level Set Method for Structural Topology Optimization, Evolutionary Algorithms, 2011, pp.565-584. 2. Xie Y.M., Steven G.P., Shape and layout optimization via an evolutionary procedure, Proceedings of International Conference on Computational Engineering Science, Hong Kong University of Science and Technology, Hong Kong, 1992, pp.17-22. 3. Xie Y.M., Steven G.P., A simple evolutionary procedure for structural optimisation, Computers and Structures, Vol.49, 1993, pp.885-896. 4. Steven G.P., and Xie Y.M., Evolutionary structural optimization with FEA, Computational Mechanics, Vol.1, 1993, pp.27-34. 5. Steven G.P., Querin O.M., and Xie Y.M., Evolutionary structural optimization applied to aerospace structures, Proceedings of Pacific International Conference on Aerospace Science and Technology, Tainan, Taiwan, 1993, pp.349-354. 6. Xie Y.M., Steven G.P., Optimal design of multiple load case structures using an evolutionary procedure, Engineering Computations, Vol.11, 1994, pp.295-302.

80


7. Steven G.P., Querin, O.M., Xie Y.M., Evolutionary structural optimization on small computers, Proceedings of the International Conference on Education, Practice and Promotion of Computational Methods in Engineering Using Small Computers, Vol.2, 1995, pp.1075-1080. 8. Hinton E.,

Sienz J., Fully stressed topological design of structures using an evolutionary

procedure, Engineering Computations, Vol.12, 1995, pp.229-244. 9. Ghannadiasl A., Younesi S.S., Taghizadieh N., Optimization of support conditions to minimize the weight of plate structures, 5th National Conference on Earthquake and Structure, Kerman, Iran, 2014, Paper No-15117.

OPTIMIZATION OF HOLE SIZE AND LOCATION TO MAXIMIZE THE STRUCTURAL STRENGTH OF CONCRETE BEAMS WITHOUT ADDITIONAL REINFORCEMENT IN OPENING REGION Amin Ghannadiasl1, Said Ebadi2 1

2

Faculty of Engineering, University of Mohaghegh Ardabili, Ardabil, Iran Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran e-mail: [email protected]

This paper investigates the optimization of the opening size and location on the structural behavior of reinforced concrete beam. The reinforced concrete beams with and without vertical hole is analyzed in this paper. The software ANSYS is used to handle the finite element analysis. The shear strength, ultimate strength, stiffness, deformed shape, and cracking around the hole region are investigated. The effect of different boundary conditions is assessed. The present study concludes that the hole location has much more effect on the structural strength than the hole size. Finally, some numerical examples are presented to illustrate the optimization of the opening size and location on the structural behavior of reinforced concrete beam without additional reinforcement in the opening region. Introduction. In modern building construction, the vertical holes in reinforced concrete beams are often applied to the passage of pipes and utility ducts. These ducts are necessary in order to accommodate essential services such as air ducts, electricity, telephone, water supply, and computer network. The presence of holes will transform concrete beam behavior into a more complex behavior, as the dimension of the cross section of the beam is induced suddenly. Therefore, the shear strength, ultimate strength, cracks and stiffness may be seriously affected in opening region. Also, the hole produces disturbances or discontinuities in the normal flow of stresses, thus leading to stress concentration and cracking around the hole region. For any 81


discontinuity, special additional reinforcement or enclosing of the hole close to its periphery can be provided in sufficient quantity to control crack and prevent premature failure of the reinforced concrete beam [1-7]. Problem statement. Openings can be classified as small or big openings and the best location of the hole is decided based on its size. The vertical holes in reinforced concrete beams have been found to take many shapes such as rectangular, circular, diamond. However, rectangular and circular holes are the most common ones in practice [2]. Somes and Corley present that a circular opening may be considered as large when its diameter exceeds 0.25 times the depth of the web [4]. On the other hand, when the hole is small enough to maintain the beamtype behavior if the usual beam theory applies, then the opening may be termed as small. Also, when beam-type behavior ceases to exist due to the provision of the hole, then the opening may be classified as a large opening. Objectives and scope of study. The purpose of this study is to investigate the effect of sizing and location of the vertical circular holes in reinforced concrete beams on the behavior of beams without strengthening of the opening by additional reinforcement. This paper focuses on the effect of different diameters of circular holes on the behavior of beams and the effect of location the hole on the behavior of the reinforced concrete beam. The scope of this study is to simulate simply supported reinforced concrete rectangular beams with and without circular holes via ANSYS software package to obtain the ultimate strength, crack and the load-deflection. All beams have a cross section of 300 mm × 400 mm and 4000 mm in length with the circular holes in diameter 50 mm to 200 mm provided at an optimum distance from the support of the concrete beam. Results. By introducing the circular vertical hole with diameter less than 50% of the depth of the reinforced concrete beam without special reinforcement in opening region at L/2 distance means at the center span of the beam has no effect on the ultimate load capacity of the reinforced concrete rectangular beams, meaning that these beams behave similar to the beams without opening. Also, the mode of failure is flexure at mid-span in these beams. But, by introducing circular vertical hole with a diameter of 50% of depth at L/4 distance decrease the strength at least about 50% compared to reinforced concrete beam without the circular vertical hole and by introducing the circular vertical hole with a diameter of 37.5% of depth decrease the strength at least about 20%.

82


Keywords: reinforced concrete beams, vertical hole, finite elements, ANSYS. AMS Subject Classification: 49Q12, 65K10, 74K10.

References 1. Mansur M.A., Tan K.H., Lee S.L., Collapse loads of R/C beams with large openings, Journal of Structural Engineering, Vol.110, No.11, 1984, pp.2602-2618. 2. Ahmed A., Fayyadh M.M., Naganathan S., Nasharuddin K., Reinforced concrete beams with web openings: A state of the art review, Materials and Design, Vol. 40, 2012, pp.90-102. 3. Ramadan O.M., Abdelbaki S.M., Saleh A.M., Alkhattabi A.Y., Modeling of reinforced concrete beams with and without opening by using ANSYS, Journal of Engineering Sciences, Vol.37, No.4, 2009, pp.845-858. 4. Somes N.F., Corley W.G., Circular openings in webs of continuous beams, ACI Special Publication SP-42, 1974, pp. 359–398. 5. Amiri S., Masoudnia R., and Ameri M.A., A review of design specifications of opening in the web for simply supported RC beams, Journal of Civil Engineering and Construction Technology, Vol.2, No.4, 2011, pp.82-89. 6. Amin H.K.M., Agarwal V.C., Aziz O.Q., Effect of Opening Size and Location on the Shear Strength Behavior of RC Deep Beams without Web Reinforcement, International Journal of Innovative Technology and Exploring Engineering, Vol.3, No.7, 2013, pp.28-38. 7. Saksena N.H., Patel P.G., Effect of Opening Size and Location on the Shear Strength Behavior of RC Deep Beams without Web Reinforcement, International Journal of Advanced Engineering Technology, Vol.4, No.2, 2013, pp.40-42.

A FEEDFORWARD BACKPROPAGATION NEURAL NETWORK METHOD FOR THE BEST ESTIMATION OF OUTRAGE RATE IN MEDIUM VOLTAGE Reza Ghasemi1 1

Shahed University, Tehran, Iran e-mail: [email protected]

Development and establishment of electricity emergency centers aka the OMS (Outage Management System) is an important step toward further satisfaction of customers, controlling outages with low-voltage and mid-voltage plans, regulating operations for dealing with unwanted outages, identification of problematic and critical points across the grid, regulating the removal of passages lighting blackouts and effective monitoring of utilization activities of distribution companies. Doing these important jobs is not possible without using valuable 83


information and data recorded in these centers. The ability to identify outages in the grid helps greatly in optimal repair and maintenance, analyzing weaknesses in the network and improving intelligent and reasonable plans. By using artificial neural networks (ANNs) as one of data mining techniques, this paper provides a new method for identification and prediction of future outages based on the prediction of weather and environmental conditions like temperature, moisture, speed of wind and peak load, etc. Advantages of this method involves substituting complex classic calculations with quick efficient and cost effective computations. The software module created has the capability to be added to Electricity Emergency Center’s software.

Keywords:

ANN failure pattern discovery, Electricity Emergency Center (OMS), Climatic and

meteorological conditions, average load. AMS Subject Classification: 35J05, 49J20.

ON A DETERMINATION OF THE RIGHT HAND SIDE OF THE STRING OSCILLATIONS EQUATION IN THE MIXED PROBLEM H.F. Guliyev1, G.G. Ismailova2 1

Baku State University, Baku, Azerbaijan Sumgait Branch of Azerbaijan Institute of Teachers, Sumgait, Azerbaijan e-mail: [email protected]

2

Inverse problems for differential equations are one of the intensively studied classes of mathematical problems [1]. Note that such problems arise in very various fields of mathematics, geophysics, seismology, astronomy, ecology and so on. There appear new statements of inverse problems, theories for their solution, numerical algorithms and corresponding software [2]. In the present paper, we suggest a method for solving an inverse problem for the string oscillation equation. Determination of the unknown right hand side of the equation here is reduced to the minimization of the functional constructed by means of the additional information. The gradient of the functional is calculated, a necessary and sufficient optimality condition is derived. In the domain QT  ( x, t ) | 0  x  l , 0  t  T } we consider the boundary value problem:

84


  2u  2u  2  2  v( x, t ), ( x, t )  QT , x  t  u ( x,0)  u1 ( x), 0  x  l , u ( x,0)  u 0 ( x), t  u (0, t )  0, u x (0, t )  u x l , t , 0  t  T .  

(1)

Here u 0 ( x) W21 (0, l ), u1 ( x)  L2 (0, l ) – are given functions, v( x, t )  L2 (QT ) – is unknown function. To find v( x, t ), we use the additional information u( x0 , t )   (t ),

0  t  T,

(2)

where  (t )  L2 (0,T ) – is a given function, x0  (0, l ). We reduce this problem to the optimal control problem, i.e. minimization of the functional T

1 2 J (v)   u ( x0 , t; v)   (t ) dt  min, 20

(3)

on the solution of the problem (1). Where u( x, t; v) is the solution of the problem (1), that corresponds to the function v( x, t ). We call the function v( x, t ) a control. If we find the control

v( x, t ) that delivers to the functional (3) the zero value, then the additional condition (2) is fulfilled. In the paper it is shown that

inf J (v)  0.

vL2 ( QT )

Then the differential of the functional (3) is calculated and it is shown that < J (v), v > L2 (QT )    ( x, t; v) v( x, t )dxdt , QT

where  ( x, t; v) is a generalized solution of the following problem adjoint problem:   2  2 ( x, t()x , tQ )  2  2  u ( x, t ; v)   (t )  ( x  x0 ), T ,QT , x  t   ( x, T )  0, 0  x  l,  ( x, T )  0, t  0  t  T.  x (l , t )  0, (0, t )   l , t ,  

and gradient of the functional (3) is calculated as J (v)   ( x, t , v). Then by means of the solution of the adjoint problem, a necessary and sufficient optimality condition in the problem (1), (3) is obtained, i.e.

85


Theorem.

Let the conditions imposed above on the data of the problem (1)-(2) be

satisfied. Then for the optimality of the control v*  v* ( x, t )  L2 (QT ) in the problem (1), (3) it is necessary and sufficient fulfilment of the equality  * ( x, t )  0. for all ( x, t )  QT , where  * ( x, t )  is a solution of the adjoint problem (4) by v*  v* ( x, t ). Then the following iterational formula is proposed for the determination of the minimizing control vn1 ( x, t )  vn ( x, t )   n J (vn ).

Keywords: boundary problem, inverse problem, optimality condition, adjoint problem, optimal control. AMS Subject Classification: 35J05, 49J20.

References 1. Kabanikhin S.I., Inverse and Ill – Posed Problems, Novosibirsk, 2009, 457 p. (Russian). 2. Lions J.L. Optimal Control of the Systems Described by Partial Differential Equations, Mir, Moscow, 1972, 414 p. (in Russian).

OPTIMAL CONTROL PROBLEM FOR EQUATIONS OF FLEXURAL-TORSIONAL VIBRATIONS OF THE BAR H.F. Guliyev1, A.T. Ramazanova1 1

Baku State University,Baku, Azerbaijan e-mail: [email protected]

It is known that a number of problems of mathematical physics, engineering, mechanics, etc. are described by partial differential equations of the fourth order, equation of vibrations of a tuning fork, bar, equation of vibrations of rotating shafts, pitching of the vessel, equation of vibrations of plates, etc. [1,2,3 ]. In this work we study the optimal control problem in the system, describing the flexural - torsional vibrations of a bar. Formulation of the optimal control problem and the correctness of the boundary value problem is given below. Consider the vibrations of a bar, described by the system of two differential equations in domain

Q  0  x  l ,0  t  T 

86


2  2 y  2 y  2 E ( x ) I ( x )   x A x   x A x e x  f1  x, t , y, , v1, v2  ,             x 2  x 2  t 2 t 2 2 x 2

1

 2 y  2   2  E x C x               G x C x    x A x e x   x 2  x 2 t 2 



 2



 2   x  I x   Ax e 2 x  2  f 2 x, t , y,  , v1 , v 2 , t

where l  0, T  0 are given numbers, y( x, t ) -cross displacement of a bar,   x, t  - angle of rotation of cross section of a bar, E  x  -Young modulus, I  x  - polar moment of inertia of cross section around its center of gravity,   x  - density of the bar material, A  x  - area of crosssection, e  x  - distance from center of gravity to the center of torsion, C  x  - sectorial moment of inertia of the cross section, G( x) -the shear module, C  x  - geometric rigidity of free torsion,

E ( x)C  x  - rigidity of the flexural torsion, G( x)C  x  - rigidity of free torsion, v1  x, t  , v2  x, t  - operating functions, fi  x, t , y, , v1, v2  , i  1, 2 -given functions in the area Q  R 2  R 2 . Let the bar be freely clamped. Then at the points x  0 and x  l we have the boundary y x0  y xl  0,

y x



 x

x0



xl

 0,

 x 0

 x 0

y x  x

 0,

 3

0

 4

x l

x l

and the initial conditions

y  1  x  , t t 0

 5

 t

 6

y t 0  0  x  ,

 t 0  g0  x  ,

t 0

 g1  x  .

For a class of admissible controls U d we take a set of

v  x, t    v1  x, t  , v2  x, t  

measurable vector-functions

taking values from V  R 2 in Q and for almost all

 x, t   Q ,

where V is an arbitrary set. It needs to minimize functional J  v    f0  x, t , y, , v1, v2  dxdt Q

87

7


in a class U d under the conditions

1   6

, where

 y  x, t; v  ,  x, t; v 

is a solution of

problem 1   6  , by corresponding control v  x, t    v1  x, t  , v2  x, t   , f0  x, t , y, , v1, v2  is a given function in Q  R 2  R 2 . The optimal control problem 1   7  we call a problem

1   7  . We assume that, the data of the problem 1   7  satisfy the following conditions: 1) E  x  , I  x  ,   x  , A  x  , e  x  , C  x  , G( x) , C  x  are measurable, bounded and positive functions on the interval  0,l  . 2) 0 , 1, go , g1 are defined functions, and 0 W22  0, l  , g0 W22  0, l  , 1, g1  L2  0, l  . 3) Function fi  x, t , y, , v1, v2  , i  0,1, 2 is continuous in Q  R 2  R 2 and has continuous derivatives

fi fi f f f f , , and i , i , i  1, 2 bounded and i , i , i  0,1, 2 satisfy a Lipschitz y  y  y 

conditions on  y,  . Theorem. Let the problem 1  7 satisfies the conditions 1)-3). Then for the optimality of the control

v x, t , v x, t  0 1

0 2

for the 1  7 problem, the fulfiment of the following

conditions is necessary

max H x, t , y0 x, t , 0 x, t , v1 , v2 , 1 x, t , 2 x, t  

v v1 ,v2 V





 H x, t , y0 x, t , 0 x, t , v10 x, t , v20 x, t , 1 x, t , 2 x, t  ,

for almost all

 x, t   Q ,

where y0 x, t , 0 x, t  is a solution of problem

1  6

by

vx, t   v0 x, t  ,  1 x, t ,  2 x, t  is a solution of the adjoint problem 2 x 2

  2   2  2  E x I x  2 1    x Ax  2 1   x Ax ex  2 2   x  t t 

 2 x 2

H x, t , y0 x, t , 0 x, t , v10 x, t , v20 x, t , 1 x, t , 2 x, t  , y

8

  2   2  2  2 1  E x C w x  2 2   Gx C x  2 2   x Ax ex  2 2   x  I x   Ax e 2 x   x  x t t 2 











H x, t , y 0 x, t , 0 x, t , v10 x, t , v20 x, t , 1 x, t , 2 x, t  ,  88

9


1 x0  1 xl  0,

 1 x

 2 x0   2 xl  0,

 2 x

1 t T  0,

 x 0

 x 0

 1 x

x l

 2 x

x l

 0,

10

0,

11

 1  2  0,  2 t T  0, t t T t

 0.

12

t T

Here H  x, t , y, , v1, v2 ,1, 2   1 f1  x, t , y, , v1, v2   2 f 2  x, t , y, , v1, v2   f 0  x, t , y, , v1, v2  is Hamiltonian function for the problem (1)-(7). Keywords: optimal control, bar, flexural- torsional vibrations, necessary condition. AMS Subject Classification: 35J05, 49J20.

References 1. Tikhonov A.N., Samara A.A.., Equations of Mathematical Physics, Moscow, 1976, 736 p. 2. Armand J .-L.P., Applications of the Theory of Optimal Control of Systems with Distributed Parameters to Problems of Optimization of Structures, Moscow, Mir, 1977, 144 p. 3. Komkov V., Optimal Control Theory for the Damping of Vibrations of Simple Elastic Systems, Moscow, Mir, 1975, 158p.

AN OPTIMAL CONTROL PROBLEM FOR HYPERBOLIC EQUATION WITH NONLOCAL BOUNDARY CONDITION WITH CONTROLS AT THE COEFFICIENTS H.F. Guliyev1, H.T. Tagıyev1, A.A. Mehdiyev1 1

Baku State University, Baku, Azerbaijan e-mail: [email protected]

Suppose, that it is required to minimize the functional

J ( )   f 0 ( x, t , u( x, t ), ( x, t ))dxdt Q

(1)

T

subject to  m   u ( x, t )   2u( x, t )    aij ( x, t , ( x, t ))   f ( x, t , u ( x, t ), ( x, t )) , ( x, t )  Q T , (2) x  t 2 i, j  1 xi  j 

u( x,0)   0 ( x),

89

u( x,0)  1( x) , x   , t

(3)


m u ( x, t ) cos( , xi )   K ( x, y, t , u ( y, t ))dy , x   ,  aij ( x, t , ( x, t )) x j  i, j  1 S T

(4)

where u ( x, t ) is a state function of system,  ( x, t ) is a control function. As class of admissable control U ad we take set of QT    (0, T ) continious functions

 ( x, t ) with continuously derivatives

 ( x, t )  ( x, t ) so that values of controls in [ ,  ] where , xi t

 ,  given numbers. Suppose, that data of the problem satisfy to the following conditions: 1. Functions aij ( x, t ), i, j  1,2,..., m  3 are continuos on derivatives

a ij t

,

a ij xi

,

a ij 

,

 2 a ij xi 

QT [ ,  ] and have the continuous

besides this for   R m and for any

( x, t, )  Q  [ ,  ] , T

m m 2 1  aij ( x, t ) i j     i ,   const  0 ; 2.  0 ( x)  W2 (),1( x)  L2 (); i, j  1 i 1

3. Functions f ( x, t , u,  ) and f 0 ( x, t , u,  ) are continuous and have continuous derivatives f f  2 f 0 f f  2 f 2 f f , 0, 0, on Q  R  [ ,  ] . So, that derivatives , are bounded , , T u u  u u  u u

and

f satisfies to Hölder condition with respect to u with factor u

,

f 2    1 , derivative 0 , 3 u

satisfies to Lipshitz condition by u , function K ( x, y, t , u) is continuos and have continuous derivatives

K K K K on   QT  R . So, that K ( x, y, t ,0)  0 , functions satisfies to , , t u t u

Lipshitz condition by u , derivative

K u

satisfy to Hölder condition with multiplier

,

2 1     1 , besides this, suppose, that A  L  (mes) 2   1 . 3 2 

We introduce the following system:

 2 ( x, t )  t 2 

m  K ( , x, t , u ( x, t ))  ( x, t ) 0 (a ( x, t ,  ( x, t )) )    ( , t ) ds   ij 0  x  x  u   i, j  1 i j

H ( x, t , u0 ( x, t ),0 ( x, t ), ( x, t )) u

,

90

(5)


 ( x, T ) 0, t

 ( x, T )  0,  ( x, t )  A

0,

(6)

(7)

S T

where H ( x, t , u,, )  f ( x, t , u, )  f ( x, t , u, ) is a Pontryagin’s function. Theorem. Suppose, that the problem (1)-(4) satisfy to given conditions and 1 A  L (  (mes) 2 )  1 . If (u0 ( x, t ),0 ( x, t )) is an optimal pair and  ( x, t ) is a solution of the 2

corresponding conjugate problem (5)-(7), then for almost all ( x, t )  QT and for all arbitrary

  [ ,  ] the following inequality is satisfied  m u H ( x, t , u , , )     0) 0 0 ( a ( x , t ,  )    (  0 )  0 . ij 0    x  x     Q  i, j  0 i j  T

Keywords: Pontryagin’s function, Lipshitz condition, hyperbolic equation. AMS Subject Classification: 35J05, 49J20.

ON A BOUNDARY CONTROL PROBLEM FOR A THIN PLATE OSCILLATIONS EQUATION H.F. Guliyev1, Kh.I. Seyfullayeva2 1

Baku State University, Baku, Azerbaijan Sumgayit State University, Sumgayit, Azerbaijan e-mail: [email protected]

2

In the present work, we suggest a boundary optimal control problem for a linear equation of thin plate oscillations. In the work we prove a theorem on the existence and uniqueness of the optimal control, calculate the differential functional and derive necessary optimality condition in the form of an integral inequality. It is known that some processes of mathematical physics are described by partial differential equations of fourth order. For instance, equations of oscillations of a bar, camerton, elastic plate, thin plate and so on are such equations [1]-[3]. Therefore, investigation of optimal control problems in the processes described by such equations is important problems. When control functions are boundary functions, it becomes difficult to study the control problems. But

91


note that the boundary control problem is very natural compared with distributed parameters problems from theoretical and practical point of view. Let the control process be described by a thin plate oscillations equation  2u  a 2 2 u  0 в QT    (0, T ) ,   (0, l1 )  (0, l 2 ) t 2

(1)

with initial u( x1 , x2 ,0)   0 ( x1 , x2 ) ,

u ( x1 , x2 ,0)  1 ( x1 , x2 ) , ( x1 , x2 )   t

(2)

and boundary conditions u (0, x 2 , t )  0, u (l1 , x2 , t )  0,

u (0, x2 , t ) u (l1 , x2 , t )   ( x 2 , t ),  0, ( x 2 , t )  (0, l 2 )  (0, T ), x1 x1

u ( x1 ,0, t ) u ( x1 , l 2 , t ) u ( x1 ,0, t )  0, u ( x1 , l 2 , t )  0,  0,  0, ( x1 , t )  (0, l1 )  (0, T ), x 2 x2

(3)

where a 2 , l1 , l 2 , T are the given positive numbers,  ( x2 , t ) is a boundary control function,

 0 ( x1 , x2 ) W22 () , 1 ( x1 , x2 )  L2 () are the given functions,  is the Laplace operator with respect to x1 , x2 . Let’s consider a space of controls H  W24, 2 (0, l 2 )  (0, T ) . For a class of admissible controls U ad we take the set of functions  ( x2 , t ) from H , for which  (0, t )   (l 2 , t )  0,  4 x24

M, L2 ( 0 ,l2 )( 0 ,T ) 

 ( x2 ,0)  (0, t )  (l 2 , t )   0,  ( x2 ,0)   0 , moreover x2 x2 t

 2 t 2

 M ,where M is a given number. L2 ( 0 ,l2 )( 0 ,T ) 

It is supposed that the functions  0 ( x1 , x2 ) and  ( x2 , t ) satisfy the agreement conditions. We state a problem: in the set U ad find such a function that together with the solution of boundary value problem (1)-(3) it delivers minimum to the functional

J ( ) 

1  2 2 2   u ( x , x , T ) dx dx   ( x2 , t )dx2 dt , 1 2 1 2 2  2 0 0 l T

where   0 is a positive number.

92

(4)


Under the solution of problem (1)-(3) for each fixed admissible control  ( x2 , t ) we understand a function u( x1 , x2 , t )  W22,1 (QT ) , such that for any function   W22,1 (QT ) ,

 ( x1 , x2 , T )  0,  (0, x 2 , t )  0,  (l1 , x2 , t )  0,

 (0, x 2 , t )  (l1 , x 2 , t )  0,  0, x1 x1

 ( x1 ,0, t )  0,  ( x1 , l 2 , t )  0,

 ( x1 ,0, t )  ( x1 , l 2 , t )  0, 0 x 2 x 2

satisfies the integral identity  u 

  t  t

QT

  a 2 u  dx1dx2 dt   1 ( x1 , x2 ) ( x1 , x2 ,0)dx1dx2  0  

(5)

and the conditions u( x1 , x2 ,0)   0 ( x1 , x2 ) ,

u (0, x2 , t )   ( x2 , t ) x1

(6)

in the ordinary sense. In the work we obtain the following results: Theorem 1. In the optimal control problem (1)-(4) there exists a unique optimal control. Theorem 2. Let the above assumed conditions be fulfilled on the data of problem (1)-(4). Then the functional (4) is Frechet continuously differentiable on H and its differential at the point  ( x2 , t )  U ad with the increment  ( x2 , t )  H ,  ( x2 , t )   ( x2 , t )  U ad is determined by the expression J ( ), 

   2 (0, x2 , t )  2 (0, x2 , t )   ( x2 , t )dx2 dt ,     ( x2 , t )  a 2   x12 x22    0 0 T l2

H

(7)

where  ( x1 , x2 , t ) is the solution of adjoint problem:

 ( x1 , x2 ,T )  0 ,  (0, x 2 , t )   (l1 , x 2 , t )  0,

 2  a 2 2  0 в QT , 2 t

(8)

 ( x1 , x2 , T )  u ( x1 , x2 , T ) , ( x1 , x2 )   , t

(9)

 (0, x 2 , t )  (l1 , x 2 , t )   0, ( x 2 , t )  0, l 2  0, T , x1 x1

 ( x1 ,0, t )  ( x1 , l 2 , t )  ( x1 ,0, t )   ( x1 , l 2 , t )  0,  0,  0, ( x1 , t )  0, l1  0, T , x 2 x 2

93

(10)


where u ( x1 , x2 , t ) is the solution of problem (1)-(3) for the given control  ( x2 , t ) . Theorem 3. Let all the above conditions imposed on the data of problem (1)-(4) be fulfilled. Then for the optimality of the control  ( x2 , t )  U ad in problem (1)-(4) it is necessary and sufficient that the inequality 2   2  (0, x2 , t )  2     (0, x 2 , t )   ( x2 , t )   ( x2 , t ) dx2 dt  0 ,   U ad ,  ( x , t )  a  0 0  2  x12 x22   

T l2

be fulfilled, where   ( x1 , x2 , t ) is the solution of problem (8)-(10) for u  u ( x1 , x2 , t ) , while

u ( x1 , x2 , t ) is the solution of problem (1)-(3) for    ( x2 , t ) . Keywords: optimal control, existence theorem, necessary condition. AMS Subject Classification:35J05, 35J10.

References 1. Tikhonov A.N., Samarskiy A.A.,Equations of Mathematical Physics, Moscow, Nauka, 1972, 736p. 2. Komkov V. Theory of optimal control of vibrations damping of prime elastic systems, Moscow, Mir, 1975, 160 p. 3.

Arman J.-L.P. Applications of theory of optimal control of the systems with distributed parameters to the problems of optimization of constructions, Moscow, Mir, 1977, 144 p.

ON NUMERICAL SOLUTION TO AN OPTIMAL CONTROL PROBLEM WITH CONSTRAINT ON THE STATE FOR A PARABOLIC EQUATION S.I. Guseynov1, S.R. Kerimova1 1

Azerbaijan State Oil Academy, Baku, Azerbaijan e-mail: [email protected]

Determination of optimal regimes of heat conduction processes leads to optimal control problems for a linear parabolic equation only when considering small intervals of temperature changes [1]. That is why we need to solve optimal control problems not only with constraint on the temperature of the heat sources, but also on the temperature distribution at all points. Such

94


problems also arise when investigating thermodynamic processes, when overeating of material above a certain critical temperature is not allowed [2]. Therefore, development of efficient computational algorithms for solving specific optimal control problems with constraint on the state for a parabolic equation is of considerable interest. In the work we consider an optimal control problem with two control functions and constraint on the state for a linear parabolic equation. We have developed an algorithm of numerical solution to the problem with application of the gradient projection, penalty functional, and finite-difference methods with the help of the obtained analytical formula for the gradient of the functional. We consider a heat conduction process, the mathematical model of which has the following form: 1   u  u  xk ( x)   f ( x, t )   (t )u  c( x) , x x  x  t u( x, 0)   ( x), xc  x  x R ,

 1u   2 k ( x)x x

c

 v(t ),

u( x R , t )  g (t ),

xc  x  x R ,

0  t  T,

(1) (2)

0  t  T,

0  t  T,

(3) (4)

where k (x) , c(x) , g (x) ,  (t ) are given functions, xc , xR ,T , 1 , 2 given numbers,  12   22  0 . We put the following problem: to find the functions f ( x, t ) , v(t ) , and u( x, t ) , satisfying the conditions (1)-(4) and the constraints f min  f ( x, t )  f max ,

(5)

vmin  v(t )  vmax ,

(6)

u( x, t )  u max ,

(7)

such that the functional

 u( x,T )  u ( x) xdx

xR

J ( f , v) 

*

2

(8)

xc

would take on its smallest possible value under the given function u * (t ) . Here f min , f max , vmin , v max , and u max are given numbers characterizing the utmost capabilities of the heat sources and

maximum allowable value of the temperature.

95


To solve the posed problem, we use the penalty functional method. Let us introduce the following functional  k ( f , v)  J ( f , v)  Ak

T xR

  max{u( x, t )  u

max ;

0} dxdt , 2

(9)

0 xc

where { Ak } is some sequence of positive numbers, lim Ak   (for example, Ak  10k ). k 

For each k  1, 2, ... , the problem of minimization of the functional (9) under the conditions (1)-(7) is being solved by the gradient projection method, which requires calculating the gradient of the functional. By using the method of increments we have obtained an analytical formula for the gradient of the functional:   y    grad ( f , v)   f ( x, t ),  v (t )   y ( x, t ),   1 xk ( x)   2 x y( x, t )     x   x  xc  





(10)

where y( x, t ) is the solution to the following adjoint boundary-value problem:  k ( x)   y  y  x    (t ) y  2 Ak max{(u ( x, t )  u max ); 0}  c( x) ,  t  x x  x  *  2(u ( x, T )  u ( x)) , xc  x  x R  y ( x, t )   c( x)   y   1 y ( x, t )   2 k ( x)  0 x  x  xc   0  t  T.  y ( x R , t )  0,

xc  x  x R , 0  t  T

(11)

Thus the solution to the problem is reduced to building the sequences f ( n) ( x, t ) and v ( k ) (t ) of the gradient projection method, specifying some initial approximations, where the step is chosen according to the condition of monotone decrease of the functional (9) by applying the bisection (halving) method. For numerical implementation of the algorithm we use finite-difference method on nonuniform meshes. Boundary-value problems (1)-(4) and (10) are approximated by implicit twolayer schemes, linear with respect to the values u and y on a new layer; under the fixed f ( x, t ), v(t ) they are solved by the sweep method.

For approximate calculation of the functional we use the trapezoidal rule. The precision on the functional is verified by the condition  (kn)   (kn1)   , where   0 is given sufficiently small number.

96


The developed algorithm has been tested on several model problems with and without the constraint on the state. The numerical experiments showed that by applying the proposed method one can find a numerical solution to the problem (1)-(8) with the required accuracy, and it can be applied to the determination of optimal modes of heat conductivity processes.

Keywords: heat conduction process, an optimal control problem, parabolic equation, numerical solution, the gradient of the functional. AMS Subject Classification: 49J20, 74Sxx.

References 1. Guseynov S.I., Kerimova S.R., Solution an optimal control problem with respect to quasilinear heat conduction equation, The Third International Conference «Problems of Cybernetics an Informatics», 2010, Baku, Vol.III, pp.140-142. 2. Guseynov S.I., Gasimov S., Numerical Solution the optimal Speed – in action problem for a heat conduction process with phase constraints, 24th Mini Euro Conference «Continuous Optimization and Information - Based Technologies in the Financial Sector», Izmir, Turkey, 2010, pp.120-123.

INVESTIGATION OF THE OPTIMAL CONTROL PROBLEM FOR LINEAR IMPULSE SYSTEMS K.K. Hasanov1, Kh.T. Huseynova1 1

Baku State University, Baku, Azerbaijan e-mail: [email protected], [email protected]

The optimal control problem of linear impulse system with minimization of a quadratic functional is considered provided that the given time the system should reach to the desired state. For investigation the problem is reduced to the problem of moments. Such functional Hilbert space is constructed in which the minimized expression is represented as the norm for functional over this space. Here the Fredholm’s alternatives for integral equations with real symmetric kernel are used. The problem of such kind for the ordinary linear system have been considered in the works [1-8]. 1. Statement of the problem. Let the controlled process is described by a linear impulse system Dx(t )  A(t ) x(t )  B(t )u(t )  C(t ) Du(t ) ,

97

(1)


with initial condition x(0)  x 0 ,

(2)

where the matrices A(t )  (n  n), B(t )  (n  r ), C(t )  (n  r ) are continuous at t [0,T ] . Admissible controls u (t ) are right-continuous r- dimensional vector functions of bounded variation on [0, T ], each of which is defined on some open interval in the neighborhood of the interval [0, T ] . Each this control determines on [0, T] some measure Du(t ) . Set of admissible controls we'll denote by VBr (0,T ) . Let (t ) the fundamental matrix of homogeneous system x  A(t ) x ,

(3)

satisfying the condition (0)  I (where I – is an identity matrix). For each control u (t ) , the corresponding solution x(t ) of equation (1) with the initial condition (2), we have the formula t

t

x(t )  Ф(t ) x  Ф(t )  Ф ( s) B( s)u ( s)ds  Ф(t )  Ф 1 ( s)C ( s) Du( s) , 1

0

0

(4)

0

where the last integral is understood in the Riemann-Stieltjes sense. It should be noted that the solution x(t ) is also a function of the bounded variation . Let the control u(t ) VBr (0, T ) moves the system (1) from the state (2) in the state x(T )  x1 . Then, using the formula (4), we have: T

T

1 1 1 1 0  Ф (s) B(s)u(s)ds  Ф (s)C (s) Du(s)  Ф (T )( x  Ф(T ) x ) . 0

(5)

0

Thus, the equation (5) is a necessary and sufficient condition in order to the control u (t ) transferred system (1) from state x 0 to the state x1 . Among the admissible controls u (t ) is required to find the optimal control such that at a given moment T moves the system (1) from the state (2) to the state x1 and the functional T

J (u )  x1 (T ) Fx (T )   [ x(t )W (t ) x(t )  u(t )U (t )u (t )]dt

(6)

0

reached the least value, where F  (n  n) is an non-negative constant matrix, W (t )  (n  n) – symmetric, nonnegative and continuous, and U (t )  (r  r ) is a positive defined and continuous matrix. Method of solution. Further for simplicity assume that x 0  0 , F  0, B(t )  0, t [0, T ] . 98


Substituting in (6) the solution (4) of the problem (1), (2), we have   t     t 1   1       J (u )      ( s)C ( s) Du( s)   (t )W (t )(t )   ( s)C ( s) Du( s)  dt   0 0  0    T

T

  u(t )U (t )u (t )dt .

(7)

0

Hence, changing the integration order, we receive T



J (u )    1 (t )C (t ) Du(t ) 0

  (s)W (s)(s)   T

s

t

0

1

T  ( )C ( ) Du( ) ds   u (t )U (t )u (t )dt . 0 

(8)

Further, by using the Heaviside function  (t ) can be written T  s 1  T 1     t  (s)W (s)(s) 0  ( )C ( ) Du( ) ds  t  (s)W (s)(s) 0  (t   ) ( )C ( ) Du( ) 

T

    (  t ) ( s   ) ( )C ( ) Du( ) ds. 0  T

(9)

1

Suppose T  1   C ( t )( Ф ( t ) ) ( s)W ( s)( s)ds 1 ( )C ( ) , t   ,    t K (t , )   T  C ( )( 1 ( )) ( s)W ( s)( s)ds 1 (t )C (t ), t   .  

(10)

It is possible to be convinced directly that K (t , ) is continuous on set of variables, a symmetric, non-negative matrix. Considering (9) after the done some transformations the functional (8) can be expressed as follows: T T

T

0 0

0

J (u )    ( Du(t ))K (t , ) Du( )   u(t )U (t )u (t )dt .

Keywords: impulse system, bounded variation, functional, Hilbert space, integral equation, control. AMS Subject Classification: 49N25.

References 1. Krasovskii N.N., Theory of Motion Control Наука, M, Nauka, 1968, 476 p.

99

(11)


2. Kurzhansky A.B. Design of optimum control minimizing mean-square error, Avtomat. i Telemekh., Vol.25, No.5, 1964, pp.624-630. 3. Hasanov K.K., Mazamov A.D., Controllability of linear impulse system with aftereffect, News of Baku University, No.1, 2003, pp.77-86. 4. Gasanov K.K., Guseynova Kh, T. On existence and uniqueness of the solution to initial boundary value problem for the first order partial linear system, Advances and Applications in Mathematical Sciences, Vol.11, No.3, 2012, pp.115-124. 5. Hasanov K.K., Huseynova Kh,T., The optimal control of linear system with generalized controls, News of Baku University, No.1, 2013, pp.25-33. 6. Hasanov K.K., Gasumov T.M., Minimal energy control for the wave equation with non-classical boundary condition, Appl. Comput Math., Vol.9, No.1, 2010, pp.47-56. 7. Li E.B., Markus L., Fundamentals of Optimal Control Theory, Moscow, Nauka, 1972, 576 p. 8. Kolmogorov A.N., Fomin S.V., Elements of the Theory of Functions and Functional Analysis, Moscow, Nauka, 1972, 496 p.

INVESTIGATION OF THE OPTIMAL CONTROL PROBLEM FOR HEATING ROD PROCESS USING BOUNDARY CONTROL METHOD OF MOMENTS K.K. Hasanov1, L.K. Hasanova1 1


The problem of optimal control for heating rod process using a boundary control with minimum total cost is considered. The lateral surface and left end of rod are insulated, and right end of rod is fed heat flux, the temperature in which in the moment of time t is equal u (t ) . It is required to control the temperature of the external environment, so that to the given moment of time T the distribution of temperature in the rod is equal to the given temperature, and to minimize the total cost [1-5]. The indicated problem mathematically can be formulated as follows: It is required to find such control u (t ) from L2 (0, T ) , that the solution of the boundary problem y( x, t ) yt  a 2 y xx , ( x, t )  D(0  x  l , 0  t  T ) , y( x,0)   ( x) ,

0 xl,

y x (0, t )  0 , y x (l , t )  [u(t )  y(l , t )] ,

in the moment of time T is equal

100

(1) (2)

0 t  T

(3)


y( x,T )  y0 ( x) ,

(4)

and the functional l T

T

0 0

0

J (u )    (t ) y 2 ( x, t )dxdt    (t )u 2 (t )dt

(5)

reached the least value, where a, , l ,T - given positive number,  (x) , y0 ( x)  L2 (0,T ) and  (t ) ,

 (t ) positive continuous functions. Admissible control u (t ) is any function from L2 (0,T ) . Theorem 1. For each function u(t )  L2 (0,T ) ,  ( x)  L2 0, l  the boundary problem (1)(3) has a unique generalized solution y( x, t ) W2(1,0) ( D) , representable as a Fourier series: 

y( x, t )   e

( ak ) 2 t

k 1

l

 k    ( x) k ( x)dx, k 

k l

0

l

( k  a vk (l )  u ( )e ( ak )  d )vk ( x) , 2

2

(6)

0

,  k ( x) 

cos n x

n

, 1 ,  2 ,  3 ,... positive roots of equation l ctg    .

For simplicity we consider the case  ( x)  0 . Substituting in (5) the solution (6) and using that the function system {vk ( x)} is orthonormalized in L2 (0, l ) , we obtain 2

T  cos  n l  ( an ) 2 ( t  )  J (u )   a   (t )  u (  ) e d  dt   (t )u 2 (t )dt .    n 1   n 0 0 0  l

2



4

(7)

Let's assume  cos  n   n 1   n

2

  hn (t , ) . 

(8)

T ( an ) 2 ( t   2 s ) ds, t   ,    ( s )e t hn (t , )  T   ( s )e ( an ) 2 (t   2 s ) ds, t   .  

(9)

( a 2 ) 2 K (t , )  2



where

As hn (t , ) positive-symmetric, then K (t , ) is also a positive symmetric function. The set of functions defined on [0,T ] and summarized qudratically we'll denote by H (0,T ) . The scalar product of two function x(t ), y(t ) from H (0,T ) we define by formulae TT

T

0 0

0

( x, y )   K (t , ) x(t ) y ( )dtd    (t ) x(t ) y(t )dt . 101

(10)


The functional (5) on this space is represented in the norm form, i.e.

J (u )  u

T T

T

0 0

0

   K (t , )u (t )u ( )dtd    (t )u 2 (t )dt .

2

(11)

Let for the control u (t ) at the given moment T the distribution of temperature in the rod is equal to y( x,T )  y0 ( x) . Then, considering that the function y( x, t ) is defined by row (6), we have: 

T

y 0 ( x)   a 2  vn (l )vn ( x)  u (t )e ( an ) n 1

2

(T t )

dt .

(12)

0

Theorem 2. Let K (t , ) is defined by formulae (8). Then for each n equation T

 K (t, ) p ( )d   (t ) p (t )  a (t ) n

n

(13)

n

0

has a unique solution pn (t )  H (0,T ) , где an (t )   a 2 vn (l )e ( an ) As functions

a1 (t ), a2 (t ), a3 (t ),...

2

( T t )

, n  1,2, .

are linearly independent, then the solutions

p1 (t ), p2 (t ), p3 (t ),... are also linearly independent.

Indicated optimal problem is reduced to the finding the function u (t ) , for which the conditions would be satisfied T   u (t )dt  y0 n , n  1,2,... , K ( t ,  ) p (  ) d    ( t ) p ( t ) n n 0  0  

T

u

2

T T

T

0 0

0

   K (t , )u (t )u ( )d dt    (t )u 2 (t )dt  min ,

(14)

(15)

l

where y0 n   y0 ( x)vn ( x)dx , n  1,2,... . 0

The initial problem is reduced to the problem (14), (15), which is the l - problem of moments [1]. Theorem 3. Let u j (t ), j  1,2,... be the solution of the problem of moments T  0 0 K (t, ) pk ( )d   (t ) pk (t )u j (t )dt   kj , j  1,2,... .

T

102

(16)


u (t ) 

Then



y

0 j u j (t )

is a solution of the problem of the moments (14)-(15), where  kj is

j 1

Kronecker's symbol. Keywords: functional, optimal control, integral equation, method of moments. AMS Subject Classification: 49-XX.

References 1. Butkovskii A.G., Methods of Control of Systems with Distributed Parameters, Nauka, 1975, 568p. 2. Gabasova O.R. On optimal control of linear hybrid systems with terminal constraints, Appl. Comput. Math., Vol.9, N.1, 2010, pp. 194-205. 3. Krasovskii N.N., Theory of Motion Control, Moscow, Nauka, 1968, 476 p. 4. Hasanov K.K., Gasumov T.M. Minimal energy control for the wave equation with non-classical boundary condition, Appl. Comput. Math., Vol.9, No.1, 2010, pp.47-56. 5. Hasanov K.K., Hasanova L.K. An existence of the time optimal control for the object described by the heat conductive equation with non-classical boundary condition, TWMS Jour. of Pure and Appl. Math., Vol.4, No.1, 2013, pp.44-51.

ON A NUMERICAL ALGORITM FOR THE SOLUTION OF THE INVERSE PROBLEM WITH RESPECT TO POTENTIAL N.Sh. Huseynova1, M.M. Mutallimov1, M.Sh. Orucova2 1

Institute of Applied Mathematics, Baku State University, Baku, Azerbaijan 2 Azerbaijan State University of Economics e-mail: [email protected], [email protected], [email protected]

It is known that the motion of a particle in a central field is described by the following equation



a d  2 dR  bR  q(r ) R  ER . r  r 2 dr  dr  r 2

(1)

Here a  0 and b are given numbers, q(r ) - is the energy of interaction. If we multiply the equation (1) by the r 2 and introduce such substitution Q(r )  b  q(r )r 2 ,

we obtain the following: 103


a

d  2 dR  2 r   Q(r ) R  Er R . dr  dr 

(2)

The analytical solution of the equation (2) for different potentials is very interesting. But it is not always possible to obtain the analytical solutions. In addition, the solution of equation (2), finding of the potential Q(r ) with respect to the energy eigenvalues, i.e., solution of the inverse problem is also very interesting. Assume that R(r0 )  z 0 , R(r1 )  z1 , R(r2 )  z 2 ,..., R(rn )  z n

(3)

here 0  r0  r1  ...  rn ; n  2 . We consider the equation (2) on the interval [r1 , rn ] . In the work the primary aim is finding the potential Q(r ) in the interval [r1 , rn ] . We also need to show that the solution of the equation (2) R(r ) - function satisfies the equation (3). 

We will assume that the solution of the equation (2) is R(r ) and the condition  R(r )dr   0

is satisfied. Then the solution of the iverse problem is finding the potential Q(r ) , which it is necessary that by r  0 , the function Q(r ) would be continuously differentiable. No we will find the minimum of the following functional J (Q) 

n 1

 [ R( r )  z ] i

i

2

 min,

(4)

i 1

from the equation (2) we obtain the following conditions: R(r0 )  z 0 , R(rn )  z n .

(5)

Assume that U  {Q  Q(r )  L2 (r0 , rn ) : Q0  Q(r )  Q1 , r [r0 , rn ]} .

(6)

Here 0  Q0  Q1 - are given numbers. We suppose that the function    (r ) is solution of the following equation: a

n 1 d  2 d  2 r   Q(r )  Er   2  [ R(r )  zi ] (r  ri ) i 1 dr  dr 

 (r0 )  0,

 (rn )  0 .

(7) (8)

Applying the traditional technique [1, 2], one can show that the functional (4) is differentiated and the formula for the gradient J (Q)   R

104

(9)


is true. If R  R(r ) ,    (r ) , Q  Q(r ) , then the formulae (9) is the solution of the equations (2), (5), (7), (8). Algorithm 1. Consider the arbitrary initial potential Q0  Q(r ) U . 2. Found the solution of the equations (2), (5) with the potential Q0 (r ) , denote this solution R0  R0 (r ) .

3. Substituting the solution R0  R0 (r ) to the sweep problem (7), (8), solving this problem we find the function 0   0 (r ) . 4. Using the solutions R0  R0 (r ) and  0   0 (r ) , we found the gradient of the functional (4). 5. Minimize the linear functional rn

I 0 (Q)   0 (r ) R0 (r )Q(r )dr  min

(10)

r0

in the set U and find the helper function Q0  Q0 (r ) . 6. The new potential is constructed as follows: Q1 (r )  Q0 (r )  (1   )Q0 (r ), 0    1.

7. The accuracy criterion is checked. It may be either such

max Q1 (r )  Q0 (r )   , or such r r r 0

n

J (Q1 )  J (Q0 )   .

In the 6-th step the parameter  should be chosen thus that the obtained new values of functional with corresponding  were smaller than previous one J (Qk 1 )  J (Qk ) or J (Qk  (1   ))Qk  J (Q) .

These conditions are called the monotoncity conditions. From the monotoncity conditions can be seen that finding the parameter  from the condition J (Qk  (1   )Qk )  min, 0    1 is advantaged. However, finding  from these conditions creates additional difficulties. Therefore it is important to give another method, which is important from a practical point of view. We assume  

1 and check the monotoncity condition. If the monotoncity condition is 2 1 1 4 8

satisfied, then the corresponding  iteration is continued. Otherwise, assuming   , ... the monotoncity condition is checked.

105


From the algorithm can be seen that on each 5-th step of iteration the linear functional is minimized in the set U. The set U has a simple structure and it is solution doesn't create the difficulties. Therefore, the functional (10) is discretized and is operated within constraints to the linear function of minimization, in other words is reduced to the linear programming problem. Keywords: inverse problem, variation method, optimal potential. AMS Subject Classification: 45Q05, 47A75, 49R05.

References 1. Vasilev F.P., Numerical Methods for Solving Extremal Problems, М.: Nauka, 1981, 400p. 2. Huseynova N.Sh., Niftiyev A.A., Mutallimov M.M. Numerical algorithm for an inverse problem with respect to the potential, Proceedings of the Institute of Applied Mathematics, Vol.3, No.2, 2014, pp.185-195.

ON A DISCRETE-CONTINUOUS CONTROL PROBLEM WITH FUNCTIONAL CONSTRAINED RIGHT END OF TRAJECTORY G.A. Huseynzadeh1 1


In the report we consider a problem on minimum of the functional S0 u   0 xN ,t1 

(1)

at constraints

uk , t U ,

k , t   D  k , t : 1  k  N ;

t0  t  t1,

Si u   i xN , t1   0 , i  1, p , dxk , t   f k , t , xk , t , xk  1, t , uk , t  , dt

x0, t   g t , t  t 0 , t1 ,

xk , t 0   hk , 1  k  N .

k , t  D ,

(2) (3) (4) (5)

Here k is a natural number, f k , t , x, y, u  is the given n -dimensional vector-function continuous with respect to t , x, y  together with partial derivatives with respect to x, y  for any

k , N is the given natural number, hk  is the given discrete vector-function, g t  is the given

106


continuous vector-function, U is the given nonempty and bounded set, u k , t  is r - dimensional piecewise-continuous (with finite number of first kind discontinuity points) with respect to t for each k vector of control actions, and i x  , i  0, p is the given continuously differentiable scalar functions. By means of the modification of the increments method in the problem under consideration we get a first order necessary optimality condition. Keywords: discrete continuous systems, necessary optimality conditions. AMS Subject Classification: 49K10.

OPTIMAL CONTROL PROBLEMS WITH MIXED CONTROL-STATE CONSTRAINTS: OPTIMAL EXPLOITATION OF RENEWABLE RESOURCES M.H. Imanov1 1

Institute of Applied Mathematics, Baku, Azerbaijan e-mail: [email protected]

In the report the optimal control problem tf



J (u, x)  exp(rt )(P  E (t ) x(t )  C E E (t )  C I I (t ))dt  max 0

with mixed control-state constraints 0   x  F ( x)  qEx, x(0)  x  0  K  I    K , K (0)  K

0  E(t )  K (t ), 0  I (t )  I max , t [0, t f ]

is considered [1]. We investigate the possibility of application of the method of similar solutions to the problem using the methods offered in [2]. Keywords: control-state constraints, optimal control problem. AMS Subject Classification: 49J15.

References 1. Clark C.W., Clarke F.H., Munro G.R., The optimal explotation of renewable Problem of irreversible investment, Econometric, Vol.47, No.1, 1979. 107

resource stocks:


2. Imanov M.H., Regularity analysis for nonlinear terminal optimal control problems subject to state constraints, International Journal of Applied Mathematics and Statistics, Vol.30, No.6, 2012, pp.80-92.

OPTIMAL CONTROL PROBLEM WITH CONTROLS IN COEFFICIENTS OF QUASILINEAR ELLIPTIC EQUATION A.D. Iskenderov2, R.K. Tagiyev2 1

Lenkaran State University, Lenkaran, Azerbaijan 2 Baku State University, Baku, Azerbaijan [email protected], [email protected]

Let the domain   En (n  2) is a full-sphere, a spherical stratum, a parallelepiped or can be transformed to one of these domains by means of regular transformation from C 2 () ,  is a continuous Lipschitz boundary of domain  , x  ( x1 ,, xn )   is an arbitrary point. Let the controlling system be described by the following quasilinear elliptic equation in the domain  n



  aij ( x, k ( x))u x j i , j 1



xi

 q( x)a( x, u )  f ( x), x  ,

(1)

with boundary condition

u( x)  0, x  , where

aij ( x, k ) (i, j  1, n) ,

a( x, u) ,

f (x)

(2) are

given

functions,

k ( x)  (k1 ( x),, k r ( x)), q( x) are controlling functions. Let v( x)  (k ( x), q( x)) is a control,

u  u (x , v ) is a solution of the problem (1), (2) and is a state of system corresponding to control v(x) . Let us introduce the set of admissible controls Vad  K  Q , where



K  k ( x)  (k1 ( x),..., kr ( x))  (W1 ())r : 0   i  ki ( x)  i , kix j ( x)  di( j ) (i  1, r; j  1, n), a. e. on } ,

Q  q( x)  L () : 0  q0  q( x)  q1 a. e. on ,

(3)

where i   i  0, di( j )  0 (i  1, r , j  1, n), q1  q0  0 – are given numbers, a.e. denotes a property ''almost everywhere''. Let us formulate the following optimal control problem: among of all admissible controls v( x)  (k ( x), q( x))  Vad , satisfying constraints

108


J l (v)   [ Fl ( x, u( x, ), ux ( x, ), k ( x))  q( x)Gl ( x, u( x, ), ux ( x, ))]dx  0, (l  1, l0 ),

(4)



finding control v ( x)  (k ( x), q ( x))  Vad , minimizing functional J 0 (v)   [ F0 ( x, u( x, ), u x ( x, ), k ( x))  q( x)G0 ( x, u( x, ), u x ( x, ))]dx

(5)



(l  0, l0 ) are given functions, p  ( p1 ,..., pn ), k  (k1 ,..., kr ) .

where F l( x, u, p, k ), Gl ( x, u, p))

We call this problem as problem (1)-( 5). Let us suppose, that following conditions are satisfied 1) f ( x)  L2 () ; the functions aij ( x, k ) and their partial derivatives with respect to

xm m  1, n are measurable with respect to x   and continuous with respect to k  K 0 , where

K0  {k  (k1,..., kr )  Er : i  ki  i (i  1, r )}; 2) at almost all x   and for all   (1,...,n )  En , k ( x)  K , the ellipticity inequality n

  i2  i 1

n

n

i , j 1

i 1

 aij ( x, k ( x))i j    i2 ,

is valid, ,  const  0 and at all k ( x)  K the inequalities take place aijxm ( x, k ( x))

n 1, 

  (i, j, m  1, n);

3) the function a( x, u ) are measurable with respect to x   and continuous with respect to u  R , at almost all x   and for all u1, u2  R the relations

a( x,0)  0, 0  [a( x, u1 )  a( x, u2 ](u1  u2 )  L(u1  u2 ) 2 , L  const  0 are valid. 4) functions F l ( x, u, p, k ), Gl ( x, u, p) are measurable with respect to x   and continuous with respect to u  R , p  En , k  K0 ; at n  2 and n  3 for any h  0 there exist such functions  ( h) ( x),  ( h) ( x)  L1 (), and constants M 3 , M 4  0 , that at almost all x   , and for all u  [h, h] , p  En , k  K 0 the inequalities take place r

r

F l( x, u, p, k )   ( h ) ( x)  M 3 p 2 , G l ( x, u, p)   ( h ) ( x)  M 4 p 2 , (l  0, l0 ), for n  4 there exist such functions  ( x),  ( x)  L1 () and constants M 5 , M 6  0, that at almost all x   and for all u  R , p  En , k  K 0 the inequalities take place

F l ( x, u, p, k )   ( x)  M 5 ( u

r1

r

 p 2 ), G l ( x, u, p)   ( x)  M 6 ( u 109

r1

r

 p 2 ), (l  0, l0 ),


where r1* , r2* are some numbers that satisfy the following conditions

r1 [2, ) at n  4, r1 [2,2n /( n  4)) for n  5, r2* [2, ) at n  2 r2 [2,2n /(n  2)) for n  3, 5) set of admissible controls, satisfying constraints (4) is not empty, i.e.

W  { ( x)  Vad : J l ( )  0 (l  1, l0 )}   . Let us introduce the space B  (Ws11 ()) r  Ls2 (), where s1  n for n  2,

s2  2 at

n  2 and n  3, s2  n / 2 for n  4 . Theorem 1. Let conditions 1)-5) are satisfied. Then there is at least one optimal control

  ( x)  (k ( x), q ( x))  Vad of the problem (1)-(5), i.e.

J 0  inf{J 0 ( ) :   ( x) Vad }  ,

V  {  ( x) Vad : J 0 (  )  J 0 }  . Set of optimal controls V in the problem (1)-(5) is weakly compact on B and arbitrary minimizing sequence { ( m) ( x)}  {(k ( m) ( x), q ( m) ( x))}  Vad of the functional J 0 ( ) converges weakly to the set V in B . Let following conditions are satisfied: 6) the functions aij ( x, k ) (i, j  1, n) , a( x, u) , Fl ( x, u, p, k ) , Gl ( x, u, p) (l  0, l0 ) have aijkm ( x, k ) (i, j  1, n; m  1, r ) ,

partial derivatives

au ( x, u ) ,

Flu ( x, u, p, k ) ,

Flpi ( x, u, p, k ) ,

Flkm ( x, u, p, k ) , Glu ( x, u, p) , Glpi ( x, u, p) (l  0, l0 ; i, j  1, n; m  1, r ) that are measurable with

respect to x   , and are continuous with respect to x  , u  R, p  En , k  K0 ; ak ( x, u)  0

x   and for all u   ; the operators generated by functions aijkm

at almost all

(i, j  1, n; m  1, r ) , au , Flu , Flpi , Flkm , Glu , Glpi (l  0, l0 ; i, j  1, n; m  1, r ) continuously



   L  L  W  ,  , L   L  , L   L  in L  , L  , L  ,

W  , 1 

operate from



Lr1  Lr2  W1 n

r

Lr1  ,



Lr1   Lr2   W1  , r

n

n

r1

r1

r

1 

r2

n

r2

r1



r2

2

s

L2  , L1  , L2  , L2  accordingly, where s  2 , at n  2 and n  3 , s 

n for n  4 . 2

For the problem (1)-(5) we introduce conjugate state  l   l x, v  as a solution to the following problem 

 a x, k x   n

i , j 1

ij

lxi

xj

n



 qx au x, u  l  Flu  qx Glu    Flpi  qx Glpi i 1





 l x, v   0 , x   l  0, l0 , 110



xi

x  ,

(6) ( 7)


where u  ux, v  is a solution to the problem (1),(2). Theorem 2. Let conditions 1)-6) are satisfied, v x   k x , q x   Vad is optimal control in the problem (1)-( 5), u x   ux, v  ,  l  x    l x, v  l  0, l0  , are solutions to the problems (1), (2) and (6), (7), corresponding to the control v x  . Then there are numbers l  0

l  0, l0 

simultaneously unequal to zero such that at almost all x   and for any

k x   K , q  q0 , q1  the following inequality is satisfied 

        a x, k x u l0

l 0

 l

r

n

 m 1 i , j 1

ijk m



 l  x  Flk

x j

i



m

x, u x , u x x , k x km x   km x dx  

  x, u ( x) l ( x)  Gl x, u ( x), ux ( x)q  q ( x)  0

ON A LINEAR PROBLEM OF CONTROL OF ROSSER-TYPE HYBRID SYSTEM A.Ya. Jabbarova1 1


A linear problem of optimal control of Roesser type-hybrid system with a linear quality test is considered. Necessary and sufficient optimality condition in the form of the Pontryagin’s maximum is proved. The case of complex functional is considered separately. Assume that the controlled process is described by the system of linear equation of the form z t , x   At , x  z t , x   Bt , x  yt , x   f t , x, ut , x , t  T  t0 ,t1  , x  x0 , x0  1, ...x1 , t

(1)

yt , x  1  Ct , x zt , x  Dt , x yt , x  g t , x, ut , x, t  T , x  X  x0 , x0  1..., x1  1 with boundary condition z t0 , x   ax  ,

yt , x0   bt ,

Here

x  X  x1 ,

(2)

t T.

At , x  , Bt , x  , C t , x  , Dt , x  are the given matrix-functions of appropriate

dimensions continuous with respect to t for all x , zt , x , yt , x  is the

n  m -dimensional

state vector, ax  - is the given discrete n -dimensional vector-function, bt  is the

given

continuous m - dimensional continuous vector-function, t 0 , t1 , x0 , x1 are given and difference , 111


x1  x0 is a natural number, f t , x, u 

g t, x, v

function continuous with respect to t, u 

t, v 

is the given n

m -

dimensional vector-

x , u  ut , x  is the r -dimensional

for all

vector-function of control actions being piecewise continuous with respect to t (with finite number of first kind discontinuity points) for all x with the values from the non–empty bounded set U  R r , i.e.

ut , x  U  R r ,

t T ,

x  X  x1 .

(3)

Such control function are called admissible controls. On the solutions of system (1)-(2) generated by all possible admissible controls, define the functional S u  

x1 1

t1

 cx zt1 , x   d t yt, x1  dt .

x  x0

(4)

t0

Here cx  и d t  are the given n and m -dimensional discrete and continuous vectorfunction. Consider a problem on minimum of functional (4) at restraints (1)-(3). We call the admissible control delivering minimum to functional (4) at restraints (1)-(3) an optimal control, the appropriate process ut , x , zt , x , yt , x  – an optimal process. In the paper necessary and sufficient optimality condition in the form of the discrete principle of maximum is proved.

Keywords: linear control problem, Roesser-type systems, necessary and sufficient condition, Pontryagin’s maximum principle. AMS Subject Classification: 49K10, 49K20.

OPEN OPTIMAL PROBLEMS IN VOTING THEORY M.M. Konstantinov1, P.H. Petkov2 1

UACEG, Sofia, Bulgaria TU-Sofia, Sofia, Bulgaria e-mail: [email protected], [email protected] 2

Voting theory is not very inspiring from theoretical point of view but may have (and really has) great political impact [1]. An electoral system in broad sense is the set of law regulations that govern carrying out of elections and referenda. In a narrow sense the electoral 112


system is the mathematical algorithm for transforming votes cast into mandates, or seats, in political, academic and other type of elections. Electoral systems for proportional representation (ESPR) as well as mixed systems are more interesting from mathematical point of view. Among plain ESPR the D’Hondt (or Jefferson) family of algorithms as well as the Hare-Niemeyer (or Hamilton) family of methods are most used. In turn, ESPR algorithms may often be formulated as optimal control problems. In contrast, such formulation of bi-proportional seat allocation algorithms may not be straightforward. Bulgaria has a long history of very well documented parliamentary elections in the period 1991-2014. The ESPR used therein is bi-proportional and has varied during the above mentioned period. In the present paper we describe a set of open optimal problems arising in the implementation of bi-proportional seat allocation algorithms. The results are illustrated by examples from the new Bulgarian political history. Keywords: elections, voting theory, proportional systems, optimization. AMS Subject Classification: 91B12, 91B14.

References 1. Balinski, M., Young, H., Fair Representation: Meeting the Ideal of One Man, One Vote., Brookings Institute Press, Hanover, PA 2011, ISBN 978-0-8157-0090-6.

CONJUGATE OPERATOR METHOD AND ITS APPLICATION Andrey L. Karchevsky1 1

Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia e-mail: [email protected]

Often in practice, inverse problems are solved using the optimization method. For this well-known and standard approach the gradient of the residual functional is searched with the help of solving conjugate problem. The computational complexity of this approach: the computation of the gradient requires solutions of the direct and conjugate problems, and the conjugate problem can be solved after solving the direct problem only. Thus, certain amount of time must be spent on an iteration of the minimization process.

113


Application of the conjugate operator method will be demonstrated on the example of the following problems: 1. coefficient hyperbolic inverse problem, 2. retrospective inverse problem of heat conduction, 3. the Cauchy problem for the Laplace equation which can be reduced to the inverse problem on determination of unknown boundary conditions, 4. the inverse problem of determining the right-hand part of the elliptic equation, 5. numerical solution of the Fredholm integral equation of the first (second) kind, 6. solution of a system of linear algebraic equations. It will be shown that for first and second inverse problems the calculation time can be reduced about two times. This will be achieved due to the fact that the direct and conjugate problems can be solved in parallel. The third and fourth inverse problem can be reduced to the problem type the moment problem. If the optimization method applied to solving these problems, it is not necessary solving the direct and conjugate problems. Application of the conjugate operator method to the fifth and sixth problems allows us to get cost-effective technology to solve these problems, if they must be solved repeatedly for different right-hand parts. This technology is that solving the original problem is divided into two steps: 1) solving the subsidiary problems, knowledge of the right parts is not required, 2) finding solution for the specific right-hand part reduces to the calculation of several scalar products. The first four problems are presented in the work [1]. The application of the conjugate operator method for solving applied problems on real data can be found in the works [2,3]. The work was supported by RFBR grant 14-01-00208. Keywords: conjugate operator method, inverse problem, Fredholm integral equation, system of linear algebraic equations AMS Subject Classification: 65K10, 35R30, 45B05, 65F05.

References 1. Karchevsky A.L., Reformulation of an inverse problem statement that reduces computational costs, Eurasian Journal of Mathematical and Computer Applications, Vol.1, No.2, 2013, pp.5-20. http://ejmca.enu.kz/images/stories/arkhiv/karchevsky.pdf 2. Karchevsky A.L., Marchuk I.V., Kabov O.A., Calculation of the heat flux near the liquid-gassolid contact line, Applied Mathematical Modelling, (in press). 114


3. Marchuk I.V., Karchevsky A.L., Surtaev A.S., Ajaev V.S., Kabov O.A., Heat flux at the surface of metal foil heater under evaporating sessile droplets, Langmuir, (in press).

ON WEAK SUBGRADIENTS IN NONCONVEX OPTIMIZATION AND OPTIMALITY CONDITIONS Refail Kasimbeyli1 1

Department of Industrial Engineering, Anadolu University, Eskisehir, Turkey e-mail: [email protected]

Consider the problem of minimizing function 𝑓: 𝑆 → 𝑅 over the set 𝑆 ⊆ 𝑅 𝑛 . The following is a well-known optimality condition in nonsmooth convex analysis which states that [1, Proposition 1.8.1, page 168] if 𝑓: 𝑅 𝑛 → 𝑅 is a convex function then vector 𝑥 minimizes 𝑓 over a convex set 𝑆 ⊆ 𝑅 𝑛 if and only if there exists a subgradient 𝑥 ∗ ∈ 𝜕𝑓(𝑥) such that 𝑥 ∗ (𝑥 − 𝑥) ≥ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑆,

(1)

𝜕𝑓(𝑥) = {𝑥 ∗ ∈ 𝑅 𝑛 ∶ 𝑓(𝑥) − 𝑓(𝑥) ≥ 𝑥 ∗ (𝑥 − 𝑥) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑅 𝑛 }

(2)

where is a subdifferential of 𝑓at 𝑥.

Although the above condition is valid for both convex and

nonconvex functions, for a variety of reasons, if the function 𝑓 is not convex, the subdifferential 𝜕𝑓(∙) is not a particularly helpful tool. This makes it very tempting to look for definitions of generalized derivatives and subdifferentials for a nonconvex function. The concepts of generalized differentiability appropriate for applications to optimization were defined in convex analysis: first geometrically as the normal cone to a convex set that goes back to Minkowski [2], and then – much later – analytically as the subdifferential of an extended real-valued convex function. The latter notion, inspired by the work of Fenchel [3], was explicitly introduced by Moreau [4] and Rockafellar [5]. It is well known that every convex function 𝑓: 𝑋 → 𝑅 on a Banach space admits the classical directional derivative 𝑓(𝑥 + 𝑡ℎ) − 𝑓(𝑥) 𝑓 ′ (𝑥; ℎ) = lim ( ) 𝑡↓0 𝑡 in any direction ℎ ∈ 𝑋 at any point of its efficient domain 𝑑𝑜𝑚(𝑓). This notion was generalized by many researchers, such as Clarke [6], Rockafellar [7,8] and others. By using a general notation 𝑓 𝑔 for the directional derivatives mentioned above, the corresponding subdifferential of 𝑓 at 𝑥 is defined by 𝜕 𝑔 𝑓(𝑥) = {𝑥 ∗ ∈ 𝑅 𝑛 ∶ 𝑓 𝑔 (𝑥; ℎ) ≥ 𝑥 ∗ (ℎ) 𝑓𝑜𝑟 𝑎𝑙𝑙 ℎ ∈ 𝑅 𝑛 } 115


This is a standard way to introduce subgradients via directional derivatives. For convex functions it is equivalent to the classical subdifferential of convex analysis: 𝜕𝑓(𝑥) = {𝑥 ∗ ∈ 𝑅 𝑛 ∶ 𝑓 ′ (𝑥; ℎ) ≥ 𝑥 ∗ (ℎ) 𝑓𝑜𝑟 𝑎𝑙𝑙 ℎ ∈ 𝑅 𝑛 }. One of the main purposes of introducing all these generalizations was to obtain optimality conditions for nonconvex problems. Note that only the necessary part of the optimality condition given in (1) could be obtained for nonconvex case by using different subdifferential and normal cone generalizations. Since these generalizations do not satisfy the main property of the classical subgradient given in (2) for nonconvex functions, one cannot expect to obtain the optimality condition similar to that given in (1) in nonconvex case. For example, let function 𝑓: 𝑅 → 𝑅 be defined as 𝑓(𝑥) = |𝑥|. Then, Clarke's directional derivative is 𝑓 0 (0; ℎ) = |ℎ| for all ℎ ∈ 𝑅 and 𝜕 𝑜 𝑓(0) = [−1,1]. It is clear that hyperplanes with normal vectors (slopes in this case) from this subdifferential cannot be used to support the epigraph of the function. A similar interpretation is also valid for Mordukhovich's subdifferential, which is defined for this function as 𝜕 𝑀 𝑓(0) = {−1,1}. The relation given in (2) for the classical subdifferential means the existence of the supporting hyperplane to epigraph of 𝑓 at the point under consideration. It is obvious that to use such a hyperplane as a supporting surface for nonconvex functions may not be possible in general. This means that the investigation of nonconvex case can be made available by changing the supporting philosophy and using suitable nonlinear supporting surfaces. By using a cone of concave functions (instead of linear ones used in convex analysis), Gasimov [9] investigated duality relations and obtained optimality conditions for some classes of nonconvex optimization problems in both single-objective optimization and vector optimization. Azimov and Gasimov [10,11] constructed a duality scheme by using a special class of concave functions. They used a class of superlinear conic functions. A graph of such a function is a conical surface which can be used as a supporting surface for a certain class of nonconvex sets. By using the mentioned class of superlinear functions, they introduced the concept of weak subdifferential and derived a collection of optimality conditions and duality relations for a wide class of nonconvex optimization problems. The superlinear conic functions were applied to construct the so-called sharp augmented Lagrangian functions for nonconvex constrained optimization problems and to derive zero duality gap conditions. In this paper, we study optimality conditions for nonconvex problems involving a special class of directionally differentiable functions. By using the weak subgradient notion, we generalize the necessary and 116


sufficient optimality condition given in (1) to a nonconvex case. We show that the point 𝑥 minimizes 𝑓 over a set 𝑆 ⊆ 𝑅 𝑛 if and only if there exists a weak subgradient (𝑥 ∗ , 𝛼) ∈ 𝜕 𝑤 𝑓(𝑥) such that 𝑥 ∗ (𝑥 − 𝑥) + 𝛼‖𝑥 − 𝑥‖ ≥ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑆. Keywords: directional derivative, weak subgradient, nonconvex optimization, duality, optimality conditions. AMS Subject Classification: 90C26, 90C30, 90C46.

References 1. Bertsekas D.P., Nedic A., Ozdaglar A.E., Convexity, Duality and Lagrange Multipliers, Lecture Notes, MIT, 2001. 2. Minkowski H., Theorie der Konvexen Körper, In: Gesammelte Abhandlungen, II, B.G. Teubner (Eds.), Insbesondere Begründung inhres Ober Flächenbegriffs, Leipzig , 1911. 3. Fenchel W., Convex Cones, Sets and Functions, in: Lecture Notes, Princeton University, Princeton, New Jersey, 1951. 4. Moreau J.-J., Fonctionelles sous-differentiables, C. R. Acad. Sci. 257, 1963, pp.4117–4119. 5. Rockafellar R.T., Convex Functions and Dual Extremum Problems, Ph.D. dissertation, Department of Mathematics, Harvard University, Cambridge, Massachusetts, 1963. 6. Clarke F.H., Generalized gradients and applications, Trans. Amer. Math. Soc. 205, 1975, pp.247– 262. 7. Rockafellar R.T., Directional Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. 39, 1979, pp.331–355. 8. Rockafellar R.T., The Theory of Subgradients and Its Applications to Problems of Optimization: Convex and Nonconvex Functions, Helderman Verlag, Berlin, 1981. 9. Gasimov R.N., Duality in nonconvex optimization, Ph.D. Dissertation, Department of Operations Research and Mathematical Modeling, Baku State University, Baku, 1992. 10. Azimov A.Y., Gasimov R.N., On weak conjugacy, weak subdifferentials and duality with zero gap in nonconvex optimization, Int. J. Appl. Math. 1, 1999, pp.171–192. 11. Azimov A.Y., Gasimov R.N., Stability and duality of nonconvex problems via augmented Lagrangian, Cybernet. Systems Anal. 3, 2002, pp.120–130.

117


CONSTRUCTION OF CONTROL FOR ONE-DIMENSIONAL HEAT EQUATION WITH A DELAY D.Y.Khusainov1, E.I.Azizbayov2, I.A.Dzhalladova3 1

Taras Shevchenko National University of Kyiv, Ukraine 2 Baku State University, Baku, Azerbaijan 3 Kyiv National Economic University, Ukraine e-mail: [email protected]

In the present work we consider a problem of controllability for one-dimensional heat equation with a delay. An explicit representation of the control function taking the system to the given final state is obtained. It is shown that subject to certain conditions imposed on the rate of decrease of the appropriate Fourier coefficients, the control function and the solution are regular in the classical sense, both with respect to time and spatial variables. We consider a non-homogeneous one-dimensional heat equation with delay ut ( x, t )  a12u xx ( x, t )  a22u xx ( x, t   )  c1u( x, t )  c2u( x, t   )  f ( x, t ) ,

(1)

with initial and boundary conditions u( x, t )   ( x, t ) ,  ( x, t )  e x ( x, t ) , 0  x  l ,    t  0 ,

(2)

u( x,0)  1 (t ) , 1 (t )  1 (t ) , u( x, l )  2 (t ) , 2 (t )  e l1 (t ) ,    t  0 ,

(3)

and

respectively. Definition. We call afunction of the form 0 ,    t   ,   1 ,    t  0,  t  1 b , 0  t ,  1!  t (t   ) 2 exp  {b, t}   1  b  b2 ,   t  2 , 1! 2!  ... , ...  k t  k [t  ( k  1) ] , (k  1)  t  k 1  b 1!    b k!  ............................................................................

(4)

a delayed exponential. Using the written function, we can represent the classical solution of the initial-boundary value problem (1) - (3) as

118


x u ( x, t )  S1[ , 1 , 2 ]( x, t )  S2[ f , 1 , 2 ]( x, t )  1 (t )  [ 2 (t )  1 (t )] , l

(5)

with linear operators

S1[ ( x, t ), 1 (t ), 2 (t )]  0    n   e Ln (t  ) exp  {Dn , t} n ( )   e Ln (t s ) exp  {Dn , t    s}[n ( s)  Ln  n ( s)]ds  sin x , l n1   

 t  n S2 [ f ( x, t ), 1 (t ), 2 (t )]    e Ln (t  s ) exp  {Dn , t    s}Fn ( s)ds  sin x , l n 1  0 

(6)

where

2 n Fn (t )   f ( , t ) sin d  M n [ 1 (t ), 2 (t )] , l 0 l l

(7)

2  d     n M n [ 1 (t ), 2 (t )]    1 (t )  [ 2 (t )  1 (t )]  sin d  l 0  dt  l l  l

 c1mn [1 (t ), 2 (t )]  c2mn [1 (t   ), 2 (t   )] ,

mn [ 1 (t ), 2 (t )] 

2    n 1 (t )  [ 2 (t )  1 (t )] sin d ,   l 0 l l  l

(8)

2 n  n (t )    ( , t ) sin d  mn [ 1 (t ), 2 (t )] , l 0 l l



(9)

2  n   a1   l  

2  c    n    1   n  Ln  c1   a1  , Dn  c2   a2  e   l   l    2

.

(10)

Theorem 1. Let T  0 , m : T  and   0 , the functions F , Ft  C 0 ([0, l ]  [0,T ]) , ,  t ,  tt   C 0 ([0, l ]  [0,   ]) be such that their Fourier coefficients Fn (t ) and  n (t ) satisfy

the conditions





lim n2 m 1 max n (s)  n2 n (s)  n4  n (s)  0 ,

n 

s[  , 0 ]

lim max n2( m  k ) 1

n  1 k  n

max

( k 1)  s  k

 F (s)  n n

2



Fn (s)  0 .

(11) (12)

Then the problem (1) - (3) has a unique classical solution u  C 0 ([0, l ]  [ ,T ]) , satisfying the conditions t u,  xxu  C 0 ([0, l ]  [0,T ]) . Furthermore, the functions u ,  t u and  xxu are defined uniformly and absolutely convergent Fourier series. 119


Remark. Conditions of the theorem may be interpreted as the requirement that the functions  , 1 ,  2 , f

to the certain Sobolev space with a sufficiently large amount of

generalized derivatives. Further a control problem is considered. It is required to find a control function U ( x, t )  f ( x, t ) , at which the solution u ( x, t ) at time T  0 reaches a predetermined value, i.e.

u( x,T )  ( x) , 0  x  l .

(12)

The control function U ( x, t ) is sought in the form of expansion 

n

n 1

l

U ( x, t )  U n (t ) sin

x.

Theorem 2.Let the functions  ( x, t ), 1 (t ), 2 (t ),  ( x) be such that the conditions of theorem 1 are fulfilled. Then the coefficients of the control functions (13) have the form U n (t )  e  Ln (T t )

Rn (T ) Dn , exp

Rn (T )  n  s1n (T )  s2n (T )  mn [1 (T ), 2 (T )] ,

2 n (v) sin sds ,  l 0 l l

n 

2 l  n s1n (t )  e exp  {Dn , t}   ( ,0) sin d   l l 0  Lnt

0 2 l  n  Dn  e Ln (t s ) exp  {Dn , t  2  s}   ( , s) sin d  ds , l  l 0 

t

s2 n (t )   e Ln (t  s ) exp  {Dn , t    s}M n [ 1 ( s), 2 ( s)]ds 0

is the solution of the control problem. Keywords: heat equation, classical solution, delayed exponential, constant delay. AMS Subject Classification: 35K05, 49J20.

120

(13)


ON A STRENGTHENING OF THE DISCRETE MAXIMUM PRINCIPLE M.J. Mardanov1, T.K. Melikov1,2 1

Institute of Mathematics and Mechanics, ANAS, Baku, Azerbaijan 2 Institute of Control Systems of ANAS, Baku, Azerbaijan e-mail: [email protected], [email protected]

It is known that immediate transfer of the Pontryagin’s maximum principle to the discrete system is generally not possible [1]. So, naturally both theoretical and practical interests arise to the finding of the new and wide classes of problems for which stronger optimality conditions in the form of the maximum principle hold true. This work is devoted to the investigation of the optimality of admissible controls in such formulation. 1. Let it needs to minimize the functional 𝒮(𝑢) = Φ(𝑥(𝑡1 ))

(1)

on the trajectories of the discrete system 𝑥(𝑡 + 1) = 𝑓(𝑥(𝑡), 𝑢(𝑡), 𝑡 ), 𝑡 ∈ 𝑇 , 𝑥(𝑡0 ) = 𝑥 ∗ ,

(2)

with constraints 𝑢(𝑡) ∈ 𝑈(𝑡) ⊂ 𝐸 𝑟 , 𝑡 ∈ 𝑇,

(3)

where 𝑥 = (𝑥1 , … , 𝑥𝑛 )′ is a state vector; ′- stands for the operation of transpose; 𝑢 = (𝑢1 , … , 𝑢𝑟 )′ in a control vector; 𝑡-discrete time; 𝑥 ∗ is given vector; 𝑇 = {𝑡0 , 𝑡0 + 1, … , 𝑡1 − 1}; U (t ), t  T is a set from 𝑟-dimensional Euqulidian space 𝐸 𝑟 ; ( x), x  E n , f ( x, u, t ), x, u, t   E n  E r  t0 , t1  are contionously differentiable functions.

The control u(t ), t  T satisfying to the condition (3) we call an admissible control. Admissible control 𝑢(𝑡), 𝑡 ∈ T and corresponding solution x(t ), t  T  t1 of the system (2) that gives minimum to the functional (1) we call optimal. Then each pair (𝑢(𝑡), 𝑥(𝑡))is called an optimal process. 2. In the work using the method given in [2] the necessary optimality conditions are derived and the following theorem has been proved. Theorem. Let (𝑢0 (𝑡), 𝑥 0 (𝑡)) be some process and the sets 𝑓(𝑥 0 (𝑡), 𝑈(𝑡), 𝑡) = {𝑧: 𝑧 = 𝑓(𝑥 0 (𝑡), 𝑢, 𝑡), 𝑢 ∈ 𝑈(𝑡)} are convex for all 𝑡 ∈ T\{𝑡1 − 1}. Then for the optimality of this process the fulfillment of the following inequalities is necessary: ∆𝑢̃ Φ(𝑓(𝑥 0 (𝑡1 − 1), 𝑢0 (𝑡1 − 1), 𝑡1 − 1)) ≥ 0, ∀𝑢̃ ∈ 𝑈(𝑡1 − 1);

121

(4)


𝜓 𝑇 (𝑡1 − 1; 𝑢̂)𝑓𝑥 (𝑥 0 (𝑡1 − 1), 𝑢̂, 𝑡1 − 1)∆𝑣 𝑓(𝑥 0 (𝑡1 − 2), 𝑢0 (𝑡1 − 2), 𝑡1 − 2) ≤ 0, ∀𝑢̂ ∈ 𝑈 0 (𝑡1 − 1), ∀𝑣 ∈ 𝑈(𝑡1 − 2);

(5)

𝜓 𝑇 (𝜃1 ; 𝑢0 (𝜃), 𝑢̂)𝑓𝑥 (𝑥 0 (𝜃1 ), 𝑢̅, 𝜃1 )∆𝑣 𝑓(𝑥 0 (𝜃), 𝑢0 (𝜃), 𝜃) ≤ 0,

(6)

∀𝑢̂ ∈ 𝑈 0 (𝑡1 − 1), ∀𝑢̅ ∈ 𝑈[𝑥 0 (∙)](𝜃1 ), ∀𝑣 ∈ 𝑈(𝜃), ∀𝜃 ∈ 𝑇 ∗ = {𝑡0 , … , 𝑡1 − 3}, 𝑇 ∗ ≠ ∅, where 𝜓(∙) is a solution of the system [3] 𝜓(𝑡1 − 1; 𝑢0 (𝑡), 𝑢̂) = 𝜓𝑇 (𝑡; 𝑢0 (𝑡 + 1), 𝑢̂)𝑓𝑥 (𝑥 0 (𝑡), 𝑢0 (𝑡), 𝑡), 𝑡 ∈ {𝑡0 + 1, … , 𝑡1 − 2}, 𝜓(𝑡1 − 2; 𝑢0 (𝑡1 − 1), 𝑢̂) = 𝜓𝑇 (𝑡1 − 1; 𝑢̂)𝑓𝑥 (𝑥 0 (𝑡1 − 1), 𝑢̂, 𝑡1 − 1), 𝜓(𝑡1 − 1; 𝑢̂) = −Φ𝑥 (𝑓(𝑥 0 (𝑡1 − 1), 𝑢̂, 𝑡1 − 1)), and the set 𝑈[𝑥 0 (∙)](𝑡), 𝑡 ∈ 𝑇, 𝑈 0 (𝑡1 − 1), following to [2, 3] is defined as 𝑈 0 (𝑡1 − 1) = {𝑢̂: ∆𝑢̂ Φ(𝑓(𝑥 0 (𝑡1 − 1), 𝑢0 (𝑡1 − 1), 𝑡1 − 1)) = 0, 𝑢̂ ∈ 𝑈(𝑡1 − 1)}, 𝑈[𝑥 0 (∙)](𝑡) = {𝑢̅: ∆𝑢̅ 𝑓(𝑥 0 (𝑡), 𝑢0 (𝑡), 𝑡) = 0, 𝑢̅ ∈ 𝑈(𝑡)}, 𝑡 ∈ 𝑇. It should be noted that the theorem in differ from the known ones [4-7 and etc.] is proved without the assumption on the convexity at the end point of the set 𝑇 and in the terms of the set 𝑈[𝑥 0 (∙)](𝑡), 𝑡 ∈ {𝑡0 , … , 𝑡1 − 2}, since may be considered as an improvement of those ones. The optimality condition (4), was firstly obtained in [2], and (5), (6) are principally new. Keywords: optimal control, discrete system, maximum principle, necessary optimality condition. AMS Subject Classification: 35J05, 35J25.

References 1. Butkovskiy A.G. On necessary and sufficient conditions of optimality for the impulse control systems, Automation and Remote Control, Vol. 24, No.8, 1963. 2. Misir J.Mardanov, Telman K. Melikov. A method for studying the optimality of controls in discrete systems, Proceedings of the Institute of Mathematics and Mechanics, ANAS, Vol.XXXX, 2014, pp.5-13. 3. Misir J. Mardanov, Telman K. Melikov. Strengthened optimality condition of the first type in discrete systems of control, Transactions of the Institute of Mathematics and Mechanics, ANAS, Vol.XXXIV, No. 4, 2014, pp.5-13. 4. Halkin H. A maximum principle of the Pontryagin’s type for systems described by nonlinear difference equations, SIAM J. Control, 4, 1966, pp.90-111. 5. Holtzman j. M. Convexity and the maxsimum principle for discrete systems, IEEE Trans. Automat. Control, Vol.11, No.1, 1966, pp.30-35.

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6. Propoy A.I. On a maximum principle for the discrete control systems, дискретных систем управления, Automation and Remote Control, Vol.26, No.7, 1965. 7. Gabasov R. On a theory of optimal processes in the discrete systems, J. Cal. Math., Math., Ph., Vol.4, No.8, 1968.

4D OPTIMAL CONTROL PROBLEM FOR A VOLTERRA-HYPERBOLIC INTEGRO-DIFFERENTIAL EQUATION OF BIANCHI TYPE WITH NONCLASSICAL GOURSAT CONDITIONS I.G. Mamedov1 1

Institute of Control Systems of NAS of Azerbaijan, Baku, Azerbaijan e-mail: [email protected]

Let the controlled object be described by the 4D (four-dimensional) volterra-hyperbolic integro-differential equation of Bianchi type: (V1,1,1,1u )(x, y, z, t )  u xyzt ( x, y, z, t ) 

A

i , j , k ,l i  j  k l  4 i , j , k ,l 0,1

( x, y, z, t ) D xi D yj D zk Dtl u ( x, y, z, t ) 

x y z t



  { T x0 y0 z0 t0

 ,  , ,  ; x, y, z, t ) Dxi Dyj Dzk Dtl u ( ,  , ,  )}dddd 

i , j , k ,l ( i  j  k l  4 i , j , k ,l 0,1

  ( x, y, z, t , v( x, y, z, t )),

( x, y, z, t )  G  ( x0 , x1 )  ( y0 , y1 )  ( z0 , z1 )  (t0 , t1 ) ,

(1)

under the following non-classical Goursat conditions [10]: V0,0,0,0 u  u ( x0 , y 0 , z 0 , t 0 )   0,0,0,0  (V1,0,0,0 u )( x)  u x ( x, y 0 , z 0 , t 0 )  1,0,0,0 ( x)  (V0,1,0,0 u )( y )  u y ( x 0 , y, z 0 , t 0 )   0,1,0 ( y ) (V u )( z )  u z ( x0 , y 0 , z , t 0 )   0,0,1,0 ( z )  0,0,1,0 (V0,0,0,1u )(t )  u t ( x0 , y 0 , z 0 , t )   0,0,0,1 (t )  (V1,1,0,0 u )( x, y )  u xy ( x, y, z 0 , t 0 )  1,1,0,0 ( x, y ) (V u )( x, z )  u xz ( x, y 0 , z , t 0 )  1,0,1.0 ( x, z )  1,0,1,0  (V1,0,0,1u )( x, t )  u xt ( x, y 0 , z 0 , t )  1,0,0,1 ( x, t )  (V0,1,1,0 u )( y, z )  u yz ( x 0 , y, z , t 0 )   0,1,1,0 ( y, z ) (V u )( y, t )  u yt ( x0 , y, z 0 , t )   0,1.0,1 ( y, t )  0,1,0,1 (V0,0,1,1u )( z , t )  u zt ( x0 , y 0 , z , t )   0,0,1,1 ( z , t )  (V1,1,1,0 u )( x, y, z )  u xyz ( x, y, z , t 0 )  1,1,1,0 ( x, y, z ) (V u )( x, y, t )  u xyt ( x, y, z 0 , t )  1,1,0,1 ( x, y, t )  1.1,0,1 (V0,1,1,1u )( y, z , t )  u yzt ( x0 , y, z , t )   0,1,1,1 ( y, z , t )  (V1,0,1,1u )( x, z , t )  u xzt ( x, y 0 , z , t )  1,0,1,1 ( x, z , t )

123

(2)


where D   /  is a generalized differentiation operator in the Sobolev sense; u( x, y, z, t ) is the desired function;

Ai , j ,k ,l ( x, y, z, t )

are the given

measurable functions on

G;

Ti , j ,k ,l ( ,  , , ; x, y, z, t )  L (G  G);  ( x, y, z, t , v( x, y, z, t )) are the given functions on G  R r

satisfying the Caratheodory conditions;

v( x, y, z, t )  (v1 ( x, y, z, t ),, vr ( x, y, z, t ))

is r-

dimensional controlling vector-function; i , j ,k ,l are the given elements. Let the function v( x, y, z, t ) be measurable and bounded on G and almost at all points of ( x, y, z, t )  G accept its values from some given set   R r . Then this vector-function is said to

be an admissible control. A set of all admissible controls is denoted by   . In the paper we consider the following linear non-local problem of optimal control: find admissible control v( x, y, z, t ) from   , for which the solution

u( x, y, z, t ) Wp(1,1,1,1) (G)  u( x, y, z, t ) : Dxi Dyj Dzk Dtl u( x, y, z, t )  Lp (G), i, j, k , l  0,1, (1  p  ) of 4D Goursat boundary-problem (1)-(2) delivers the least value to the 4D linear

multi-point functional: N

S (v)   ak u ( xk(1) , yk(1) , zk(1) , tk(1) )  min ,

(3)

k 1

where ( xk(1) , yk(1) , zk(1) , tk(1) )  G are the given points; ak  R are the given numbers. The necessary and sufficient conditions for optimal process for ordinary differential equations and partial differential equations under local conditions sufficiently thoroughly studied by many mathematicians. The results obtained in this area studied in detail in monographs such as [1]-[6] and others. Also highlighted the work of [7] et al., which studied various classes of problems of optimum control. The 4D optimal control problem (1)-(3) was studied by means of a new variant of the increment method. The method essentially uses the notion of an integral form conjugation equation and allows also to cover the case when the coefficients of the equation (1) are, generally speaking, non-smooth functions. In other words, this variants is more natural than classic increment variants developed for example by [8]. Note that the optimal control problem (1) - (3) is investigated using the following 4D integral representation in space Wp(1,1,1,1) (G) [9]:

124


x

y

x0

y0

u x, y, z, t   u x0 , y0 , z0 , t0    u x ( , y0 , z0 , t0 )d   u y ( x0 ,  , z0 , t0 )d  z

t

x y

z0

t0

x0 y0

  u z ( x0 , y 0 ,  , t 0 )d   ut ( x0 , y 0 , z 0 , )d    u xy ( ,  , z 0 , t 0 )dd  y z

x z

x t

x0 z 0

x0 t 0

y t

z t

x y z

y0 t 0

z0 t0

x0 y 0 z 0

   u xz ( , y0 ,  , t0 )dd    u xt ( , y0 ,z0 , )dd    u yz ( x0 ,  ,  , t0 )dd  y0 z 0

   u yt ( x0 ,  , z0 , )dd    u zt ( x0 , y0 ,  , )dd     u xyz ( ,  ,  , t0 )ddd  x y t

y z t

    u xyt ( ,  , z0 , )ddd 

  u

x0 y0 t0

y0 z 0 t 0

x z t

x y z t

x0 z0 t0

x0 y0 z0 t0

yzt

( x0 ,  ,  , )ddd 

    u xzt ( , y0 ,  , )ddd      u xyzt ( ,  ,  , )dddd .

Keywords: optimal control problem, non-local problem, integro-partial differential equations, integral representations. AMS Subject Classification: 35L25, 35R09, 31B10.

References 1. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F., Mathematical Theory of Optimal Processes, M.: Nauka,1969, 384 p. (in Russian). 2. Krasovsky N.N., Theory of Motion Control, M.: Nauka,1968,476 p. (in Russian). 3. Dubovitsky A.Y., Milyutin A.A., Necessary Conditions for a Weak Extremum in the General Optimal Control Problem, M .: Nauka,1971,113p (in Russian). 4. Vasilyev F.P., Methods for solving extreme problems, M .:Nauka,1981,400p (in Russian). 5. Lions J.-L., Optimal Control of Systems Described by the Partial Differential Equation, M.: Mir,1972, 416 p. (in Russian). 6. Gabasov R., Kirillova F.M., Maximum Principle in the Theory of Optimal Control, Minsk: Science and Technology, 1974, 271 p. (in Russian). 7. Akhmedov K.T., Akhiev S.S., Necessary optimality conditions for some problems of optimal control theory, Dokl. Azerb. AN SSR, 1972, Vol.28, issue 5, pp.12-16 (in Russian). 8. Egorov A.I., On optimal control of processes in some systems with distributed parameters, Avtomatika i Telemekhanika,1964, Vol.25, issue 5, pp.613-623 (in Russian).

125


9. Akhiev S.S., About the general view of linear-bounded functionals in one anisotropic space of S.L. Sobolev type, Dokl. Azerb. AN SSR, 1979, Vol.35, issue 6, pp.3-7 (in Russian).

THE OPTIMUM CONTROL OF FLUACTING SYSTEM A.C. Mammadov1, B.M. Yusifov1, I.Z. Aliyev1 1

Sumgayit State University, Sumgayit,Azerbaijan e-mail: [email protected]

The report examines the construction of an approximate solution of the Cauchy problem and optimal control in a Hilbert space. Let flu action of operated system be described but the differential equation with partial derivatives 2  2 y( x, t )  4 y( x, t ) 2  y ( x, t )  a    f ( x)u (t ),   0 t 2 x 2 x 2 t 2

(1)

with initial dates y( x,0)   ( x), yt ( x,0)   ( x) ,

(2)

and boundary value conditions y(0, t )  0, y(, t )  0 ,

(3)

here  ( x), ( x), f ( x) the giving functions in [0, ] , u (t ) operating function from set of admissible controls, to find control u(t ) U  {u(t )  L2 (0,T ) : u(t )  1} , that at the solvability of a problem (1)-(3) delivers the at least possible value to the functional 

J (u )   {[ y( x, t )  Q0 ( x)]2  [ yt ( x, T )  Q1 ( x)]2 }dx .

(4)

0

In work [1] it is shown that under conditions f ( x), ( x)  L2 (0, ) there is a unique solution of a problem (1)-(3), which is represented in form.

ak



y ( x, t )  [ k cos

 2  a(k ) 2

k 1



 2  a(k ) 2 ak

t

 f u( ) sin k

0

t

 2  a(k ) 2 ak  k  sin t ak  2  a(k ) 2

ak  2  a(k ) 2

where

126

 (t   )d ] sin

k x, 


2 k 2 kx 2 k  k    ( x) sin xdx,  k   ( x) sin dx , f k   f ( x) sin xdx, k  1,2,... 0  0  0  





Using analytical representation at the solution of a problem (1)-(3) and orthogonality of system of functions {sin

k x} on [0, ] after some equivalent transformations, functional (4) it 

is possible to present in a form T

TT

0

0 0

J (u )  J  2  (t )u (t )dt   R(t , s)u (t )u ( s)dtds ,

(6)

here  (t ) and R(t , s) is defined by help dates of system (1)-(3). Moreover easy can show that functions  (t ) and R(t , s) is continuously on [0,T ] and [0  t , s  T ] correspondingly. Consequently, R(t , s) is positive kernel in [0  t , s  T ] .

Following theorems are proved. Theorem 1. There is at least one u(t ) U , which at solutions of a problem (1)-(3) delivers the least possible value to the functional J (u ) . Theorem 2. a) If u0 (t ) is the solution of the integral equation T

 (t )   R(t , s)u ( s)ds  0 ,

(7)

0

satisfying to a condition u0 (t ) U almost everywhere on [0,T ] , that this control is optimum. T

b) If u0 (t ) the optimum control, satisfying to a condition  u02 (t )dt  1 , that it is the solution 0

of the equation (7). Theorem 3. Let the integral equation (7) has no solution u (t ) , satisfying to condition u(t ) U almost everywhere. Then that control u0 (t ) U to be optimum, it is necessary and

sufficiently, that it to be the solution of a following nonlinear integral equation: t

u (t ) 

 (t )   R(t , s)u ( s)ds 0

(8)

T

 (t )   R(t , s)u ( s)ds 0

L2 [ 0 ,T ]

From optimal control existence, follows that the integral equation (8) has the solution every time when the equation (7) has no solution in U . 127


Hence optimum control u0 (t ) is defined or as the solution of the nonlinear integral equation (8), or as the solution linear integral equation. Easy con show, if  (t )  R(t , s) , the equation (7) has no solution in U . Keywords: optimal control, functional, time problems, Hilbert spaces. AMS Subject Classification: 35J05.

References 1. Weinberg M., Variation Methods for the Study of Nonlinear Operators, Moscow, 1956.

APPROXIMATE SOLUTION OF A MIXED PROBLEM, AND THE OPTIMAL CONTROL PROBLEM FOR SYSTEMS WITH DISTRIBUTED PARAMETERS IN HILBERT SPACE A.C. Mamedov1, B.M. Yusifov1 , H.H. Alieva1 1

Sumgayit State University, Sumgayit, Azerbaijan e-mail: [email protected]

In the report the minimization of a quadratic functional for oscillatory systems. Let V and H the Hilbert spaces above R , with the norm  V and  H . V n and H n Hilbert spaces, defined by Hilbert spaces V and H : V n  {a; a *  (a1 , a 2 ,..., a n ); a i V ; i  1, n}, H n  {b; b *  (b1 , b 2 ,...,b n ); b i  H ; i  1, n}

with scalar products n

n

i 1

i 1

(a1 , a 2 )   (a1i , a 2i )V ; a1 , a2  V n ; (b1 , b2 )   (b1i , b2i ) H ; b1 , b2  H ,

and V n  H n  V n* . The state of the control system described by a differential equation of the second order with respect to time d2y   Ay  Qu(t ) , dt 2

(1)

y(0)  y0 , y(0)  y1 ,

(2)

with the initial conditions

where y(t; u)  L2 ([0, T ];V n ) and describes the state of the system over time; t ; y 0 and y1 - the given elements of the V n and H n respectively, A -linear, symmetric positive definite n  n 128


dimensional matrix operator defined in the everywhere dense set in H n ; u (t ) - r - dimensional control vector function from of the set of admissible controls 

U  {u (t )  L (0, T ); u (t )  r 2

r

 u 0 i 1

2 i

(t )dt  1} ;

Q - given n r - dimensional matrix whose elements are the elements of the Hilbert space H .

The paper [1] considers the following optimal control problem: from the set of admissible controls U find a control u (t ) , so that the solutions of the problem (1) - (2) delivers the smallest possible value of the functional J (u)  y(T ; u)  y2

1 Vn

 y(T ; u)  y3

2 Hn

,

(3)

where y 2 and y 3 are given elements T - a fixed time. Let

z0  Q, z1  A1 z0 ,..., z k 1  A1 z k ,... . Suppose that for any m  2 the system

z0 , z1 ,..., z m1 is linearly independent in H mn . For each m constructed matrix operator Am as a

solution to the following problem moments in the Hilbert space H mn , k   z k  Am z0 ; k  0,1,..., m  1,  m  Em z m  Am z 0 ,

(4)

where Em - the projection operator in the subspace H mn with the basis z 0 , z1 ,..., z m1 . Along with the problem (1) - (2) consider the problem d 2 ym Am  y m  A 1Q  u (t ), y m (0)  y m (0)  0 . 2 dt

(5)

The solution of (5) is called an approximate solution of the problem (1) - (2). Along with the functional (3) consider the functional in the form J m (u)  ym (T ; u)  y2

2 Vn

 yn (T , u)  y3

2 Vn

(6)

Control u m (t )  V delivering the minimum value of the functional (6) is called "approximate optimal control." It is proved the following. Theorem 1. For each given control u(t ) U sequence of solutions y m (t; u ) of problem (5) and the sequence

dy m (t ; u ) derivatives of these solutions are uniformly over t strongly dt

129


converges to the solution y(t; u) of (1) - (2) and its derivative,

dy (t ; u ) in V n dt

and H n

respectively. Analogously as in [1], it can be shown that the optimal control u (t ) minimizes the functional (4) at solutions the problem (1) - (2) is determined by the solution of the equation. t

 (t )   R(t , s)u ( s)ds  0 ,

(7)

0

or as a solution of the equation T

 (t )   R(t , s)u ( s)ds 0

.

T

 (t )   R(t , s)u ( s) 0

(8)

L12 ( 0 ,T )

It is easy to show that the "approximate optimal" control is a solution of the equation. T

m (t )   Rm (t , s)u ( s)ds  0 ,

(9)

0

or a solution of the equation T

 m (t )   Rm (t , s)u ( s)ds 0 T

 m (t )   Rm (t , s)u ( s)ds 0

 u (t )

(10)

L12 ( 0 ,T )

Theorem 2. Suppose that u 0 (t ) the solution of (7), and u m0 (t ) solution of equation (9). Then for any m inequality T

 R(t, s)[u

0

( s)  u m0 ( s)]ds   (t )  m (t )  R(t , s)  Rm (t , s) ,

0

is hold, i.d.at m   solution of equation (9) converges to the optimal control u (t ) with weight R(t , s) in space L12 (0, T ) . Keywords: optimal control, quadratic functional, time problem, Hilbert spaces. AMS Subject Classification: 35J05.

130


References 1. Mamedov A.C. Optimal control of systems with distributed parameters in Hilbert space, Proceedings of the Universities, Mathematics, Deposition RISTI, No.2754-79, 1979, 9 p.

FINITE-APPROXIMATE CONTROLLABILITY OF EVOLUTION EQUATIONS N.I. Mahmudov1 1

Department of Mathematics, Eastern Mediterranean University, Gazimagusa, T.R. North Cyprus, Mersin 10, Turkey e-mail: [email protected]

We study the finite-approximate controllability for the abstract evolution equations in Hilbert spaces [1-4]. Assuming the finite-approximate controllability of the corresponding linearized equation we obtain sufficient conditions for the approximate controllability of the semilinear evolution equation. The result we obtained is generalization and continuation of the recent results on this issue [3.5]. Keywords: approximate controllability, fractional differential equations, compact operators, semigroup theory AMS Subject Classification: 93B05 (49J45 93C25).

References 1. Lions J .L., Zuazua E., The cost of controlling unstable systems: Time irreversible systems. Rev. Mat. UCM, Vol.10, No.2, 1997, pp.481-523. 2. Zuazua E., Finite dimensional null controll ability for the semilinear heat equation, J. Math. Pures et. 3. Bashirov A.E., Mahmudov N.I., On concepts of controllability for deterministic and stochastic systems. SIAM J. Control Optim. Vol.3, No.6, 1999, pp.1808-1821. 4. Li X., Yong J., Optimal control theory for infinite dimensional systems, Birkhauser, 1994. 5. Mahmudov N.I., Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J. Control Optim., 42, 2003, No.5, pp.1604-1622.

131


ON ONE GASLIFT PROBLEM DESCRIBED BY THE SYSTEM OF DELAY DIFFERENTIAL EQUATIONS M.M. Mutallimov1, U.Z. Imanova2 1

Baku State University, Baku, Azerbaijan Azerbaijan University of Architecture and Construction, Baku, Azerbaijan e-mail: [email protected], [email protected]

2

System approach to the solution of practical problems reveals new features of tasks. Here such kind of approach is applied to the gas-lift process. As it is known [1], a mathematical model of gas lift chink (rift) approximately described by the following linear system of differential equations in partial derivatives:  P c 2 Q    t F x ,   Q   F P  2aQ  t x

(1)

where t  0, x  0,2L . Breaking [2] L into N to equal parts for the system l  L / N (1) reduces to a system of ordinary differential equations. N  2 for annulus (ring) space we get a system 2 2  c1 c Q1 (t )  1 Q0 (t ),  P1 (t )   F1l F1l   F F Q 1 (t )   1 P1 (t )  1 P0 (t )  2a1Q1 (t ),  l l  2 2 c1 c1  P ( t )   Q ( t )  Q1 (t ), 2  2 F1l F1l  F1 F1  Q2 (t )   l P2 (t )  l P1 (t )  2a1Q2 (t ).

(2)

Here, Q0 t , P0 t  functions correspond to a volumetric flow rate and pressure of injected gas into the annulus (ring) space, which is controlled via a gas lift process. Clearly, downhole pressure and flow rate, which are formed on the bottom lift depend from their respective values at the bottom of the annular space with a certain delay argument t value, i.e.  Q2 (t )  Q2 (t   ), P2 (t )  P2 (t   ).

Using (3), will get for a lift

132

(3)


2 2  c2 c2  P ( t )   Q ( t )  Q2 (t   ),  3 3 F2 l F2 l   F F Q 3 (t )   2 P3 (t )  2 P2 (t   )  2a 2 Q3 (t ),  l l  2 2 c2 c2  P ( t )   Q ( t )  Q3 (t ), 4 4  F2 l F2 l  F2 F2  Q4 (t )   l P4 (t )  l P3 (t )  2a 2 Q4 (t ).

(4)

Thus, we obtain a system of differential equations (2), (4). The corresponding initial conditions will be as follows: Pk t 0   Pk0 , Qk t 0   Qk0

k  1,4

(5)

Then, denoting  x(t )  P1 (t ), Q1 (t ), P2 (t ), Q2 (t ), P3 (t ), Q3 (t ), P4 (t ), Q4 (t ) ,  x 0  P10 , Q10 , P20 , Q20 , P30 , Q30 , P40 , Q40 ,  u (t )  Q0 (t ), P0 (t )





and   0   F1  l   0   F  1 A l   0    0   0    0 

0 0  0  0 B  0   0  0  0

2

c1 F1l

0

0

 2a1

0

0

2

0

0

0

0

0

2

c1 F1l 

0

0

0



c1 F1l

0

0

0

F1 l

 2a1

0

0

0

0



c2 F2 l

0

F2 l

 2a 2

0

2

0

0

0

0

0

0



2

0

0

0

0

c2 F2 l

0

0

0

F2 l

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 2 c2  F2 l

0  0

F2 l 0

0

0

0 0 0

0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0  0 0 0 0 ,  0 0 0 0  0 0 0 0  0 0 0 0

133

0 

F2 l

 0    0   0    0  ,  0    0  2  c  2  F2 l    2a 2  

 c1 2   F1l   0  0  G 0   0  0   0  0 

 0  F1  l  0  0,  0 0  0 0 


from (2), (4) and (5) will get x(t )  Ax(t )  Bx(t   )  Gu(t ) ,

x0  x 0 .

(6) (7)

Here u (t ) is regarded as a control parameter. Demanding minimizing the function J



1 Q4 T   Q 2



2

T



1 u (t ) Ru(t )dt 20



(8)

will get linear - quadratic optimal control problem. Keywords: gas lift process, mathematical model, linear-quadratic problem, optimal control. AMS Subject Classification: 49J15, 93A30.

References 1. Aliev F.A., Ilyasov M.Kh., Jamalbayov M.A. Mathematical simulation of gas lift well running. Reports of Azerbaijan National Academy of Science, Vol.LXIV, No.4, 2008, pp.30-41. 2. Mutallimov M.M., Askerov I.M., Ismailov N.A., Rajabov M.F. An asymptotical method to construction a digital optimal regime for the gaslift process, Appl. Comput. Math., Vol.9, No.1, 2010, pp.77-84.

NECESSARY OPTIMALITY CONDITION IN ONE DISCRETE TWO-PARAMETER SYSTEM K.B. Mansimov2, S.T. Aliyeva1, Zh.B. Ahmadova2, A.I. Agamaliyeva1 1

Baku State University, Baku, Azerbaijan Institute of Control Systems of ANAS, Baku, Azerbaijan e-mail: [email protected], [email protected], [email protected], [email protected] 2

In the paper we consider a problem on the minimum of the functional x1

S u    z t1 , x 

(1)

x  x0

at restraints ut , x U  R r ,

t  t0 , t0  1, ..., t1  1 , x  x0 , x0  1,..., x1  1 ,

(2)

x1

z t  1, x    K t , x, s, z t , s , u t , s  , x  x0

zt0 , x   ax  , x  x0 , x0  1,..., x1 . 134

(3)


Here  z  is the given twice continuously differentiable scalar function, U is the given bounded and open set, K t , x, s, z, u  is the given n -dimensional vector-function continuous in totality of variables together with partial derivatives with respect to z, u  to second order, respectively, ax  is the given discrete vector-function, t 0 , t1 , x0 , x1 are given, and the differences t1  t 0 , x1  x0 are natural numbers, and u t , x  is r - dimensional discrete vectorfunction of control actions. In the problem under consideration, necessary first and second order optimality conditions are obtained. The case of the existence of functional restraints of inequality type was studied separately.

Keywords: optimal control, discrete two-parameter system, necessary optimality conditions. AMS Subject Classification: 49K10.

ON OPTIMALITY OF QUASISINGULAR CONTROLS IN STOCHASTIC CONTROL PROBLEM K.B. Mansimov1, R.O.Mastaliyev1 1


In the report we consider a stochastic optimal control problem whose mathematical model is given by the Ito stochastic differential equation 1, 2. Let (Ω, 𝐹, 𝑃) be a complete probability space. 𝑊(𝑡) be 𝑛 − dimensional standard Winer process determined on the space (Ω, 𝐹, 𝑃). 𝐿2𝐹 (𝑡0 , 𝑡1 ; 𝑅 𝑛 ) be a space of measurable with respect 𝑡

by (𝑡, 𝜔) random processes 𝑥(𝑡, 𝜔): [𝑡0 , 𝑡1 ] × Ω → 𝑅 𝑛 , for which 𝐸 ∫𝑡 1 ‖𝑥(𝑡)‖2 < +∞. 0

Here and in what follows the sign 𝐸 is mathematical expectation. Let on a fixed time interval 𝑇 = [𝑡0 , 𝑡1 ] the control process be described by the following stochastic differential system: 𝑑𝑥(𝑡) = 𝑓(𝑡, 𝑥(𝑡), 𝑢(𝑡))𝑑𝑡 + 𝜎(𝑡, 𝑥(𝑡))𝑑𝑊(𝑡), 𝑡 ∈ 𝑇,

(1)

𝑥(𝑡0 ) = 𝑥0 . Here 𝑓(𝑡, 𝑥, 𝑢) is the given 𝑛 −dimentional vector-function continuous in totality of variables together with partial derivatives with respect by (𝑥, 𝑢) to second order inclusively,

135


𝜎(𝑡, 𝑥): 𝑇 × 𝑅 𝑛 → 𝑅 𝑛×𝑛 is a matrix function of sizes (𝑛 × 𝑛), continues in totality of variables together with partial derivatives with respect by 𝑥 to second order inclusively. 𝑢(𝑡) ∈ 𝑈𝑑 ≡ {𝑢(. ) ∈ 𝐿2𝐹 (𝑡0 , 𝑡1 ; 𝑅 𝑟 )/𝑢(𝑡) ∈ 𝑈 ⊂ 𝑅 𝑟 },

(2)

where 𝑈 is the given nonempty, bounded and convex set. Call 𝑈𝑑 the set of admissible controls. Our goal by minimize the functional 𝑆(𝑢) = 𝐸{ℎ(𝑥(𝑡1 ))}

(3)

on the set of admissible controls. Here ℎ(𝑥) is the given twice continuously differentiable scalar function. By means of the stochastic analogue of the method suggested and developed in the papers of [3,4], we get a linearized necessary optimality condition 5,

and also study the quasi- singular case 6.

Necessary optimality condition of quasi-singular controls is established. Then investigated particular cases. Keywords: stochastic control problem, necessary optimality conditions, quasi-singular control, optimal control. AMS Subject Classification: 49K05, 49K15.

References 1. Gikhman I.I., Skorokhod A.V. Controllable Random Processes. Kiev, Naukova dumka, 1977, 250p. 2. Chernousko F.A., Kolmanovski V.B., Optimal Control of Random Perturbations, Moscow, Nauka, 1978, 352 p. 3. Mansimov K.B., Singular Controls in Systems with Delay, Baku, ELM, 1999, 176 p. 4. Mansimov K.B., Mardanov M.J. Quality Theory of Optimal Control of Goursat-Darboux Systems, Baku, ELM, 2010, 336 p. 5. Gabasov R., Kirillova F.M., Maximum Principle in Theory of Optimal Control, Moscow, Nauka i Tekhnika, 1974, 274p. 6. Gabasov R., Kirillova F.M., Singular Optimal Controls, Moscow, Nauka, 1973, 256 p.

136


NECESSARY OPTIMALITY CONDITION IN A PROBLEM OF OPTIMAL CONTROL OF ONE CONTINUOUS SYSTEM K.B. Mansimov1, Sh.M. Rasulova1 1


In the report we consider a problem on the minimum of the functional x1

S u, v   1  yx1    2 x, z t1 , x  dx

(1)

x0

at constraints ut , x U  R r ,

t, x  D  T  X  t0 , t1  x0 , x1  ,

v x   V  R q ,

x X ,

(2)

z t , x   f t , x, z t , x , ut , x  , t , x   D , t

z t0 , x   yx ,

(3)

x X ,

dyx   g x, yx , vx  , dx

(4)

yx0   y0 .

Here

f t , x, z, u 

g x, y, v

is the given n -dimensional vector-function continuous in

totality of derivatives together with partial derivatives with respect to z

y

to second order,

inclusively, t 0 , t1 , x0 , x1 , y0 are given, U and V are the given nonempty and bounded sets,

ut , x  , vx  are piecewise continuous r and q -dimensional control vector-functions, 1  y 

2 x, z  is the given scalar function continuous in totality of variables together with

1  y  , y

 21  y    2 x, z   2 2 x, z   . ,  ,  z z 2 y 2  

For the problem of type (1)-(4), in the paper 1, 2, A.I.Moscalenko has obtained several necessary optimality conditions in the form of the Pontryagin maximum principle. In the present paper, we study the case of degeneration of the Pontryagin maximum principle in problem (1)(4) (special case).

137


Necessary optimality conditions singular in the sense of Pontryagins maximum principle 3, 4 were established. Keywords: optimal control problem, necessary optimality condition, Pontryagin maximum principle, singular controls. AMS Subject Classification: 49K15, 49K20.

References 1. Moscalenko A.I., On a class of optimal regulation problems, Zhurnal Vichisl. Mat.: Mat. Fizika. No.1, 1969, pp.68-95. 2. Moscalenko A.I., Some problems of theory of optimal control, Author’s Them for PhD degree, Tomsk, TSU, 1971, 21 p. 3. Gabasov R., Kirillova F.М., Singular Optimal Control, Moscow, Nauka, 1973, 256 p. 4. Mansimov К.B., Mardanov М.J. Quality Theory of Optimal Control of Goursat-Darboux Systems. Baku, Elm, 2010, 360 p.

QUEEUING MODEL WITH INSTANTANEOUS AND DELAYED FEEDBACK Agassi Melikov1, Anar Rustamov2 1

Institute of Control Science, Baku, Azerbaijan 2 Qafqaz University, Baku, Azerbaijan e-mail: [email protected], [email protected]

Among models of queueing systems with feedback two kinds of models should be distinguished: 1) models with instantaneous feedback (i.e. models without orbit) and 2) models with delayed feedback (i.e. models with orbit). In the available literature both kinds of models have been investigated separately. Models with instantaneous feedback were investigated in papers [1]-[6] while models with delayed feedback are examined in papers [7]-[11]. But models of queueing systems with simultaneously instantaneous and delayed feedback have not been investigated. In this paper, the Markov models of multichannel queueing systems with instantaneous and delayed feedback are examined. Let’s note that taking into account of both types of feedback mechanisms leads to increase of vector dimension which describe the state of the system. As a result an approach to investigate the models based on the system of balance equations (SBE) for steady-state probabilities becomes inefficient especially for the large scale models. So, 138


developing efficient methods for asymptotic analysis of models with a large number of channels and large size of orbit is highly desired. In given paper both exact and asymptotic methods to calculate the quality of service (QoS) metrics of the queueing models with instantaneous and delayed feedback are proposed. We consider a model of queueing system with N > 1 identical channels and without buffer. These channels are used by Poisson flow of primary calls (p-calls) with intensity p . Distribution function of channel occupancy time of p-calls is assumed to be exponential with parameter µp . At the completion of the servicing of the each p-call the following decisions might be accepted: (a) p-call leaves the system (with the probability 1 (x)); (b) p-call instantaneously feedback (with the probability2 (x)); p-call feedback via orbit after some delay (with the probability 3 (x)). Here parameter x denotes the state of some external random environment. It is assumed that orbit has finite size R, 0 < 𝑅 < ∞. Inter arrival times between repeated calls from orbit (r-calls) have exponential distribution with mean 1⁄r . Note that r-calls are not persistent, i.e. if upon arrival of the r-call all channels of the system are busy then it is lost. Distribution functions of channel occupancy time of feedback calls (both instantaneous and delayed) are assumed to be exponential with same parameter µr ; generally speaking µp ≠ µr . Main characteristics of the investigated models are (i) loss probability of p-calls (Pp); (ii) loss probability of r-calls from the orbit (Pr); (iii) average number of p-calls in channels (Lp); (iv) average number of r-calls in channels (Lr); (v) average number of r-calls in orbit (Lo). It is shown that the mathematical model of the investigated queueing system with feedback is three-dimensional Markov chain (3-D MC). Both exact and asymptotic methods to calculate the characteristics of the proposed model are developed. Exact method is based on the system of balance equations (SBE) for steady-state probabilities of appropriate 3-D MC while asymptotic method uses the new hierarchical space merging algorithm for 3-D MC [12]. Results of numerical experiments are demonstrated. Keywords: queueing model, instantaneous and delayed feedback, three-dimensional Markov chain, exact analysis, asymptotic analysis. AMS Subject Classification: 90B05, 90B22.

139


References 1. Takacs L., A singleserver queue with feedback, Bell System Technical Journal, Vol. 42, 1963, pp. 505-519. 2. Wortman M.A., Disney R.L., Kiessler P.C., The M/GI/1 Bernoulli feedback queue with vacations, Queueing Systems., Vol.9, No.4, 1991, pp. 353-363. 3. D'Avignon G.R., Disney R.L., Queues with instantaneous feedback , Management Sciences, Vol. 24, No.2, 1977, pp.168-180. 4. Berg J.L., Boxma O.J., The M/G/1 queue with processor sharing and its relation to feedback queue, Queueing Systems, Vol. 9, No.4, pp. 365-402. 5. Hunter J.J. Sojourn time problems in feedback queue , Queueing Systems, Vol. 5, Issue 13, 1989, pp. 55-76. 6. Dudin A.N., Kazimirsky A.V., Klimenok V.I., Breuer L., Krieger U., The queueing model MAP/PH/1/N with feedback operating in a Markovian random environment, Austrian Journal of Statistics, Vol. 34, No.2, 2007, pp. 101-110. 7. Takacs L., A queueing model with feedback , Operations Research, Vol. 11, Issue 4, 1997, pp. 345-354. 7. Pekoz E.A., Joglekar N., Poisson traffic flow in a general feedback , Journal of Applied Probability, Vol. 39, No.3,2002, pp. 630-636. 8. Lee H.W., Seo D.W., Design of a production system with feedback buffer , Queueing Systems, Vol.26, No.1, 1997, pp.187-200. 9. Lee H.W., Ahn B.Y., Analysis of a production system with feedback buffer and general dispatching time, Math. Prob. Eng., Vol.5, 2002, pp. 421-439. 10. Foley R.D., Disney R.L., Queues with delayed feedback, Advances in Applied Probability, Vol.15, No.1, 1983, pp. 162-182. 11. Ponomarenko L., Kim C.S., Melikov A., Performance Analysis and Optimization of Multitraffic on Communication Networks, Heidelberg, Dortrecht, London, New York, Springer, 2010, 208p.

140


FEEDBACK OPTIMAL BOUNDARY CONTROL OF THE PROCESS OF HEATING THE MATERIAL TAKING INTO ACCOUNT THE IMPACT OF THE EXTERNAL ENVIRONMENT R.S. Mammadov1 1


Let the controlled process be described by the vector-function u(t , x)  ( y(t ), u(t , x)) , which satisfies the following equation ut  a 2u xx  b( x)[u(t , x)  y(t )]  p(t , x)

(1)

within the domain Q  {0  x  1, 0  t  T } and the following initial and boundary conditions u(0, x)  u 0 ( x) ,

(2)

ux (t ,0)  0, ux (t ,1)  u(t ,1)  0,   0

(3)

on the boundary of Q . Here b(x) , u 0 ( x) are given functions from L2 (0,1) ; p(t , x) is the controlling parameter; y(t ) is the temperature of the external environment, which is assumed to be the solution to the following Cauchy problem: y(t )  a0u(t ,1)  b0 y(t )  p0 (t ) , y(0)  y0 ,

(4) (5)

where a0 ,b0 are given real numbers and p0 (t ) is the controlling parameter as well. The vector-functions p(t , x)  ( p0 (t ), p(t , x)) are the admissible controls, for which p0 (t )  L2 (0, T ) , p(t , x)  L2 (Q) .

The considered optimal control problem consists in finding the control in the form of feedback such that the following functional 1

T

T 1

I [ p]  [ y(T )  0 ]   [u (T , x)  ( x)] dx   (  p (t )dt    p 2 (t , x)dxdt ) , 2

2

2 0

0

0

(6)

0 0

  const  0 , would take its minimal possible value. Here T is the fixed point of time;  0 the given number;

 (x) the given function from L2 (0,1) . Applying the dynamic programming method, the solution to the posed optimal control problem in the form of feedback is sought for as follows: 141


1   p0 (t )  q0 (t ) y (t )   K 0 (t , s)u (t , s)ds  r0 (t ),  0  1  p(t , x)  q(t , x) y (t )  K (t , x, s)u (t , s)ds  r (t , x). 0  

(7)

Here the coefficients are known functions which are determined from the corresponding boundary problems. Keywords: Feedback optimal boundary control, dynamic programming method, Cauchy problem. AMS Subject Classification: 93B52, 49J20, 49K20.

DEFINITION OF THE SUBOPTIMIST SOLUTION AND ITS FINDING IN THE BULL PROGRAMMING PROBLEM WITH INTERVAL DATA K.Sh. Mammadov1, A.H. Mammadova1 1

Institute of Control Systems, ANAS, Baku, Azerbaijan e-mail: [email protected]

Consider the following problem 𝑛

(1)

∑[𝑐𝑗 , 𝑐𝑗 ]𝑥𝑗 → 𝑚𝑎𝑥 𝑗=1 𝑛

∑[𝑎𝑖𝑗 , 𝑎𝑖𝑗 ]𝑥𝑗 ≤ 𝑏𝑖 ,

(𝑖 = 1, 𝑚 ),

(2)

𝑗=1

𝑥𝑗 = 0 ∨ 1 ,

( 𝑗 = 1, 𝑛 ).

(3)

Here is assumed that 𝑐𝑗 , 𝑐𝑗 , 𝑎𝑖𝑗 , 𝑎𝑖𝑗 , 𝑏𝑖 , 𝑏𝑖 , 𝑏𝑖 ∈ [𝑏𝑖 , 𝑏𝑖 ] (𝑖 = 1, 𝑚 , 𝑗 = 1, 𝑛 ) are given nonnegative integers. The problem (1)-(3) is called Bull programming problem with interval initial data. The stated problem was insively studied by different authors since 1980 [1; 2; 4; 5]. Definition 1. As a solution of the problem (1)-(3) we understand 𝑛 −dimensional vector 𝑋 = (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) that satisfy the conditions ∑𝑛𝑗=1 𝑎𝑖𝑗 𝑥𝑗 ≤ 𝑏𝑖 , (𝑖 = 1, 𝑚) by the integers ∀ 𝑎𝑖𝑗 ∈ [𝑎𝑖𝑗 , 𝑎𝑖𝑗 ] , ∀ 𝑏𝑖 ∈ [𝑏𝑖 , 𝑏𝑖 ] , (𝑖 = 1, 𝑚 ; 𝑗 = 1, 𝑛) . Definition 2. The admissible solution 𝑋 𝑜 = (𝑥1𝑜 , 𝑥2𝑜 , … , 𝑥𝑛𝑜 ) of the problem (1)-(3) is called to be an optimist solution if the number 𝑓 𝑜 = ∑𝑛𝑗=1 𝑐𝑗 𝑥𝑗𝑜 is maximal. Then the number 𝑓 𝑜 is called an optimist value for the problem (1)-(3). 142


Definition 3. The admissible solution 𝑋 𝑠𝑜 = (𝑥1𝑠𝑜 , 𝑥2𝑠𝑜 , … , 𝑥𝑛𝑠𝑜 ) of the problem (1)-(3) is called to be a suboptimist solution if the number

𝑓 𝑠𝑜 = ∑𝑛𝑗=1 𝑐𝑗 𝑥𝑗𝑠𝑜 gets a maximal value

defined by some criteria. The number 𝑓 𝑠𝑜 is called a suboptimist value for the problem (1)-(3) . In the problem (1)-(3) we assume that 𝑎𝑖𝑗 ≥ 0, 𝑎𝑖𝑗 ≥ 0, 𝑏𝑖 ∈ [𝑏𝑖 , 𝑏𝑖 ] , 𝑏𝑖 > 0, 𝑏𝑖 > 0, 𝑐𝑗 > 0, 𝑐𝑗 > 0, (𝑖 = 1, 𝑚 , 𝑗 = 1, 𝑛) 𝑏𝑖 (𝑖 = 1, 𝑚 , 𝑗 = 1, 𝑛)

are given numbers and the conditions 𝑎𝑖𝑗 ≤ 𝑎𝑖𝑗 ≤

are satisfied.

Let us write the problem (1)-(3) in the following equvivalent form 𝑛

(4)

∑[𝑐𝑗 , 𝑐𝑗 ]𝑥𝑗 → 𝑚𝑎𝑥 𝑗=1 𝑛

∑[𝛼𝑖𝑗 , 𝛼𝑖𝑗 ]𝑥𝑗 ≤ 1 ,

(𝑖 = 1, 𝑚 ),

(5)

𝑗=1

𝑥𝑗 = 0 ∨ 1 , Here

( 𝑗 = 1, 𝑛 ).

(6)

𝛼𝑖𝑗 = 𝑎𝑖𝑗 ⁄𝑏𝑖 , 𝛼𝑖𝑗 = 𝑎𝑖𝑗 ⁄𝑏𝑖 ( 𝑖 = 1, 𝑚, 𝑗 = 1, 𝑛 ) and 0 ≤ 𝛼𝑖𝑗 ≤ 1, 0 ≤ 𝛼𝑖𝑗 ≤ 1, (𝑖 =

1, 𝑚, 𝑗 = 1, 𝑛 ). We introduce the definitions 𝑃𝑗 = (𝛼1𝑗 , 𝛼2𝑗 , … , 𝛼𝑚𝑗 )𝑇 ,

𝑃𝑗 = (𝛼1𝑗 , 𝛼2𝑗 , … , 𝛼𝑚𝑗 )𝑇 , 𝑃𝑜 = (1,1, … ,1)𝑇 ,

(𝑗 = 1, 𝑛).

As a penalty of use of the remaining resource 𝑃𝑜 we take 𝑞𝑖 =

1 , (𝑖 = 1, 𝑚 ) 1 − 𝛼𝑖𝑗∗

(7)

As a criteria of choosing of each 𝑥𝑗𝑠𝑜 = 1 we take ∗ 𝑗∗ = 𝑎𝑟𝑔 max {𝑓𝑗 }. 𝑗∈𝛺

Here 𝛺 = {1,2, … , 𝑛},

(8)

𝑓𝑗 = 𝑐𝑗 ⁄𝑄𝑗 , ( 𝑗 ∈ 𝛺),

𝑚

𝑄𝑗 = ∑ 𝛼𝑖𝑗 , 𝑞𝑖 ( 𝑗 ∈ 𝛺) ,

𝑞𝑖 = 1⁄(1 − 𝑟𝑖 ) , ( 𝑖 = 1, 𝑚 ) ,

𝑖=1

𝑟𝑖 = ∑ 𝛼𝑖𝑗 , ( 𝑖 = 1, 𝑚 ),

𝜔 𝑠𝑜 = {𝑗 | 𝑥𝑗𝑠𝑜 = 1} .

𝑗∈𝜔 𝑠𝑜

In the beginning we accept 𝑏𝑖 ≔ 𝑏𝑖 , (𝑖 = 1, 𝑚). Then we get 𝜔 𝑠𝑜 = ∅ , 𝑟𝑖 = 0, 𝑞𝑖 = 1 (𝑖 = 1, 𝑚). The by sequental using the above formulas we find some index 𝑗∗ and take 𝛺\{𝑗∗ } . If 𝑃𝑗∗ ≤ 𝑃0 , then we accept 𝑃0 ≔ 𝑃0 − 𝑃𝑗∗ , 𝑥𝑗𝑠𝑜 : = 1 və 𝜔 𝑠𝑜 ≔ 𝜔 𝑠𝑜 ⋃{𝑗∗ } . It is clear that the ∗

143


found by those formulas numbers 𝑟𝑖 , 𝑞𝑖 , (𝑖 = 1, 𝑚), 𝑄𝑗 , 𝑓𝑗 ( 𝑗 = 1, 𝑛) and the index 𝑗∗ will get new values. Otherwise, i.e. if the relation 𝑃𝑗∗ ≤ 𝑃0 will not true at least for one index we take 𝑥𝑗𝑠𝑜 : = 0 , 𝛺: = 𝛺\{𝑗∗ } and find new index 𝑗∗ from (8). By this way the process of construction ∗ of the solution will be continued till reaching 𝛺 = ∅ . In result we obtain a suboptimist solution 𝑋 𝑠𝑜 = (𝑥1𝑠𝑜 , 𝑥2𝑠𝑜 , … , 𝑥𝑛𝑠𝑜 ) and corresponding suboptimist value 𝑓 𝑠𝑜 = ∑𝑛𝑗=1 𝑐𝑗 𝑥𝑗𝑠𝑜 for the problem (1)-(3) . Note that for the particular case 𝑚 = 1 of the problem (1)-(3) this algorithm was given in [3]. Keywords: Bull programming, interval initial data, optimist value. AMS Subject Classification: 35Q90.

References 1. Emelichev V.A., Podkopaev D.P., Quantitative stability analysis for vector problems of 0-1 programming, Discrete Optimization, No.7, 2010, pp.48-63. 2. Devyaterikova M.V., Kolokolov A.A., Kolosov A.P., The algorithms of chice of 𝐿 −classes for the Bull knarck problem with interval data, The 3rt Russian Conference “The problems of optimization and economic applications”, Omsk, 2006, pp. 87. 3. Mamedov K.Sh., Memedova A.N., Huseynov S.Y. The definition of suboptimist and subpessimist solutions in the knarck problem with interval coefficients and methods for thier construction, Reports of ANAS, 2013, No.6, pp.125-131. 4. Mashenko S.O., Bavsunovski A.N. Effective alternatives to the decision making problems with fuzzy set of preference relations, Problems of Control and Informatics, No.6, 2013, pp.50-60. 5. Roshin V.A., Semenova N.V., Sergienko I.V Decompositional approach to the solution of some integer programming problems with non-exact data, Computational Mathematics and Mathematical Physics, Vol.30, No.5, 1990, pp.786-791.

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OPTIMAL DOSING STRATEGIES AGAINST SUSCEPTIBLE AND RESISTANT BACTERIA Mudassar Imran1 1

Gulf University of Science and Technology Kuwait e-mail: [email protected]

Antibiotic modelling is concerned with the problem of finding efficient and successful dosing techniques against bacterial infections. In this study, we model the problem of treating a bacterial infection where the bacteria are divided into two sub-populations: susceptible and resistant. The susceptible type may acquire the resistance gene via the process of conjugation with a resistant bacterium cell. Efficient treatment strategies are devised that ensure bacteria elimination while minimizing the quantity of antibiotic used. Such treatments are necessary to decrease the chances of further development of resistance in bacteria and to minimize the overall treatment cost. We consider the cases of varying antibiotic costs, different initial bacterial densities and bacterial attachment to solid surfaces, and obtain the optimal strategies for all the cases. The results show that the optimal treatments ensure disinfection for a wide range of scenarios.

Keywords: mathematical model, antibiotic costs, bacterial densities, optimal strategies. AMS Subject Classification: 49N90.

COMPUTATION OF DERIVATIVE TENSORS USING HYPERCOMPLEX-STEPS AND APPLICATION TO OPTIMIZATION PROBLEMS H.M. Nasir1 1

Department of Mathematics and Statistics, Sultan Qaboos University Muscat, Oman e-mail: [email protected], [email protected]

Optimization algorithms such as Line Search and Newton Methods require the Jacobian and Hessian of the target differentiable function [1]. Computing them for functions of several variables is a challenging task. Approximation methods such as finite difference do not produce satisfactory results [2]. Exact methods such as symbolic differentiation involve time and memory consuming data structures and implementation. Another exact method known as the automatic differentiation or algorithmic differentiation involves user interference of rewriting to partition

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the function into simple expressions involving intermediate variables and intrinsic functions[3]. This makes the automatic differentiation not fully automatic[4]. Complex-step differentiation is a recently popular method to compute a real valued function f(x) and its first derivative at a real value by considering a small step size h as an imaginary part[5,6]. The Taylor expansion of the complex function f(x + ih) =: u+iv gives f(x) = u and f’(x) = v/h. For a function of several variables, the function and its first partial derivatives 𝜕𝑓/𝜕𝑥𝑖 are computed by using imaginary step in the variable 𝑥𝑖 . Although this method can give results to near machine precision accuracy by choosing the imaginary step size very small, it requires separate computation of complex valued functions for each partial derivatives which means the function is computed several times resulting in superfluous computations. Besides, it does not give higher and mixed order derivatives for the Hessian. Moreover, the complex-step method gives result with accuracy order 2 only. In this paper, we propose a new generalized complex-step type method to compute the function and its Jacobian, Hessian and any higher order derivative tensors simultaneously with arbitrary accuracy order of choice. For this, we consider a hypercomplex number system and compute the function f(x), where x is multivariable vector, only once at a hypercomplex number. The derivative tensors are then extracted from the components of the computed hypercomplex valued function. Definition 1 [7] We define a n-hypercomplex system ℂ(n, 1)as a commutative algebra over the complex field with one indeterminate e satisfying en = 1.

The n-hypercomplex

numbers 𝐚 ∈ ℂ(n, 1) has the form of polynomial in e as 𝐚 = a0 + a1 e + ⋯ + an−1 en−1 = : (ai ), (a0 , a1 , ⋯ , an−1 ) ∈ ℂn with 𝐚 + 𝐛 = (ai + bi ), 𝐚𝐛 = ((𝐚 ∗ 𝐛)i ), where (𝐚 ∗ 𝐛)i is the i-th component of the cyclic convolution (𝐚 ∗ 𝐛)i = ∑n−1 k=0 a k b(i−k) with b(j) = bj mod n . Definition 2 [7] We define multi-hypercomplex system as the product of copies of finite number of n-hypercomplex systems: 𝐂(𝐧, 1) = ℂ(n1 , 1) × ℂ(n2 , 1) × ⋯ × ℂ(nK , 1), where 𝐧 = (n1 , n2 , ⋯ , nK ). We denote 𝐂(𝐧, 1) = 𝐂(n, 1) when all the ni = n. The elements of the multi-hypercomplex system are in the form of a multi variable polynomial with indeterminates 𝑒𝑖 ∈ ℂ(𝑛𝑖 , 1), 𝑖 = 1,2, ⋯ , 𝐾 expressed as 𝑎 = ∑𝑨 𝑎𝑨 𝒆𝑨, where 𝑎𝑨 ∈ ℂ and the summation covers all index tuples𝑨 = (𝑘1 , 𝑘2 , ⋯ , 𝑘𝐾 ) ∈ 𝐴1 × 𝐴2 × ⋯ × 𝐴𝐾 with 𝑘

𝑘

𝑘

𝐴𝑖 = {0,1,2, ⋯ , 𝑛𝑖 } and 𝒆𝑨 = 𝑒1 1 𝑒2 2 ⋯ 𝑒𝐾𝐾 .

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Hypercomplex numbers can be expressed in a canonical form with another basis of indeterminates 𝑢0 , 𝑢2 , ⋯ , 𝑢𝑛−1 satisfying the orthogonal property 𝑢𝑖 𝑢𝑗 = 𝛿𝑖𝑗 with Kronecker delta 𝛿𝑖𝑗 :

𝒂 = 𝑎0 + 𝑎1 𝑒 + ⋯ + 𝑎𝑛−1 𝑒 𝑛−1 = 𝑎̂0 𝑢0 + 𝑎̂1 𝑢1 + ⋯ + 𝑎̂𝑛−1 𝑢𝑛−1 . In fact, the

coefficients 𝑎̂𝑖 are the eigenvalues and base indeterminates 𝑢𝑖 are formed by eigenvectors of the circulant matrix defined by the coefficient vector 𝒂 = (𝑎𝑖 ). The canonical coefficients are indeed the discrete Fourier transform (DFT) of the vector 𝒂. If we denote the coefficient vector ̂ = [𝑎̂𝑗 ], we have the following results : 𝒂̂ ̂+ of the canonical representation as 𝒂 +𝒃=𝒂 ̂ , 𝒂𝒃 ̂ =𝒂 ̂ , where the operations on the right are performed elementwise. This observation ̂. 𝒃 𝒃 prompts one to conclude that for a differentiable function 𝑓(𝑥), the n-hypercomplex valued ̂ = 𝑓(𝒂 ̂) by using DFT and inverse DFT. function 𝑓(𝒂) can be computed via the relation 𝑓(𝒂) Returning to the goal of this paper, let 𝑓(𝑧) be an analytic function of a single complex variable. The function at the n-hypercomplex number 𝑧 + ℎ𝑒 ∈ ℂ(𝑛, 1) gives, by Taylor series expansion, 𝑓(𝑧 + ℎ𝑒) = 𝑓(𝑧) + ℎ𝑒𝑓

′ (𝑧)

= (𝑓(𝑧) +

ℎ2 𝑓 ′′ (𝑧) 2 ℎ𝑛−1 𝒆𝑛−1 𝑓 (𝑛−1) (𝑧) 𝑛−1 + 𝑒 + ⋯+ 𝑒 +⋯ (𝑛 − 1)! 2!

ℎ𝑛 𝑓 (𝑛) (𝑧) ℎ𝑛+1 𝑓 (𝑛+1) (𝑧) + ⋯ ) + (𝑓 ′(𝑧) + +⋯)𝑒 + ⋯ (𝑛 + 1)! 𝑛!

+ (𝑓 (𝑛−1) (𝑧) +

ℎ2𝑛−1 𝑓 (2𝑛−1) (𝑧) + ⋯ ) 𝑒 𝑛−1 . (2𝑛 − 1)!

If the computed 𝑓(𝑧 + ℎ𝑒) = 𝑓0 + 𝑓1 𝑒 + ⋯ + 𝑓𝑛−1 𝑒 𝑛−1 , the function

and its first 𝑛 − 1

derivatives are obtained by equating the coefficients of 𝑒 𝑘 : 𝑓 (𝑘) (𝑧) =

𝑘! 𝐼𝑚 𝑓(𝑧 + ℎ𝑒), ℎ𝑘 𝑘

𝑘 = 0,1,2, ⋯ , 𝑛 − 1.

For a function of several variables 𝑓(𝒛) = 𝑓(𝑧1 , 𝑧2 , ⋯ , 𝑧𝐾 ), consider computing the multihypercomplex valued function 𝑓(𝒛 + ℎ𝒆) = 𝑓(𝑧1 + ℎ𝑒1 , 𝑧2 + ℎ𝑒2 , ⋯ , 𝑧𝐾 + ℎ𝑒𝐾 ). Multiple Taylor series expansion gives the partial derivatives by equating the coefficients of the indeterminate expression 𝑒𝑨 , 𝑨 = (𝑘1 , 𝑘2 , ⋯ , 𝑘𝐾 ) ∈ 𝐴1 × 𝐴2 × ⋯ 𝐴𝐾 : 𝜕 |𝑨| 𝑓 𝑘

𝑘

𝑘

𝜕𝑥1 1 𝜕𝑥2 2 ⋯ 𝜕𝑥𝐾𝐾

𝜕 𝑨𝑓 𝒌! =: = 𝑓 (𝐴) (𝒛) = |𝑨| 𝐼𝑚𝑨 𝑓(𝒛 + ℎ𝒆), 𝜕𝑥𝑨 ℎ

where 𝒌! = 𝑘1 ! 𝑘2 ! ⋯ 𝑘𝐾 ! , |𝑨| = 𝑘1 + 𝑘2 + ⋯ 𝑘𝐾 and 𝐼𝑚𝑨 indicates the coefficient of 𝒆𝑨 .

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The m-th order derivative tensor is then obtained by collecting the derivatives 𝑓 (𝑨) (𝒛), |𝑨| = 𝑚. Thus, the Jacobian and the Hessian are given by 𝑓 (𝑨) (𝒛), |𝑨| = 1

and 𝑓 (𝑨) (𝒛), |𝑨| = 2

respectively with the entries arranged according to their matrix indices. We present numerical results of optimization problems to demonstrate the efficiency and accuracy of the method. We also present implementation which is fully automatic that requires no user interference. Keywords: Jacobian, Hessian, complex-step differentiation, hypercomplex systems, optimization. AMS Subject Classification: 47N10.

References 1. Fike J.A., Jongsma S., Alonso J. A., Van der Weide E., Optimization with gradient and hessian information calculated using hyper-dual numbers, 29th AIAA Applied Aerodynamics Conference, pp. 27-30 (June 2011), Honolulu, Hawaii. 2. Ramesh Kumar M., Uthra G., A study on numerical stability of finite difference formulae for numerical differentiation and integration, Annals of Pure and Applied Mathematics, 2014, Vol.8, No.2, pp.27-36. 3. Griewank A., Walther A., Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation (2nd Edition) SIAM, 2008. 4. Walter S. F., Lehman L., Algorithmic differentiation with AlgoPy, J. Comp. Science, Vol.4, 2013, pp.334-344. 5. Squire W., Trapp G., Using complex variables to estimate derivatives of real functions, SIAM Review, Vol.40, No.1, 1998, pp.110-112. 6. Martins J. R. R. A., Sturda P., Alonso J. J., The complex-step derivative approximation, ACM Transactions on Mathematical Software, Vol.29, No.3, 2003, pp.245-262. 7. Nasir H.M., A new class of multicomplex algebra with applications, Mathematical Sciences International Research Journal, Vol.2, No.2, 2013, pp.163-168.

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ANALOGUE OF THE DISCRETE MAXIMUM PRINCIPLE AND INVESTIGATION OF SINGULAR CONTROLS IN A DISCRETE TWO-PARAMETER CONTROL PROBLEM М.М. Nasiyyati1 1


Assume that the controlled system is described by the following discrete two-parameter system of equations z1 t  1, x  1  f1 t , x, z1 t , x , z1 t  1, x , z1 t , x  1, u1 t , x  ,

t , x   D1t , x :

t  t0 , t0  1, ..., t1  1, x  x0 , x0  1, ..., X  1,

z 2 t  1, x  1  f 2 t , x, z 2 t , x , z 2 t  1, x , z 2 t , x  1, u 2 t , x ,

t , x  D2 t , x : t  t1 , t1  1,..., t 2  1, x  x0 , x0  1,..., X  1,

(1)

z3 t  1, x  1  f 3 t , x, z3 t , x , z3 t  1, x , z3 t , x  1, u3 t , x ,

t , x  D3 t , x : t  t 2 , t 2  1,..., t3  1, x  x0 , x0  1,..., X  1, with boundary conditions z1 t0 , x    x ,

x  x0 , x0  1,..., X , z1 t , x0   1 t ,

t  t0 , t0  1,..., t1 ,

 x0   1 t0  , z2 t1 , x   z1 t1 , x, x  x0 , x0  1, ... , X , z2 t , x0    2 t , t  t1 , t1  1, ... , t2 ,

(2)

z1 t1 , x0    2 t1  , z3 t2 , x   z2 t2 , x ,

x  x0 , x0  1, ... , X ,

z3 t , x0   3 t ,

t  t2 , t2  1, ... , t3 ,

z2 t2 , x0   3 t1  .

Here f i t , x, zi , ai , bi , ui  , i  1, 3 are the given n -dimensional vector-functions continuous in totality of variables together with partial derivatives with respect to zi , ai , bi  , i  1, 3 to second order, inclusively,  x  ,  i t  , i  1, 3 are the given n - dimensional discrete vectorfunctions, x0 , ti , i  0, 3 , X are given, and ui t , x  , i  1, 3 are r - dimensional vector-functions of control actions with the values from the nonempty and bounded sets U  R r , i  1, 3 , i.е. ui t , x U i  R r ,

t, x Di ,

149

i  1, 3 .

(3)


 The triple ut , x   u1 t , x , u2 t , x , u3 t , x  with the above properties is called an

 admissible control, and the appropriate solution zt , x   z1 t , x , z2 t , x , z3 t , x  of boundary value problem (1)-(2), an admissible state of the process. The pair ut , x , zt , x  is called an admissible process. The problem is in minimization of the functional 3

S u    i zi ti , X  ,

(4)

i 1

at restraints (1)-(3). Here i zi  , i  1, 3 − are the given continuously differentiable scalar functions. In what follows, the problem on minimum of the functional (4) at restraints (1)-(3) will be called problem (1)-(4). The admissible process, ut , x , zt , x  , being the solution of problem (1)-(4), is an optimal process. The analogue of the discrete condition of maximum is proved. The case of degeneration of the discrete condition of maximum is studied. Keywords: discrete two-parameter system, discrete maximum principle, singular control, necessary optimality conditions. AMS Subject Classification: 49K10.

HEURISTIC GRID METHOD FOR TRAVELING SALESMAN PROBLEM Fidan Nuriyeva1,2, Gözde Kizilates3 1

Institute of Control Systems of ANAS, Baku-Azerbaijan Dokuz Eylul University, Faculty of Science, Department of Computer Science, Izmir-Turkey 3 Ege University, Faculty of Science, Department of Mathematics, Izmir-Turkey e-mail: [email protected],[email protected]

2

The Traveling Salesman Problem (TSP) is perhaps the most famous optimization problem in the NP-hard class. Many problems having natural applications in computer science and engineering can be modeled using the TSP. This paper presents a new heuristic algorithm called grid method based on finding the central point in cycle for solving the traveling salesman problem. Many benchmark examples from TSPLIB (traveling salesman problem library) were

150


solved with NN, proposed algorithm, and NND algorithms. The experimental results show that the proposed algorithm is more efficient than both NN and NND algorithms. Traveling Salesman Problem. The most important member of the large set of combinatorial optimization problems is undoubtedly the traveling salesman problem (TSP), which involves the task of determining a route among a given set of nodes with the shortest possible length. The study of this problem has attracted several researchers in various fields such as artificial intelligence, biology, mathematics, physics, and operations research, and there is a large literature on the problem. No exact polynomial time algorithms exist for solving the TSP, however many approximate algorithms applying various heuristic approaches have been proposed in the last decades. The algorithms for solving the TSP can be divided into two classes: exact algorithms and heuristic algorithms. The exact algorithms are guaranteed to find the optimal solution in a bounded number of steps. However, these algorithms are quite complex and very demanding of computer power. The heuristic algorithms can obtain good solutions but they cannot guarantee that the optimal solution will be found [1]. The TSP is stated as follows: given a complete graph G=(V,E) and a cost ci j associated with each edge in E. The value ci j is the cost incurred when traversing from vertex i ∈ V to vertex j ∈ V. Given this information, a solution to the TSP must return the cheapest Hamiltonian cycle of G. A Hamiltonian cycle is a cycle that visits each node in a graph exactly once. This is referred to as a tour in TSP terms. The new heuristic algorithm we propose is based on our previous algorithm given in [2] and well-known Nearest Neighbour algorithm. NN and NND Algorithms The nearest neighbor algorithm (NN). Among the tour construction heuristics, the nearest neighbor heuristic is the simplest one. The nearest neighbor (NN) algorithm for determining a traveling salesman tour is as follows. The salesman starts with a city then visits the city nearest to the starting city. Afterwards, he visits the nearest unvisited city and repeats this process until he has visited all the cities, in the end, he returns to the starting city [3]. The nearest neighbor algorithm from both end points (NND) . The algorithm starts with a vertex chosen randomly in the graph. Then, the algorithm continues with the nearest unvisited vertex to the chosen vertex. We will have two end vertices. We add a vertex to the tour

151


such that this vertex has not visited before and it is the nearest vertex to these two end vertices. We update the end vertices. The algorithm ends after visiting all vertices [2]. A proposed algorithm. The steps of the proposed algorithm are as follows: Step 1. Find the central vertex O(x0,y0) in the graph. Step 2. Find the farthest vertex A1(x1,y1) to the central vertex among all vertices and add it to the list ofselected vertices. Step 3.Find the farthest vertex A2(x2,y2) to the vertices in the selected list and add it to the list of selected vertices. Step 4. If the number of the selected vertices is less than 4, go to Step 3. Consequently, by finding the vertices A3(x3,y3) and A4(x4,y4) we will have 4 marginvertices. Step 5. The midpointB1(z1,t1) of thevertices O, A1 and A3, is found as follows: z1 = (x0+x1+x3) /3, t1 = (y0+y1+y3) / 3 Similarly, the midpoint B2 of the vertices O, A3 and A2,the midpoint B3 of the vertices O, A2 and A4, the midpoint B4 of the vertices O, A4 and A1 are found. Step 6.Find the intermediate vertices. The midpoint C1(u1,v1) of the vertices O, B1, A1 and B4, is found as follows: u1 = (x0+z1+x1+z4) / 4, v1 = (y0+t1+y1+t4) / 4 Similarly, the midpointC2(u2,v2) of the vertices O, B2, A3 and B1,the midpoint C3(u3,v3) of the vertices O, B3, A2 and B2 and the midpointC4(u4,v4) of the vertices O, B4, A4 and B3 are found. Finally, we have central vertex O, end vertices A1, A2, A3, A4, middle vertices B1, B2, B3, B4 and intermediate vertices C1, C2, C3, C4. Step 7. By assuming these verticesas starting point and applying NND algorithm to them we find a tour for the TSP.By taking these 13 points as “Selected Vertices”, we will find nearest neighbour to each one.Add the nearest neighbour to the list of the selected vertices and add the edge, whichconnects the nearest neighbour to the selected vertices list, to the list of selected edges.In this way, the algorithm proceeds until the tour is found that includes all vertices. Moreover, a tour is constructed by adding 9 vertices (O, A1-A4, B1-B4) to the selected vertices list. Conclusıon. In this paper, we present a heuristic algorithm for solving the TSP based on finding a central point in a graph. The proposed algorithm implemented in C++. Then the article 152


compares the experimental results of many TSP examples from TSPLIB using the three algorithms (NN, NND and proposed method). The results show that the proposed method is able to produce better solutions in reasonable time than NN and NND. Keywords: traveling salesman problem, heuristic algorithms, nearest neighbour algorithm. AMS Subject Classification: 90C27, 90C59.

References 1. Appligate D.L., Bixby R.E.,Chavatal V., Cook W.J., The Travelling Salesman Problem, A Computational Study, Princeton University Press, Princeton and Oxford, 2006, 593p. 2. Kızılateş G., Nuriyeva F., A Parametric hybrid method for travelling salesman problem, Mathematical and Computational Applications, Vol.18, No.03, 2013, pp.459-466. 3. Nuriyeva F., Kızılateş G., Berberler M.E., Improvements on heuristic algorithms for solving traveling salesman problem, International Journal of Advanced Research in Computer Science, Vol.4, No.11, 2013, pp.1-8.

DESIGN OF LOW ORDER CONTROLLERS FOR UNSTABLE INFINITE DIMENSIONAL PLANTS Hitay Ozbay1 1

Bilkent University, Ankara, Turkey e-mail: [email protected]

A design method for low order controller design is reviewed for a class of unstable infinite dimensional plants, including systems with time delays, fractional order systems, and systems represented by PDEs. The approach is based on the small gain theorem and requires minimization of an H-infinity norm over a low number of parameters. Gain margin optimization for PD controllers and integral action gain margin optimization for PI controllers are discussed and their relations to non-fragile controller design are illustrated by examples. Keywords: fractional order systems, small gain theorem, PI controllers, PD controllers. AMS Subject Classification: 49K20.

153


ASYMPTOTIC BEHAVIOUR OF EIGENVALUES OF HYDROGEN ATOM EQUATION Etibar S. Panakhov 1, Ismail Ulusoy2 Firat University, Elazig, Turkey Adiyaman University, Adiyaman, Turkey e-mail: [email protected], [email protected] *

2

Consider the singular Sturm-Liouville equation [1,2]  2 2  y ''  2   y  q( x) y   y,  x x

x   0,1

ve   C

(1)

Let’s give the solutions of this equation by integral equation representations [3]:

 ( x,  , q)  x 2 

x 1  x2 t 2   2      q  t       (t ,  , q)dt  30 t x  t 

 ( x,  , q)  c1 x 2  c2 x 1 

1 1  x2 t 2   2      q  t      (t ,  , q)dt  3 x t x  t 

The mapping (, q)   ( x, , q) is analytic from (C1 , C2 , , q)  x ( x, C1 , C2 , , q) is analytic from

3

 L2 0,1  C 0,1 . The mapping

 L2 0,1  C 0,1 .

The main result of this paper is the following theorem: Assume that 0  x1  x2  1 . Theorem

1:If

y

is

a

nontrivial

solution

of

the

equation

(1)

with

2 x2 2  2 y( x1 )  y '( x2 )  by( x2 )  0 , then       b  2 q dx   where q( x)  L 0,1 .   x1 x  1 

Theorem

2 2: If y is a nontrivial solution of the equation (1) where q( x)  L 0,1 , and

if y '( x1 )  y '( x2 )  0 then 2 1 1  x2 q dx   1 / 2  x  x   x2 q dx   2 2 1  x  x1 1   x22 x2

Keywords: hydrogen atom ; asymptotic behaviour; integral equation representations. AMS Subject Classification: 34A55, 34B24, 34D20.

References 1. Poeschel J., Trubowitz E., Inverse Spectral Theory, Academic Press, San Diego, CA, 1987. 2. Carlson R., Inverse spectral theory for some singular Sturm-Liouville problems, J.Differential Equations, 106, 1993, pp.121-140. 154


3. Panakhov E.S., Yilmazer R., On the Determination of the Hydrogen Atom Equation from Two Spectra, World Applied Sciences Journal, Vol.13, No.10, 2011, pp.2203-2210.

DEVELOPMENT OF THE METHOD FOR SOLVING COMBINATORIAL OPTIMIZATION PROBLEMS AND ITS APPLICATION FOR DATA COMPRESSION S.D. Pogorilyy1, A.V. Potebnia1 1

Kyiv National Taras Shevchenko University, Kyiv, Ukraine e-mail: [email protected], [email protected]

Introduction. In many spheres of the modern scientific research the combinatorial optimization problems are widely spread which require choosing the optimal solution among the given set of variants. The most important of them include the travelling salesman problem, minimum Steiner tree, graph partition problem and many others. The developing of the artificial molecular complexes and designing of the complicated computer networks are united by the necessity to solve the combinatorial optimization problems [1, 2]. The most part of the combinatorial optimization problems belongs to the NP-hard class (the calculating expenses for their solution are exponential). So there appears the necessity to develop the new methods of the effective distribution of resources used for the optimal solution search. For the solution of the NP-hard problems there is a great number of methods and artificial intellectual systems. They include heuristic algorithms, neuron networks, methods of the bee colonies, genetic algorithms, etc. As a result, during the last decades the dimensionality of the combinatorial

optimization

problems

which

may be

effectively solved has largely increased. Forming of the mathematical model for the differential data compression. Let us imagine that all the

Fig. 1 – Example of the graph’s vertex set splitting into the hulls

values

ui

of the original file possess the same structure and

consist of the m fields

155

f1 , f 2 ,..., f m . The scheme of the


differential compression proposed by Ernvall needs sorting of the input records set to increase the number of the similar fields between them. After compression the structure only of the first line from the original file remains unchanged. And every next line contains the meaning of those fields which differ from the previous record. Let us consider the simplest case in which every field f i consists of one bit, and the nearby lines of the compressed file differ with meanings of only one field. In this case the result file is the set of the Grey’s codes C n forming the cycle of the travelling salesman in the graph Qn , the vertices of which correspond to the separate records, and the weights of edges are equal

to the Hemming distance between them. The result of compression is the transitional sequence

 (Cn )  t1 , t2 ,..., tn , where t i identifies the position in which the lines u i and ui 1 are different. For the realization of the differential data compression the important stage is the applying of the heuristic methods of the travelling salesman problem solution which have the polynomial complexity. The purpose of this article is to develop the new approach for the NP-hard problems solving by the way of the graph’s vertex set partition to a number of hulls and their junction. The development of the new method for the combinatorial optimization problems solution. Let the undirected graph G = (V, E) is specified. The main stage of the new method is the splitting of the initial graph’s vertex set V into the hulls O  O1 , O2 ,..., On , Oi  V according to the respective rules. All the vertices should be distributed among the formed subsets, n

O

i

i 1

 V аnd Oi   . Any vertex cannot be included to two different hulls: Oi  O j  , i  j .

For the separation of the respective subsets the concept of the minimal convex hull (MCH) is used. It is known that the convex hull of the graph’s vertex set Conv V is the least convex set which includes V. The ways of the minimal convex hulls formation include the Jarvis’s march, Graham’s Scan, Quick Hull and many others [3]. For example, the emphasized vertices of the graph in Fig.1a belong to the minimal convex hull. The proposed procedure of the hulls allocation can be represented by the equation Oi  Conv(V \ O), O  O1 , O2 ,..., Oi1 . This means that at the every stage the formation of the next hull is realized by the searching for MCH of vertex set which belongs to V, but is not included to the previous hulls. The algorithm finishes its execution if all the vertices are distributed among the hulls, V \ O  .

156


Let us consider the example this method applying. Formation of the external hull O1 is realized through MCH construction for the graph’s vertex set (Fig. 1a). Calculation of the next vertex sets is conducted by analogy (Fig. 1b). The received hulls system is shown in Fig. 1c. The next stage requires “sewing together” of the nearby hulls to form the solution. The internal and Fig. 2 – Dependence of the algorithm execution

external cycles Oi and Oi 1 have the smallest length

time on the graph dimensionality and length of

for the respective sets of vertices. The aim of the

the local optimization sections

sewing procedure is the creation of the united hull

Oi  Oi 1 with preservation of the corresponding characteristics. Application of the local search optimization algorithms by means of the consecutive scanning provides the substantial improvement of the received routes [3]. Fig.2 shows the dependence of the algorithm execution time on the graph dimensionality and length of the local optimization sections. Conclusions. The graph’s vertices distribution among the hulls allows to increase the speed of the routes formation substantially. The experimental investigations of the algorithm confirm the expediency of its application in the cycles processing for the graphs of a large dimensionality. The quadratic complexity of the developed method is defined by the operations of the hulls allocation. Such level of complexity is inherent to the simplest “greedy” algorithms which realize the route formation on the basis of the nearest neighbour choosing. However, if the reward for the “greediness” is the creation of the distorted solutions, application of the new method allows receiving the qualitative cycles during small time intervals.

Keywords: combinatorial optimization, differential data compression, minimal convex hull, NP-hard problems, travelling salesman problem, graph. AMS Subject Classification: 05C85, 68R10, 68W10, 68Q25, 97K30.

References 1.

Daniel Lemire, Owen Kaser, Eduardo Gutarra, Reordering rows for better compression: Beyond the lexicographic order, ACM Trans. Database Syst. 37, 3, Article 20 (September 2012), 29 p.

157


2.

Anisimov A.V., Pogorilyy S.D., Vitel D.Yu., About the Issue of Algorithms formalized Design for Parallel Computer Architectures, Appl. Comput. Math., Vol.12, No.2, 2013, pp.140-151.

3.

Potebnia A.V., Pogorilyy S.D., Exploration of data coding methods in wireless computer networks, Fourth International Conference on Theoretical and Applied Aspects of Cybernetics (TAAC), Kyiv, 24 – 28 November, 2014, pp.17 – 31.

QUADRATIC TRANSFORMATIONS WITH APPLICATIONS TO OPTIMIZATION B. Polyak1 1

Institute for Control Science and Skoltech Center for Energy, Moscow, Russia e-mail: [email protected]

Quadratic transformations arise in numerous fields of optimization and control: general quadratic programming, relaxations for discrete optimization, multiobjective minimization, Stheorem for absolute stability, mu-analysis, ellipsoidal techniques and so on. Recent applications in physics (quantum mechanics and power systems) are highly promising. The key problem is convexity of quadratic maps. It has been established for some particular classes of problems, mainly for maps into low-dimensional space. We address a different approach to the problem: given a quadratic transformation, how to recognize if the corresponding image is convex or not. We provide some «convexity/nonconvexity certificates» which can be effectively verified. Another issue is an approximate description of the image or its boundary even for nonconvex case. Some applications to optimization problems arising in power systems will be presented. Keywords: optimization, convexity, nonconvexity, quadratic transformations. AMS Subject Classification: 90C26.

A META-HEURISTIC ALGORITHM FOR THE TWO-DIMENSIONAL STRIP PACKING PROBLEM Furkan Sabaz1, Hakan Kutucu1 1

Karabuk University, Department of Computer Engineering, Karabuk, Turkey e-mail: [email protected]

Cutting and packing problems are combinatorial optimization problems having many applications in the wood, glass, metal and leather industries, as well as in VLSI design, newspaper paging, and container and truck loading. Cutting and packing problems can be classified using different criteria such as dimension (1D, 2D and 3D), shape of items (regular and 158


irregular), orientation (items can be can be rotated by 90o) and guillotine cut constraint. In this paper, we consider two-dimensional regular strip packing problem with a fixed orientation. We are given a set of n rectangular items j=1,…,n, each defined by a width , wj, and a height, hj. The Two-Dimensional Strip Packing Problem (2SP) consists of placing all the items into a bin of width W and infinite height (hereafter called strip) without overlap so as to minimize the height of the strip. In this paper, we propose a meta-heuristic algorithm for solving the 2SP. Our algorithm includes two stages: search and placement. Permutation of bin numbers is obtained with the help of genetic algorithms. A chromosome represents a permutation of rectangles. Initially, a matrix having all entries -1 is created for each chromosome. This matrix is called “Screen matrix”. The Screen matrix values changes according to replacement of bins. Searching is held by processing the values of the matrix. In the search stage, minimum bin height and bin width are important. The placement stage is provided with replacement of a bin into first empty and proper place in the x-axis and y-axis. The reasons why our algorithm gives better results are the fact that searching algorithm is fast and can find even the tiniest space via Screen matrix. This algorithm could be adapted to 3D easily that is our future work. In order to test our genetic algorithm, we have used small and large benchmarks of Hopper and Turton [1]. The parameters used in the genetic algorithm are the population size is 50, the selection rate is 80, the mutation rate is 15 and the maximum number of generations is considered as 1000. Using different crossover methods, better results can be obtained. Our genetic algorithm has been compared with BLF+SA (Bottom Left Fill + Simulated Annealing) algorithm in [2]. Table 1 shows the results for the benchmarks from 60 to 240 size width. The column Relative Error2 indicates the relative distances between the lowest packing height found by BLF+SA in [2] and the height of the optimal solution. Our algorithm outperforms BLF+SA. Table 1: Computational Results.

Problem

Number of

Width of

Optimum

Best

Relative

Relative

Time

rectangles

the Strip

Height

Height

Error

Error2

(min.)

C4-p1

49

60

60

61

1.6

3

3

C4-p2

49

60

60

61

1.6

3

3

C4-p3

49

60

60

61

1.6

3

3

C5-p1

73

60

90

91

1.1

3

5

159


C5-p2

73

60

90

91

1.1

3

5

C5-p3

73

60

90

91

1.1

3

5

C6-p1

97

80

120

122

1.6

3

9

C6-p2

97

80

120

122

1.6

3

9

C6-p3

97

80

120

122

1.6

3

9

C7-p1

196

240

164

167

1.82

4

14

C7-p2

197

240

163

166

1.84

4

15

C7-p3

196

240

164

167

1.82

4

15

Keywords: two-dimensional bin packing, genetic algorithms, heuristics, efficient implementation. AMS Subject Classification: 52C15, 90C59 90C27.

References 1. Hopper, E and Turton, BCH, An empirical investigation of metaheuristic and heuristic algorithms for a 2D packing problem, Eur. J. Oper. Res., Vol.128, 2001, pp.34-57. 2. http://dip.sun.ac.za/~vuuren/repositories/levelpaper/HopperAndTurtonData[1].htm

ON AN EXTREMAL PROBLEM FOR GOURSAT-DARBOUX TYPE INCLUSION IN INFINITE DOMAIN M.A. Sadygov1, J. J. Mamedova1, H.S. Akhundov1 1


In the work the necessary condition of an extremum for an extreme problem of differential inclusions of Goursat-Darboux type in infinite area is obtained. There are also studied continuous dependence of the solution on perturbation of differential inclusion of Goursat-Darboux type. In the work the existence of the generalized solution of problem of Goursat-Darboux are also studied. Let a : [0,  )  [0,  )  R n  compRn , b1 : [0,  )  R n  compRn , n b2 : [0,  )  R n  compRn , M 0  R n be non-empty, where compR be the set of all non-empty

compact subsets R n . Let's designate the set of all absolutely continuous functions defined in [0,  )  [0,  ) with the finite norm 160




u (  )  u (0,0) 







u t ( ,0) d 

0



u s (0, ) d 

0

u

 , ) dd

ts (

0 0

by An ([0,  )  [0,  )) . Further the equalities and the inclusions connected with measurable functions or mapping are understood as almost everywhere. Let's consider a problem u ts (t , s)  a(t , s, u (t , s)), u t (t ,0)  b1 (t , u (t ,0)), u s (0, s)  b2 ( s, u (0, s)), u (0,0)  M 0

(1)

at (t , s)  [0,  )  [0,  ) . The function u(  )  An ([0,  )  [0,  )) satisfying the problem (1) is called the solution of a problem (1). We designate set of solutions of a problem (1) by A . Let g : [0,  )  [0,  )  R n  R be normal integrant. The solution of inclusion (1) minimizing a functional 

J (u ) 

  g (t, s, u(t, s))dtds

(2)

0 0

among all solutions of a problem (1) will be called optimal. It is required to find necessary conditions of optimality of the solution of a problem (1), (2). Let the following conditions be satisfied: 1)

There

are

functions

 (  )  L1 ([0,  )  [0,  )),

1 (  )  L1[0,  ) ,

( , )  f ( , , z), ( , )  F ( , , z),   c1 ( , x), , where  (t , s)  0, 1 (t )  0 and  2 (s)  0 at t  [0,  )

and

s  [0,  )

such

that

a(t , s, u)   (t , s)(1  u ) ,

b1 (t , u)  1 (t )(1  u ) ,

b2 (s, u)   2 (s)(1  u ) at u  R n .

2) M 0 is non-empty and there exists the number r  0 such that M 0  sup x  r . xM 0

3) Multi-valued mappings a, b1 and b 2 satisfy a Caratheodory condition. 4) The integrant g satisfies a Caratheodory condition. Let 1 (t )  1 (t )(1  r1e 1 )  e t ,  2 (s)   2 (s)(1  r2 e 2 )  e  s , where 



0

0

1    1 ( )d ,  2    2 ( )d , r1  r  1 , r2  r   2 . Let's put

161


t

s

0

0

1 (t )   1 ( )d ,  2 ( s)    2 ( )d , 1  11 : [0, d1 )  [0,) ,  2   21 : [0, d 2 )  [0,) , where d1  (1  r1e 1 )1  1 , d 2  (1  r2 e  2 ) 2  1 .

~

Let   u  (11 , 21 )  u  (1 ,  2 ) . Let's note that functions 1  [0, d1 ]  R and ~ ~ ~  2  [0, d 2 ]  R are absolutely continuous at d1  [0, d1 ) and d 2 [0, d 2 ) . Putting t  1 ( ) at

  [0, d1 ) , s   2 ( ) at   [0, d 2 ) we will receive that the problem (1), (2) is equivalent to the following problem: it is required to minimize a functional d1 d 2

  g ( ( ),  1

 ), ( , ))[1 (1 ( ))]1 [  2 ( 2 ( ))]1 d d

(3)

2(

0 0

under conditions  , ( , )  a(1 ( ),  2 ( ), ( , ))[1 (1 ( ))]1 [  2 ( 2 ( ))]1 ,

 ( ,0)  b1 (1 ( ), ( ,0))[1 (1 ( ))]1 ,  (0, )  b2 ( 2 ( ), (0, ))[ 2 ( 2 ( ))]1 ,  (0,0)  M 0

(4)

at ( , )  [0, d1 ]  [0, d 2 ] . Having designated F ( , , x)  a(1 ( ),  2 ( ), x)[1 (1 ( ))]1[ 2 ( 2 ( ))]1 , f ( , , x)  g (1 ( ),  2 ( ), x)[1 (1 ( ))]1[ 2 ( 2 ( ))]1 ,

c1 ( , x)  b1 (1 ( ), x)[1 (1 ( ))]1 ,

c2 ( , x)  b2 ( 2 ( ), x)[ 2 ( 2 ( ))]1

the problems (3), (4) can be written in the following form I ( ) 

d1 d 2

  f ( , , ( , ))d d  min

(5)

0 0

 , ( , )  F ( , , ( , )),  ( ,0)  c1 ( , ( ,0)),  (0, )  c2 ( , (0, )),  (0,0)  M 0 ,

(6)

where ( , ) [0, d1 ]  [0, d 2 ]. Let's

put

 ( , , z, )  inf{u   : u  F ( , , z)}, q1 ( , x, y)  inf{ z  y : z  c1 ( , x)} ,

q2 ( , x, y)  inf{u  y : u  c2 ( , x)}, q0 ( x)  inf{u  x : u  M 0 } .

Theorem 1. Let (, )  f (, , z), (, )  F(, , z),   c1 (, x),

  c 2 (, x) be measurable, M 0 , F(, , z), c1 (, x) and c 2 (, y) be nonempty, and compact at

(, , z) [0, d1 ]  [0, d 2 ]  R n ,

(, x) [0, d1 ]  R n ,

162

(, y) [0, d 2 ]  R n ,

there

be

functions


k(  ), k(  )  L1 ([0, d1 ]  [0, d 2 ]) , d1 d 2

  k(, )d d  1 or

k 1 (  ), k1 (  )  L1 [0, d1 ] ,

k 2 (  ), k 2 (  )  L1 [0, d 2 ] ,

where

k(, )  k1 ()k 2 () such that

0 0

 x (F(, , x), F(, , y))  k(, ) x  y ,

 x (c1 (, x), c1 (, y))  k1 () x  y ,

 x (c 2 (, x), c 2 (, y))  k 2 () x  y ,

f (, , x)  f (, , y)  k(, ) x  y

at x, y  R n . Then, if w  A n ([0, d1 ]  [0, d 2 ]) is the solution of a problem (5), (6), there is a number

m0

and

functions

v(  )  A n ([0, d1 ]  [0, d 2 ]) ,

where

v(d1 , )  v(, d 2 )

at

(, ) [0, d1 ]  [0, d 2 ] , v1 (  )  W1n,1 [0, d1 ], v 2 (  )  W1n,1 [0, d 2 ] such that 1) (v  (, ),  v(, ))  (f (, , w(, ))  m(, , w(, ), w  (, )) , 2) (v  (,0)  v 1 (),  v1 ())  mq1 (, w(,0), w  (,0)) , 3) (v  (0, )  v 2 (),  v 2 ())  mq 2 (, w(0, ), w  (0, )) , 4) v(0,0)  v1 (0)  v 2 (0)  mq 0 (w(0,0)), 5) v1 (d1 )  v(d1 ,0)  0, v 2 (d 2 )  v(0, d 2 )  0, v(d1 , d 2 )  0 , where g(x ) denotes Clarke subdifferential (see [1,2]) of the function g in a point x .

Keywords: differential inclusions, Clarke subdifferential, extreme problem. AMS Subject Classification: 05C35, 52A20.

References 1. Sadygov M. A. The Nonsmooth Analysis and its Applications to Extreme Problem for Inclusion of Goursat-Darboux Type, Baku, 1999, 135 p. 2. Clarke F. Optimization and Nonsmooth Analysis. M.:Наука, 1988, 280 p.

AN EXTREME PROBLEM FOR A VOLTERRA TYPE INTEGRAL INCLUSION M.A. Sadygov1 1


In the work, we have studied the dependencies of the solutions to integral inclusions from perturbation and investigated an extremal problem for integral inclusions. We obtained necessary

163


and sufficient minimum conditions for extremal problems of Volterra type convex inclusions. We also studied a nonconvex extremal problem for the Volterra type inclusion. We obtained a high order necessary condition in the extremal problem for the Volterra type inclusion. We also studied (see[1]) a extremal problem for the Fredholm and Hammerstein type inclusions. 1. Dependence of the solution to the integral inclusion from perturbation. Let R n be the n -dimensional Euclidean space. The set of all nonempty compact (convex compact) subsets in R n we will designate as compRn (convRn ) ; k : [t 0 , T]2  Mn is the continuous matrix function, where with M n being the set of all square n  n matrices of real elements (bij ) ; z : [t 0 , T]  R n the continuous function; F : [t 0 , T ]  R n  compRn the setvalued mapping. Let us consider a problem for inclusion t

u ( t )  F( t ,  k (t , s)u (s)ds  z( t ))

(1)

t0

The function u()  Ln1[t 0 , T] satisfying (1) we will call the solution to problem (1). n

n

Let a  max k (t, s)  max  k i , j (t, s) , if k : [t 0 , T]2  Mn is the continuous matrix t ,s[ t 0 ,T ] t ,s[ t 0 ,T ] i1 j1

function. Theorem 1. Let k : [t 0 , T]2  Mn be the continuous matrix function, z : [t 0 , T]  R n the continuous function, F : [t 0 , T]  R n  compRn the multivalued mapping, t  F(t, x) is measurable on t , and there exist a summable function M(t )  0 such that x (F(t, x), F(t, x1 )))  M(t ) x  x1 for x, x1  R n . Moreover, let ()  L1[t 0 , T] and u()  Ln1[t 0 , T] be such that t

d(u ( t ), F( t ,  k (t , s)u (s)ds  z( t )))  ( t ) t0

for t  [t 0 , T] . Then there exists such a solution u()  Ln1[t 0 , T] to problem (1) that t

t

t

t0

t0

t0

t

u (t )  u ( t )  ( t )  aM( t )  e m ( t )m (s )(s)ds

m ( t )m ( s ) (s)ds ,  k(t, s)u(s)ds   k(t, s)u (s)ds  a  e

t0

t

for t  [t 0 , T] , where m( t )  a  M(s)ds . t0

2. Convex extremal problem for integral inclusions.

Let k : [t 0 , T]2  Mn be the

continuous matrix function, z : [t 0 , T]  R n the continuous function. Hereafter we will assume

164


that f : [t 0 , T]  R n  (,] is the normal convex integrant,  : R n  (,] the convex function. Let t 0  T, F : t 0 , T R n  compRn is the multivalued mapping.

The problem of minimization of the functional T

T



J (u )   ( k (T , s)u ( s)ds  z (T ))  t0



t



f (t , k (t , s)u ( s)ds  z (t ))dt

t0

(2)

t0

is considered under the following constraints t



u (t )  F (t , k (t , s)u ( s)ds  z (t )), t [t 0 , T ] , u()  L1n [t 0 , T ] .

(3)

t0

 0, z  F (t , x), we have that problem (2) and (3) is  , z  F (t , x)

Introducing the notation  (t , x, z )  

equivalent to the minimization of the functional T



J 1 (u )   ( k (T , s)u ( s)ds  z (T ))  t0

among

T

t

T

t

t0

t0

t0

t0

 f (t,  k (t, s)u(s)ds  z(t ))dt   (t,  k (t, s)u(s)ds  z(t ), u(t ))dt

u()  L1n [t 0 , T ] .

all functions

Let the mapping

t  grFt  {( x, y) : y  F (t , x)} be

measurable on [t 0 , T ] , the set grFt be closed and convex for almost all t  [t 0 , T ] and F (t , x) be compact for all (t , x) . From here it follows that  (t,x, z) is a convex normal integrant on [t 0 , T ]  ( R n  R n ) . Let us consider the following functional T

T



Ф(u,  )   ( k (T , s)u ( s)ds  z (T ))  t0



t

T



t





t0

t0

f (t , k (t , s)u ( s)ds  z (t ))dt   (t , k (t , s)u ( s)ds  z (t ), u (t )   (t ))dt,

t0

t0

(u,  ) . The problem (2) and (3) is called stable, if where  ()  L1n [t 0 , T ] . Let h( )  inf n uL1 [t0 ,T ]

h(0) is finite and function h is subdifferentiable at zero. n

Lemma 1. Let F : [t 0 , T ]  R n  2 R ; the mapping t  F (t , x) be measurable on [t 0 , T ] ; the mapping x  F (t , x) be closed and convex for almost all t  [t 0 , T ] , i.e. grFt be closed and convex for almost all t  [t 0 , T ] ; there exist such a summable function F (t , x)   (t ) (1  x )

for

x  Rn ;

there

exist

a

t

u 0 (t )

to

the

that

problem

t



u 0 (t )  F (t , k (t , s)u 0 ( s)ds  z (t ))

such

that



x0 (t )  k (t , s)u 0 ( s)ds  z (t )

t0

dom Ft  {x : F (t , x)  }

solution

 (t )

belongs

to

t0

coupled

with

some



tube,

i.e.

{x : x0 (t )  x   }  dom Ft ;

f : [t 0 , T ]  R n  (,] the normal convex integrant;  : R n  (,] the convex function and

165


t

inf n

uL1 [t0 ,T ]

J 1 (u ) is finite; the function f (t , k (t , s)u 0 ( s)ds  z (t )  y) be summarized for y  R n , y  r ,  t0

T

where r  0 , and function  () be continuous at the point

 k (T , s)u

0 ( s ) ds

 z (T ) . Then the

t0

function h is subdifferen-tiable at zero, i.e. problem (2) and (3) is stable. Let   R n . Assume  0 (t , x, )  infn{( z  )   (t , x, z)}  inf{(z |  ) : z  F (t , x)} , where zR

inf    . n

Theorem 2. Let F : [t 0 , T ]  R n  2 R ; the mapping t  F (t , x) be measurable on [t 0 , T ] ; the mapping x  F (t , x) be closed and convex for almost all t  [t 0 , T ] ; f be the normal convex n integrant on [t 0 , T ]  R n ;  the convex function on R ; k : [t 0 , T ] 2  M n the continuous matrix

function; z : [t 0 , T ]  R n the continuous function. For the function u ()  L1n [t 0 , T ] to minimize the functional (2) among all the solutions to the problem (3), it is sufficient that there exist u1 (), u 2 ()  Ln1[t 0 , T] and b R n such that t

T

1) u1 (t )  f (t ,  k (t , s)u (s)ds  z(t )) ,

2) b   ( z (T )   k (T , s)u (s)ds) ,

t0

t0 t

T

t0

t

3) u2 (t )   0 (t ,  k (t , s)u (s)ds  z(t ),  k ( , t )t (u1 ( )  u2 ( ))d  bk(T , t )) , t

T

T

t0

t

t

4)  0 (t , z(t )   k (t , s)u (s)ds,  k ( , t )t (u1 ( )  u2 ( ))d  bk(T , t ))  (u (t )  k ( , t )t (u1 ( )  u2 ( ))d  bk(T , t ))  t



  (t , z (t )  k (t , s)u ( s)ds, u (t )), t0

and if the condition of lemma 1 is satisfied, then conditions 1)-4) become necessary, where k ( , t ) t is the transpose of the matrix k ( , t ).

3. Nonconvex extremal problem for integral inclusions. Let k : [t 0 , T ]2  M n be the continuous matrix function; z : [t 0 , T ]  R n the continuous function, i.e. z()  C n [t 0 , T ] . Hereafter we will assume that f : [t 0 , T ]  R n  R n  (,] is the normal integrant and  : R n  (,] is the function. Let t 0  T , F : t 0 , T  R n  compRn be the multi-valued mapping. We consider the following problem of minimization of the functional 166


T

T



J (u )   ( k (T , s)u ( s)ds  z (T ))  t0



t



f (t , k (t , s)u ( s)ds  z (t ), u (t ))dt ,

t0

(4)

t0

under the following constraints t



u (t )  F (t , k (t , s)u ( s)ds  z (t )), t  [t 0 , T ] , u()  L1n [t 0 , T ] .

(5)

t0

Let  (s, x, y)  inf{ z  y : z  F (s, x)} and consider the minimization of the functional T

T



J r (u )   ( k (T , s)u ( s)ds  z (T ))  t0



t

T



t





t0

t0

f (t , k (t , s)u ( s)ds  z (t ), u (t ))dt  r  (t , k (t , s)u ( s)ds  z (t ), u (t ))dt

t0

t0

among all the functions u()  L1n [t 0 , T ] . Theorem 3. If u ()  L1n [t 0 , T ] is the solution to the problem (4) and (5), F : [t 0 , T ]  R n  t

comp R n  

and t  F (t , x) are measurable on t , x (t )   k (t , s)u ( s)ds  z (t ), there exist t0

k ()  L1[t 0 , T ] ,

k1  0 ,

M ()  L1[t 0 , T ] ,





B( x (t ), )  dom Ft  x  R n : F (t , x)  

at

f (t , x1 , y1 )  f (t , x 2 , y 2 )  k (t ) x1  x 2  k1 y1  y 2 ,

k2  0 ,

t  [t 0 , T ]

and and

 0

such

that

 ( z)   (u)  k 2 z  u ,

 X ( F (t , x1), F (t , x2 ))  M (t ) x1  x2

for z, u  B( x (T ), ) , x1 , x2  B( x (t ), ) , y1 , y 2  R n . Then there exist a number r0  0 such that u (t ) minimizes the functional J r (u ) in D for r  r0 , where  D  u ()  L1n [t 0 , T ] : u ()  u () 

L1n [ t 0 ,T ]

   ,   (1  a(e m(T )  ae m(T )  M (t )dt))(a  M (t )dt  1)  a ,  t t T

T

0

0

t



m(t )  a M ( s)ds . t0

Theorem 4. Let the condition of the theorem 3 be satisfied and the function u (t ) among all solutions to the problem (5) minimizes the functional (4). Then there exists u  ()  L1n [t 0 , T ] and b  Rn such that T

t

t

t

t0

t0

1) (u (t ), k ( , t )t u ( )d  bk(T , t ))  C ( f (t ,  k (t , s)u (s)ds  z(t ),u (t ))  r (t ,  k (t , s)u (s)ds  z(t ),u (t ))) , T

2) b  C ( z(T )   k (T , s)u (s)ds) , where  C g (x ) is Clarke subdifferential of the function g at the t0

point x . 167


Keywords: Euclidean space, Volterra type inclusion, minimization. AMS Subject Classification: 05C35, 52A20. References 1. Sadygov M.A., An Extremal Problem for Integral Inclusion, Preprint No.1, Baku, 2013, 129 p.

HIGH ORDER OPTIMALITY CONDITIONS FOR P-REGULAR INEQUALITY CONSTRAINED OPTIMIZATION PROBLEM Ewa Szczepanik1, Alexey Tret'yakov2,3,4 1

Institute of Computer Science, Siedlce University of Natural Sciences and Humanities, Siedlce, Poland 2 Department of Mathematics and Physics, Siedlce University of Natural Sciences and Humanities, Siedlce, Poland 3 System Research Institute, Polish Academy of Sciences, Warsaw, Poland 4 Dorodnitsyn Computing Center, Russian Academy of Sciences, Moscow, Russia e-mail: [email protected], [email protected]

In this talk, the conjugate cone is described for the general singular case in a constrained optimization problem with p-regular constraints and Kuhn– Tucker-type optimality conditions are constructed with the use of p-factor-operators for the general mathematical programming problem as described in [1]. Previously, necessary optimality conditions were proposed for some classes of optimization problems with irregular inequality constraints. For example, the case of a singularity of only up to the second order was considered or the higher derivative operators of the constraint functions were assumed to have nontrivial p-kernel, etc. In other cases, optimality conditions were written in the form of conjugate cones with no description. Keywords: nonlinear optimization, p-factor operator, p-regularity, singularity, necessary and sufficient conditions, nonregular constraints. AMS Subject Classification: 90C46, 90C30, 49K99, 49N60.

References 1. Karmanov V. G., Denisov D. V., Tret`yakov A. A., Simulation and Analysis in Decision Making Problems, Vychisl.Tsentr Ross. Akad. Nauk, Moscow, 16:16–-24, 2003 ( in Russian).

168


PLACEMENT OF OPTIMAL CAPACITOR BY PSO ALGORITHM Farzad Mohammadzadeh Shahir1, Parisa Mohammadalizadeh 2, Morteza Farsadi 3 Farhad Nazari Heris4 1

Faculty of Electrical and Computer Engineering, Islamic Azad University, Khosroshar Branch, Khosroshahr, Iran 2 Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran 3 Faculty of Electrical and Computer Engineering, University of Urmia, Urmia, Iran 4 Faculty of Electrical and Computer Engineering, Islamic Azad University, Urmia Branch, Urmia, Iran e-mail: [email protected]

In this paper, a research regarding finding an optimal place for installing the capacitor under the algorithm of group of particles (PSO) has been presented in radial networks by considering the size and price of different capacitors. In this paper, the calculations and results for a typical radial network have been calculated by considering harmonic distortion calculation (HDF) and its results have been stated. Based on this, all the losses have been calculated and compared too. The results of this study have been calculated by MATLAB software. Keywords: capacitor, placing capacitor, PSO algorithm, radial network. AMS Subject Classification: 80M50.

OPTIMAL CONTROL OF THE MOBILE SOURCE FOR PROCESS OF INTRA SHEETED BURNING IN OIL PRODUCTION‡ Rafig A. Teymurov1 1

Institute of Mathematics and Mechanics, ANAS, Baku, Azerbaijan e-mail: [email protected]

In this article, we consider a problem of optimal control of the mobile source for process of intra sheeted burning, described by heat conductivity equation. Sufficient conditions for Frechet differentiability of the performance criterion and an expression for its gradient were determined.

The necessary conditions for optimality were established in the form of the

pointwise and integral principles of maximum. Let the state of controlled process be described by functions T (r , t ) and  ( z, t ) . We will assume that function T  T (r , t ) on a domain   (r, t ) : r0  r  R, 0  t  T  satisfies the following parabolic equation [1-2] ‡

This paper is performed with financial support of a grant of Found of Science of the State Oil Company of the Azerbaijan Republic (SOCAR) for 2014.

169


 pl

1   T  1 H  T ,  qT   (r  rf (t ))  C pl  pl r  r r  r  r r rf (t ) t

(1)

with boundary and initial conditions T (r0 , t )  Tr (t ), T ( R, t )  TR (t ), 0  t  T ,

(2)

T (r, o)  T0 (r ), r0  r  R,

(3)

where  pl is the coefficient of heat conductivity of layer; function H  H (T ) 

Qh (t )  hCh T is 2h

characterizes convective transfer of heat;  pl ,  h are the layer and air respectively density;

qT  qT (t )  2

1  ( z, t ) h

z

is the thermal losses of layer and surrounding breed;   z 1

Qh  hGN 2h

is the coefficient of intensity of a thermal emission at the front burning; G is the warmth of combustion of fuel on oxygen; N is the mass fraction of oxygen in air;

C pl is the mass

thermal capacity of layer; h is the layer thickness;  () is the Dirac's function; r0 , R are the respectively radiuses of a zone of burning and evaporation zone; Tr , TR , T0 are the given functions; rf (t ) 

1 hk

t

 Q ( )d h

is law of movement of the front of burning; k is the volume

0

of the air necessary for burning out of coke of 1m3 layer; Qh  Qh (t )  L2 (0, T ) is the control function, characterizing amount of pressurized air in layer. We also assume that the function

   ( z, t ) characterizes temperature of circumambient around layer of solid and is the solution of the following Cauchy problem

1

 2  ,  C11 2 z t

(4)

 ( z, 0)  0 ,

(5)

where  0 is the given number; 1  z   ; 1 , C1 , 1 are the heat conductivity respectively a coefficient, a mass thermal capacity and breed density. The function Qh (t ) is called the control. We will enter a set of admissible controls U  Qh  Qh (t )  L2 (0, T ) : 0  Qh  A and consider the functional r T





T





i 2 ~ ~ J (u )  1   T (r , t )  T (r , t ) drdt   2  Qh (t )  Q(t ) dt ,

rf 0

0

170

(6)


~ ~ where A  0 is the given number; 1 ,  2  0, 1   2  0 are the given parameters; T (r , t ), Q(t )

are the given functions. We pose the following problem: needed is to determine a control Qh (t ) from the set U and the functions T (r , t ) and  ( z, t ) such that under constraints (1)-(5) the functional (6) assumes the least possible value [3-5]. Here we consider the variation method for the solution of the problem of optimal control of mobile source. For the problem of optimal control considered in the present paper, the theorem of existence and uniqueness of solution was proved and the sufficient conditions for Frechet differentiability of the performance criterion and an expression for its gradient were determined, and the necessary conditions for optimality in the pointwise form and integral principles of maximum are established.

Keywords: mobile source, necessary optimality conditions, function of Hamilton-Pontryagin, maximum principle AMS Subject Classification: 49J20, 49K20.

References 1. Amelin I.D., Intra sheeted burning, Moscow, Nedra, 1980. 2. Zheltov Yu.P., Development of oil fields, Moscow, Nedra, 1998, 365p. 3. Butkovsky A.G.

Pustylnikov L.M.,

The Theory of Mobile Control of Systems with the

Distributed Parameters, Moscow, Nauka, 1980. 4. Teymurov R.A., On a problem of optimal control of mobile sources, Automation and Remote Control, 2013, Vol.74, No.7, pp.1082-1096. 5. Teymurov R.A., The problem of optimal control of moving sources for singular heat equation, Caspian Journal of Applied Mathematics, Ecology and Economics, 2013, Vol.1, No.1, pp.104113.

171


NECESSARY CONDITIONS OF OPTIMALITY OF THE GENERALIZED CONTROLS IN THE SYSTEM DESCRIBED BY THE DIRICHLET PROBLEM М. А. Yagubov1, A.A. Yagubov2 1

Baku State University, Baku, Azerbaijan National Aviation Academy, Baku, Azerbaijan e-mail: [email protected]

2

For processes described by the Dirichlet problem for elliptic equations are derived necessary conditions for optimality of generalized controls in the presence of functional limitations. Let u 0 x  measurable on D , bounded vector- function with valued from U  R r , where D  R n with a boundary from C 2 , U -is given compact set from R r .





The pair of z 0 x , u 0 x  we will name a admissibility pair, if z  z 0 x  the solution of problem





z  f x, z, z x , u 0 x  ,

1

z D  0

2

satisfying to inequalities

J i z, u    Fi x, z, z x , u dx  ci ,

3

i  1,2,..., m

D

 z z 2 2 ,...,   2  ...  2 operator of Laplace, z x   xn x1 x n  x1

where



  



If z 0 x , u 0 x  admissibility pair, then u  u 0 x  will name an admissibility controls. The set of all admissibility controls we will designate  U . The problem of determination of a minimum of functional

J 0 z, u    F0 x, z, z x , u dx

4

D

set of all admissibility pair will name an initial problem. Weakly measurable, finite family of probabilistic measures of Radon  x is weak concentrated on U we will name the generalized controls. The pair of z 0 x ,  x0  we will name the admissibility generalized pair, if z 0 x  solution of problem

172


z  f x, z, z x , u ,  x0

5

z D  0

6

satisfying to inequalities

I z,     Fi x, z, z x , u ,  x dx  ci ,

7 

i  1,2,..., m

D

and  x0 - by the admissibility generalized controls, where ,  x   d x U

The set of the admissibility generalized controls we will designate U . Determination the problem of minimum of functional I 0 z,     F0 x, z, z x , u ,  x dx D

on the set of all admissible generalized pairs we will name a convexification problem, admissibility pair of z 0 x ,  x0 , delivering minimum I 0 z,   the optimal generalized pair,  x0 by the optimal generalized controls. In this work is included the necessary condition of optimality the generalized controls. Let l be an arbitrary natural number ,  j , j  1,2,..., l arbitrary elements from U , а

 j x  non-negative measurable functions on D , satisfying conditions

l

 x   1 j 1

j

Totality of q  l ,  j , j x , j  1,2,..., l we will name an admissibly set. For set q we will consider a problem l

8

y  f x, y, y x , u ,  x0    j x  f x, y, y x , u ,  j   x0 , j 1

9

y D  0 and on the solution of this problem we consider the functional l  I j  y,      Fi x, y, y x , u ,  x0    j x  Fi x, y, y x , u ,  j   x0 j 1 D

 dx , i  0,1,2,..., m 

After the calculation of increment of the functional, is shown, that their variations look like

I i z,  0      j x  H i x, z 0 , i , u ,  j   x0 dx , i  0,1,2,..., m , l

D j 1

where

H i x, z, i , u    Fi x, z, z x , u    i f x, z, z x , u  ,

i  0,1,2,..., m is a solution of the problem 173

i  0,1,2,..., m ,

and i x 


 i   i

n Fi x, u  0 f x, u  0  , x  , x   z z k 1 xk

i

D

 Fi x, u  0 f x, u  0 , x  , x  i z xk z xk 

 , 

10 11

 0.

It is proved Theorem 1. Let z 0 x ,  x0  be an optimum generalized pair,  i x  i  0,1,2,..., m of decision of problems 10 , 11 . Then there are not all equal to the zero simultaneously numbers of o  0, 1  0, ..., m  0 , that for all   U the following inequality takes place m

  i 0

i

H i x, z 0 x , i x , u ,    x0 dx  0 ,

12

D m

m

i 0

i 0

We take H x, z, ,  , u    i H i x, z, i , u  ,  x    i i x  . Then condition 12 may be written as









13

sup  H x, z 0 x , x ,  , u ,  dx   H x, z 0 x , x ,  , u ,  x0 dx

U D

D

Condition 13 is called the integral condition of maximum. Definition. We will say that the generalized control  x0  U along z 0 x  ,  x  satisfies the condition of a maximum of Pontryagin, if z 0 x ,  x0  admissibility generalized pair and for almost all x  D









H x, z 0 x , x ,  , u ,  x0  s u p H x, z 0 x , x ,  , u ,  , where a 

supremum is taken on all probabilistic measures concentrated on U . Theorem 2. Integral condition 13 and Pontryagin maximum condition are equivalent. Keywords: admissible pair, admissible control, generalized pair, generalized controls, probabilistic measures. AMS Subject Classification: 35J05, 49J20.

174


NECESSARY CONDITIONS OF OPTIMALITY OF THE SINGULAR WITH RESPECT TO COMPONENTS CONTROLS IN THE GOURSAT-DARBOUX SYSTEMS Sh.Sh. Yusubov1 1


In this work developing the technique proposed in [1] we introduce the definition of the singular with respect to some components in the sense of the Pontryagin's maximum principle, and quasi-singular with respect to other components control and on this basis, a new scheme is offered to deriving necessary optimality conditions. The necessary conditions of optimality are obtained for the singular with respect to some components controls. Let in the domain

D  {(t , x) : t  (t 0 , t ), x  ( x0 , x1 )} the controlling process be described

by the system of hyperbolic equations z tx  f (t , x, z, z t , z x , u)

(1)

with conditions z (t , x0 )  1 (t ), t  T  [t 0 , t1 ],

z (t 0 , x)   2 ( x), x  X  [ x0 , x1 ],

1 (t 0 )   2 ( x0 ).

Here z (t , x)

(2)

is n  dimensional state vector; u(t, x) is r-dimensional control vector;

f (t , x, z, p1 , p2 , u) is a given n  dimensional vector-function, continuous relative to the set of

arguments together with partial derivatives up to the second order with respect to p ( p  z, p1 , p2 ));1 (t ),  2 ( x) are n  dimensional vector-functions continuously differentiable

on T, X correspondingly. As a set of admissible controls we consider the set of piece-wise continuous rdimensional vector-functions u(t , x), (t , x)  D , taking values from the given nonempty bounded set U u(t , x) U  R T ,

(t , x)  D.

(3)

It is supposed that to each admissible control u(t, x) corresponds the only absolutely continuous solution z(t, x) [2] of the problem (1),(2), defined on D. The problem is: minimize the functional S (u)   ( z(T1 , X 1 ),..., z(Tk , X k )),

175

(4)


subject to (1),(2), where  ( z1 ,..., z k ) is given twice continuously differentiable scalar function; (Ti , X i )  D, i  1, k , are given points, moreover t 0  T1  T2  ...  Tk  t1 , x0  X 1  X 2  ...  X k  x1 .

Admissible control u(t, x), which is a solution of the problem of minimization of the functional (4) with conditions (1)-(3), is called to be an optimal control , corresponding solution z(t, x) - an optimal trajectory and the process (u(t , x) z(t , x)) - an optimal process. For the optimality of the admissible process ( (t , x),(t , x), z(t , x)) in the problem (1)-(4) it is necessary the fulfillment of: (1) Pontryagin's maximum condition  H (t , x)  H (t , x, p(t , x), ,  (t , x), (t , x))  H (t , x, p(t , x), (t , x),  (t , x), (t , x))  0, ( ,  (t , x)) U , (t , x) [t 0 , t1 )  [ x0 , x1 ),

(5)

(2) Differential maximum principle H  (t , x)(  (t , x))  0,

( (t , x),) U ,

(t , x) [t 0 , t1 )  [ x0 , x1 ).

(6)

Here H (t , x, p, u, )   f (t , x, p, u);  (t , x) is of the adjoint variables defined by the relation k

 (t , x)    (Ti , X i ; t , x) ( z (T1 , X 1 ),..., z (Tk , X k )) / z i , i 1

where (‘) stands for transpose, and  (t , x; , s) is a solution of the integral equation t x

 (t , x; , s)  E     (t , x;  , ) f z ( , )dd   s

t

x





  (t , x;  , s) f z x ( , s)d   (t , x; , ) f zt ( , )d , s

and  (t , x; , s)  0 by t   or x < s, E is a unit n  n matrix. Definition 1. Admissible control ( (t , x),(t , x)) is called quasi-singular relative to the component w if there exists a set U 0  U , such that the condition H w (t , x)(w  w(t , x))  0,

(t , x) [t 0 , t1 )  [ x0 , x1 )

(7)

is satisfied identically relative to ( (t , x), w) U 0 , where U 0 \ ( (t , x), w(t , x))  , (t , x)  D . For the optimality of the quasi-singular with respect to the component w control ( (t , x), w(t , x)) in the problem (1)-(4) it is necessary the fulfillment of the inequality (w  w(t , x))H ww (t , x)(w  w(t , x))  0,

( (t , x), w) U 0 , (t , x) [t 0 , t1 )  [ x0 , x1 ).

(8)

Considering the possibility of degeneracy of the optimality conditions (5) and (8) we introduce:

176


Definition 2. Quasi-singular with respect to the component w control ( (t , x), w(t , x)) is called to be singular with respect to the component v in the sense of Pontryagin's maximum principle and strong quasi-singular with respect to the component w, if there exist sets U 1  U 0 and U 2  U such that the following conditions are satisfied  H (t , x)  0,

( , w(t , x)) U 2 ,

(w  w(t , x))H ww (t , x)(w  w(t , x))  0,

where U i \ ( (t , x), w(t , x))  ,

i  1,2,

(9)

( (t , x), w) U1 ,

(t , x) [t 0 , t1 )  [ x0 , x1 ),

(10)

(t , x)  D.

It is obvious that by the fulfillment of the conditions (9),(10) Pontryagin's maximum principle (5) and optimality condition (8) degenerates and do not give any additional information on the optimality of the considered control. Therefore additional investigation is needed. If

( (t , x), (t , x))

is a singular with respect to the component v in the sense of

Pontryagin's maximum principle and quasi-singular with respect to the component w optimal control in the problem (1)-(4), then among the process

( (t , x), w(t , x)), z(t , x)) are valid the

conditions a 2 (  H z x (t , x)  f (t , x)    f (t , x) M 1 (t , t , x)  f (t , x))   2ab  f (t , x)(H z x (t , x)  M 1 (t , t , x)   f  (t , x))(w  w(t , x))  b 2 ( w  w(t , x)) ( H wzx (t , x)   f  (t , x) M 1 (t , t , x)) f  (t , x)(w  w(t , x))  0,

and a 2 (  H zt (t , x)  f (t , x)    f (t , x) M 2 ( x, x, t )  f (t , x))   2ab  f (t , x)(H z x (t , x)  M 2 ( x, x, t )   f w (t , x))(w  w(t , x))  b 2 ( w  w(t , x)) ( H wzx (t , x)   f w (t , x) M 2 ( x, x, t )) f w (t , x)(w  w(t , x))  0,

for all a  0, b  0, (, w(t , x)) U 2 , ( (t , x), w) U1 , (t , x)  (t 0 , t1 )  ( x0 , x1 ). Here t1



M 1 ( , s, x)   (t , x; , x) H z x z x (t ,.x) (t , x; s, x)dt t0 x1



M 2 ( , s, x)   (t , x; t , ) H z x z x (t ,.x) (t , x; t , s)dx x0

177


Keywords: singular controls, Goursat-Darboux systems, necessary optimality conditions. AMS Subject Classification: 35J05, 49J20.

References

1. Yusubov Sh.Sh. Necessary optimality conditions of the singular controls in a system with distributed parameters, Reports of RAS, Theory and Control Systems, No.1, 2008, pp.22-27. 2. Plotnikov V.I., Sumin B.I. Stability problem in the nonlinear Goursat-Darboux systems, Differential Equations, Vol.8, No.5, 1972, pp.845-856.

178


INTELLIGENT AND FUZZY CONTROL

FUZZY MODEL AND ALGORITHM FOR SOLVING OF E-SHOP INCOME MAXIMIZATION* F.A. Aliev1, E.R. Shafizadeh1, G.R. Hasanova1, J. Zeynalov2 1

Baku State University, Institute of Applied Mathematics, Baku, Azerbaijan 2 Nakchivan State University, Nakhcivan, Azerbaijan e-mail: [email protected]

Let’s construct the economic-mathematical model and maximize the revenue of e-shop, using the fuzzy system. To do this, we need to define the basic parameters affecting the profit in this environment. We first consider the classical (crisp) model of profit maximization for the shop in the real business: n

p x i

i

 max,

(1)

i 1

n

a

ij xi

 d j , j  1, m,

(2)

i 1

xi  0, i  1, n.

(3)

Here

xi - amount of the i-th ( i  1, n ) product type;

p i - price of the i-th product type unit; a ij -j-th kind the costs for per unit of i-th type of product; d j - j-th type of costs ( j  1, m ).

We take into account that the income from the sale of the i-th product in the virtual business depends on the variety of qualitative and quantitative indicators - the price of goods and services, security parameters of the e-shop, payment issues, number of internet users in target country, legislative base of electronic trade, the market for Internet access services, long-term trend of the potential size of the market, seasonal fluctuations in business activity, the quality of Internet *

This work is supported by the Science Development Fund under the President of the Azerbaijan Republic - Grant N EİF-RİTN-MQM-2/İKT-2-2013-7(13)-29/06/1-M-24.

179


service providers, the price attractiveness providers, the influence of the speed of access to Internet resources, sources of inflow and outflow of customers, and so on., which have the fuzzy descriptions. Since many of these parameters have fuzzy description [1,2], taking them into account in the model (1)-(3) as a crisp indications, or aren’t impossible or maybe with the significant assumptions. On the basis of these arguments we can say that the fuzzy description of these parameters in the model may be more adequate than in a sense taking a crisp description. Let us consider the prices of products p i in model (1)-(3). We know that price is a very aggregate economic parameter. Since it is formulated (image) by influencing the different factors: consumer's choice, fashion, prices of fungible goods and complementary goods, age structure of consumers, consumer’s income, the cost of resources, advertising and etc. In e-commerce here adds another technical indicators that we have listed above. Based on these considerations, we take the price of the products as fuzzy parameter. If the prices of the products - p i are fuzzy then xi also be fuzzy. Then pi xi hasn’t meaning. Then it makes sense to consider ~ pi ~ xi . In this case, we use the method proposed in [3], in which the space and the scalar product of fuzzy numbers are given. We give some explanations: Let’s assume that F is the class of convex normal fuzzy numbers. For any a  F the set of  -level cuts of fuzzy number a is defined as the interval a   La ( ), Ra ( ),   (0,1].

Let a  a1 , a 2 , b  b1 , b2 , ai , bi  F , i  1,2,

 - level cuts of fuzzy number ai , bi  F we denote

  L

 ( ),   0,1

ai  Lai ( ), Rai ( ) ,   0,1

bi

bi

( ), Rbi

The scalar product a  b в F  F is defined as follows 1





 





1 a  b   La1 ( )  La2 ( ) Lb1 ( )  Lb2 ( )  Ra1 ( )  Ra2 ( ) Rb1 ( )  Rb2 ( ) d 20 Using this approach, instead of (1) we consider the following functional:





1

1 L p ( ) Lxl ( )  R pl ( ) Rxl ( ) d  max 20 l

180

(4)


Here L pl ( ) and R pl ( ) correspondingly, are the left and right boundaries of the  -level cuts of the fuzzy number p i ; Lxl ( ) and Rxl ( ) respectively are the left and right boundaries of the  level cuts of the fuzzy number xi . The condition (2) will be in the following form: 1

1 n  aij Lxi ( )  Rxi ( )d  d j , j  1, m 2 i1 0 It is clear that taking into account the fuzziness of

(5)

, we obtain the following condition:

0  Lxl ( )  Rxl ( )

(6)

Note that the functional (4) can be considered as averaged estimate of the function (1).

L

Main Results: Problem (4) - (6) is not a linear programming problem. Here the variables xl



( )  Rxl ( ) ,   0,1 , confirm that (4) is functional. For correspondence this functional to

the linear programming problem, the integrals in (4) and (5) should be discretized at the α-level cuts. Let’s divide the interval [0, 1] into N parts with step h ( h 

1 ) Then the problem (4)-(6) N

will be in the following form: 1 n  2 i 1

 L



N

k 1

( k ) Lxl ( k )  R pl ( k ) Rxl ( k )  max

pl





N 1 n a  ij  Lx ( k )  Rxl ( k )  d j , j  1, m 2 i 1 k 1 l

(7) (8)

0  Lxl ( k )  Rxl ( k )  1

(9)

Here lik  Lxl ( k ), rik  Rxl ( k ), i  1, n, k  1, N . If the parameters a ij are fuzzy, then the model (7)-(9) will be in the following form: 1 n  2 i 1 1 n  2 N i 1

 L N

k 1

pl

 L N

k 1



( k ) Lxl ( k )  R pl ( k ) Rxl ( k )  max

alj



( k ) Lxl ( k )  Ralj ( k ) Rxl ( k )  d j , j  1, m

0  Lxl ( k )  Rxl ( k )  1

(10) (11) (12)

As we see, in comparison with the classical problem the number of initial variables is increased. In the classical problem the number of variables was n, but here n × 2N. At the same time, the number of conditions is also increased. 181


In the general form these types of problems are solved using the MATLAB package. We suggest the following algorithm: Step 1. p i and aij fuzzy coefficients are given by  -level cuts   0;1 in the following form:





p i  ui  ci ; mi  yi  and aii  wij  vij ; g ij  qij , j  1, m, i  1, n i.e. Lpi  ui  ci , Rpi  mi  yi и Ld j  w j  v j , Rdj  g j  q j .( here ui , ci , mi , yi , wij , vij , g ij , qij ( j  1, m, i  1, n )-are given constants) Step 2. Take the initial step hk   at k  0 , where  is a natural number. From  (l )  0 to

 (l ) :  (l ) 

1 calculate p i (l )  ui  ci (l ); mi  yi (l ) and hk





aii  wij  vij ; g ij  qij , j  1, m, i  1, n , i.e. Lpi  ui  ci , Rpi  mi  yi и Ld j  w j  v j , Rdj  g j  q j .

Step 3. Substituting these values in the problem (10)-(12) we find the solutions Lxi(l ) and R xi (l ) , i  1, n , l  0, hk .

Step 4. On the basis of the solutions we calculate the value of the functional (10) and denote it by J (hk ) .

Step 5. Define hk : hk  h at k : k  1 Step 6. If J (hk 1 )  J (hk ) is satisfied then go to Step 2, else the process is finished. We take as solution the values of the variables of latest J (hk 1 ) which satisfies condition J (hk 1 )  J (hk ) . Keywords: mathematical economics, linear programming, fuzzy numbers, virtual business AMS Subject Classification: 91B02, 91B44. References 1. Akin O., Khaniyev T., Oruc O., Turksen I.B., Some possible fuzzy solutions for second order fuzzy initial value problems involving forcing terms, Appl. Comput. Math., Vol.13, No.2, 2014, pp.239249. 2. Aliev F.A., Niftiyev A.A., Zeynalov J.I., Optimal Synthesis problem for the fuzzy systems in semiinfinite interval, Appl. Comput. Math., Vol.10, No.1, 2011, pp.97-105.

182


3. Aliev F.A., Niftiev A.A., Zeynalov J.I., Method to solution of the fuzzy optimal control problems, Baku, 2013, 199 p.

ANALYSIS OF THE EFFECTIVENESS OF THE METHODS OF RECOGNITION OF AUTHORSHIP OF TEXTS IN THE AZERBAIJANI LANGUAGE K.R. Aida-zade1, S.G. Talibov1 1


The necessity to identify the authorship of texts arises in criminology, literature, scientific research, and many other areas. There are mainly two different formulation of the problem. One of them lies in determining the original author of the existing text from a set of potential authors. The other problem lies in confirming the authorship of a particular text of some author. The two problems are close to each other both in statement and in using the same recognition methods. Despite the relatively long history, the methods of solution to the problem of authorship recognition have gained rapid development recently. This development is due to the possibilities presented by modern computer and information technologies. It should be noted that in this direction, the results of systematic research with respect to texts in the Azerbaijani language are practically unknown. We have used the mathematical models based on artificial neural networks, Markov chains, Khmelyev’s technique, and statistical analysis to recognize the authorship of texts in the Azerbaijani language. We used different characteristics of individual words and of sentences as attributes. The analysis of the obtained results showed that the text recognition system, constructed using one of the approaches and any one set of attributes, does not always deliver reliable results. Therefore, in the text recognition system, we have had to involve several subsystems implementing different approaches and different attributes. The quality of the obtained results of such a system, implemented with the use of parallel computing technology, outperforms the results of the recognition systems built using separate classical methods in many ways.

Keywords: authorship recognition, text recognition system. AMS Subject Classification: 68T10.

183


PRESENTATION AND ANALYSIS OF FUZZY RULES PRODUCTIONS USING A MODIFIED FUZZY PETRI NETS M. A.Ahmedov1, V.A. Mustafaev1, Sh.S. Huseynzade1 1

Sumgait State University, Baku, Azerbaijan e-mail: [email protected]

Fuzzy production system is represented in the form of modified fuzzy Petri nets. An algorithm for the analysis of modified fuzzy Petri nets, which provides a visual representation of fuzzy rules of production and carried out on the basis of the output of fuzzy conclusions. Fuzzy models based on the rules for calculating with fuzzy sets , are a clear and effective presentation of interacting dynamic processes that display the data and knowledge in the form >. In general, the rules of the fuzzy production understand the expression [1]: (i):Q; X; A B, S, F, Y,

where i – the name of the fuzzy production; Q – the scope of the fuzzy characteristics of products; X – conditions for the applicability of fuzzy core products; A B - the core of the fuzzy production in which A – core condition or antecedent ; B – conclusion of the kernel or consequent ;  – a sign of the logical sequence and repetition ; S – a way to quantify the extent of truth of the core ; F – The coefficient of determination and confidence of fuzzy production; Y – postconditions products. In solving problems of fuzzy modeling of dynamic interactive processes and evaluation of the approximate reasoning using modified fuzzy Petri nets (PN). Modified fuzzy PN is defined as Np = (P, T, I, O, f, , 0),

where P and T – fuzzy sets of places and transitions; I: P × T → (0, 1, ...)

and O: T → P (0,1, ...) – function of the input and output respectively incidence; f = (f1, f2,..., fn), fj  0,1 (j  N, N – set of natural numbers), the vector of values of fuzzy switching transitions;

 = (1, 2,..., n), j  0,1 (j  N) –vector of threshold crossings; 0 = (10, 20, ... n0,) initial marking vector, each component of which is determined by the membership function of the fuzzy presence of the marker in the corresponding position of the network, while i0  0,1 (i

 N). The advantage of the modified fuzzy NP is their use to visualize fuzzy rules of products and their performance based on the output of fuzzy conclusions. In this case, the following interpretation of places and transitions of modified fuzzy NP. Fuzzy Rule product type : > seems like some transition tj  T , while A condition of this rule corresponds to the input position pj  P , a conclusion– the position of the output transition. If the condition of fuzzy rules of production consists of several sub-conditions connected fuzzy logic operation conjunctions A = A1 and A2 and ... and An, all these sub-conditions are represented as input position corresponding transition: I (ti) = {A1, A2, …, An} (i  N). If the conclusion of the fuzzy rules of production consists of a few conclusions connected fuzzy logic operations conjunctions В = В1 and В2 and ... and Вn, all these a few conclusions appears as output position corresponding transition: O (tj) = {B1, B2, …, Bn} (j  N). If the condition of fuzzy rules of production consists of several sub-conditions connected by fuzzy logic operations disjunction A = A1 or A2 or ... or An, all these sub-conditions are presented as a separate item input transitions ti : I (ti) = Ai, (i  N). If the conclusion of the fuzzy rules of production consists of a few conclusions connected transactions fuzzy logic disjunction В = В1 or В2 or ... or Вn, all these a few conclusions represented as output places the individual transitions ti : О(ti) = Вi, (i  N). Triggered transitions and change the state of modified fuzzy NP going by the rules [2]:

 transition tj  T modified fuzzy joint venture called authorization, some of the current labeling , if the following condition : min {i}  к, (i  N)  (pi, tk)

 where  - operation logical minimum к – the value threshold of transition tj  T.  if the transition tj  T modified fuzzy NP is allowed , with some labeling μ, the operation of the fuzzy transition leads to a new labeling 𝝁′𝟏 = (𝝁′𝟏 , 𝝁′𝟐 , … , 𝝁′𝒏 ) components of which determined by the following formulas :

 for each of the input positions pj  P for which I (pi, tk) > 0:

𝝁′𝒊 = 𝟎; ( pj  P)  (I (pi, tk));

 For each of the output pj  P , for which O (tk, pi) > 0 : 𝝁′𝒋 = max {j, min (j, fk)} ( pj  P)  (O (tk,pj,) > 0), (i  N)  (I (pi, tk) >0)

where fk - the value of the membership function of a fuzzy operation transition tj  T. To implement the solution of practical problems of fuzzy inference systems Fuzzy productions useful software MATLAB and Fuzzy TECH. Keywords: production system, modified Petri nets, fuzzy inference, fuzzy production, membership function, fuzzy conclusion. AMS Subject Classification: 34B99, 03B52, 46F10. 185


References 1. Borpsov V.V., Kruglov V.V., Fedulov A.S. Fuzzy Models and Networks, 2-nd ed., A stereotype, Moscow, Hotline-Telecom, 2012, 284p. 2. Akhmedov M.A., Mustafayev V.A. Development of fuzzy model for investigation functioning active elements of the flexible manufacture module, Proc. 9th Intern. Conf. on Application of Fuzzy Systems and Soft Computing, Prague, Czech Republic, 2010, Kaufering: b-Quadrat Verlag, 2010, pp.315-320.

FUZZY INITIAL VALUE PROBLEMS FOR THE SECOND ORDER DIFFERENTIAL EQUATIONS Ömer Akin1 1

TOBB University of Economics and Technology, Department of Mathematics, Ankara, Turkey e-mail: [email protected]

In the mathematical modeling of real world events, we encounter two main hinderances such as the complexity of the model and vagueness in the model. As the complexity of the system being modeled increases, our ability to make precise and relevant statements about its behavior becomes limited. Hence, we are not able to formulate the mathematical model. The second hinderance relates to the indeterminacy caused by our inability to differentiate events in a real situation exactly, and therefore to define instrumental notions in precise form. Hence such events have vagueness and it needs to work with vague notions [1]. However, classical mathematics cannot cope with such vague notions. Therefore, It is necessary to have some mathematical tools to describe vague and uncertain notions to overcome the foregoing obstacles in the mathematical modeling of imprecise real world events. In 1965, Zadeh [2] initiated the development of the modified set theory known as fuzzy set theory, which is a tool that makes possible the description of vague notions and manipulations with them. The basic idea of fuzzy set theory is simple and natural. A fuzzy set is a function from a set into into the interval. Using it, one can model the meaning of vague notions and also certain kinds of human reasoning. Hence fuzzy set theory and its applications have been extensively developed and it has become a powerful tool for modelling uncertainty and for processing vague or subjective information in mathematical models. The theoretical framework of fuzzy differential equations (FDEs) has been an active research field over the last few years. The concept of a fuzzy derivative was first introduced by

186


Chang and Zadeh [3] then it was followed up by Dubois and Prade [4] who use the extension principle. The term "fuzzy differential equation" was introduced in 1987 by Kandel and Byatt [56]. There have been many suggestions for the definition of fuzzy derivative to study fuzzy differential equations. One of the earliest suggestions was to generalize the Hukuhara derivative [7]. This generalization was made by Puri and Ralescu [8] and studied by Kaleva [9, 10]. It soon appeared that the solution has a drawback: it becomes fuzzier as time goes by. Hence, behavior of the fuzzy solution becomes quite different from that of the crisp solution. Seikkala [11] introduced the notion of fuzzy derivative as an extension of the Hukuhara derivative and fuzzy integral, which was the same as what Dubois and Prade [12] proposed. To cope with the inconveniences arising in connection with Hukuhara derivative, Hüllermeier [13] considered fuzzy differential equation as a family of differential inclusions. The main disadvantage of using differential inclusions is that it has no an adequate definition for derivative of a fuzzy number valued function. The concept of strongly generalized differentiability was introduced in [14]. In [15] a generalized concept of higher-order differentiability for fuzzy functions was presented to solve Nth-order fuzzy differential equations. Buckley and Feuring [16] and Buckley et al. [17] gave a very general formulation of fuzzy first order initial value problem. They firstly found the crisp solution, fuzzified it and then checked to see if it satisfied the FDEs. And Buckley et al [18] gave a general method to find solution of fuzzy initial value problem for second order linear differential equations. Finding the solutions of fuzzy differential equations were discussed in some other work [18-22]. However solutions were given only on a limited domain. In this talk we will present a review of some of our previous work on fuzzy initial value problems. In [23] we considered a prey-predator model with fuzzy initial values using the concept of generalized Hukuhara differentiability. We observed that if we have the α-cut [u]  [u , u ] of a fuzzy number then the inequality u   u  sometimes does not hold. Hence we proposed two alternatives for this shortcoming. And we obtained graphical solutions for the considered problems. In [24] we proposed a new algorithm based on analysis of crisp solution. We established a synthesis of crisp solution of fuzzy initial value problem and the method proposed in Kaleva [9] to solve fuzzy initial value problem. Here we investigated second order fuzzy differential equations with fuzzy coefficients, fuzzy initial values and fuzzy forcing functions. We proposed an algorithm

187


based on alpha-cut of a fuzzy set. Finally we present some examples by using our proposed algorithm. In [25], we stated a fuzzy initial value problem of the second order fuzzy differential equations. Here, we investigated problems with fuzzy coefficients, fuzzy initial values, and fuzzy forcing functions. We have extended the algorithm given in [24]. Finally, we presented some examples by using our extended algorithm. As the α-cuts may cross over each other [26] it is difficult to find the α-cuts for the solution of the fuzzy initial value problem over the domain where crisp solution of the problem exists. Since this intersection may exist at any points in the domain, it restricts the researches to a limited domain to find the α-cuts [26-28] Hence to overcome this problem in finding the solution of second order fuzzy initial value problem, we formulated a new algorithm for finding an analytical form of αcuts for the solution of the second order fuzzy initial value problem with the help of an indicator operator. This new algorithm allows us to find the solution of the fuzzy initial value problem without introducing any constraints on the domain where the crisp solution exists or without using interval conditions [24, 26-28] Our main goal is to examine the solutions of the following second order fuzzy initial value problems by using our new algorithm based on an indicator operator : y ( x)  a1 y ( x)  a 2 y ( x) 

r

 b g ( x) i

i

,

(1)

i 1

y(0)   0 ;

y (0)  y1 ,

(2)

where a 1 and a 2 are fuzzy numbers and g i ( x), (i  1,..., r ) are continuous functions on the interval [0, ) . The initial conditions  0 ,  1 and forcing coefficients bi , (i  1,..., r ) are fuzzy numbers.

Keywords: fuzzy number, fuzzy arithmetic, fuzzy initial value problem, fuzzy forcing coefficients. AMS Subject Classification: 90A, 94D. References 1. Lakshmikantham V., Mohapatra R.N., Theory of fuzzy differential equations and inclusions, ISBN 0-415-30073-8, 2003. 2. Zadeh,

L.A.,

Fuzzy

sets.

Information

and

Control,

1965, doi:10.1016/S0019-

9958(65)90241-X. 3. Chang S.L., Zadeh L.A., On fuzzy mapping and control, IEEE Transactions on Systems Man Cybernetics, 2, 1972, pp.330-340. 188


4. Dubois D., Prade H., Towards Fuzzy Differential Calculus: Part 3, Differentiation, Fuzzy Sets and Systems, 8, 1982, pp.225-233. 5. Kandel A., Byatt W.J., Fuzzy differential equations, in: Proceedings of International Conference Cybernetics and Society, Tokyo, 1978, pp.1213-1216. 6. Kandel A., Byatt W.J., Fuzzy processes, Fuzzy Sets and Systems, 4, 1980, pp.117-152. 7. Hukuhara M., Intégration des applications measurables dont la valeur est un compact convexe, Funkcial. Ekvac., 10, 1967, pp.205–223. 8. Puri M.L., Ralescu D.A., Differentials of fuzzy functions, J. Math. Anal. Appl., 91, 1983, pp.552558. 9. Kaleva O., Fuzzy differential equations, Fuzzy Sets and Systems, 24, 1987, pp.301-317. 10. Kaleva O., The Cauchy Problem for Fuzzy Differential Equations, Fuzzy Sets Systems, 35, 1990, pp.389-396 11. Seikkala S., On the fuzzy initial value problem, Fuzzy Sets and Systems, Vol.24, No.3, 1987, pp.319–330. 12. Dubois D., Prade H., Towards Fuzzy Differential Calculus: Part 3, Differentiation, Fuzzy Sets and Systems, 8, 1982, pp.225-233. 13. Hüllermeier E., An approach to modelling and simulation of uncertain systems, Int. J.Uncertain. Fuzz., Knowledge-Based System, 5, 1997, pp.117-137. 14. Bede B., Gal S.G., Almost periodic fuzzy-number-valued functions, Fuzzy Sets and Systems, 147, 2004, pp.385-403. 15. Khastan A., Bahrami F., Ivaz K., New Results on Multiple Solutions for Nth-Order Fuzzy Differential Equations under Generalized Differentiability, Boundary Value Problems, 2009, 13 p. Article ID 395714, doi:10.1155/2009/395714. 16. Buckley J.J., Feuring T., Fuzzy differential equations, Fuzzy Sets and Systems, 110, 2000, pp.43-54. 17. Buckley J.J., Feuring T., Hayashi Y., Linear Systems of First Order Ordinary Differential Equations: Fuzzy Initial Conditions, Soft Computing, 6, 2002, pp.415-421. 18. Buckley J.J., F euring, T., Fuzzy initial value problem for N th-order linear differential equations,

Fuzzy

Sets

and

Systems,

121, 2001, pp.247–255.

doi:10.1016/S0165-

0114(00)00028-2. 19. Akın et al., An Indicator Operator Algorithm for Solving A Second Order Fuzzy Initial Value Problem. 2015. 20. Bede B., Stefanini L., Generalized differentiability of fuzzy-valued functions. Fuzzy Sets and Systems, 230, 2013, pp.119–141. 21. Stefanini, L., Bede, B. Generalized Hukuhara differentiability of interval-valued functions 189


and interval differential equations. Nonlinear Analysis, Theory, Methods and Applications, Vol.71, No.3-4, 2009, pp.1311–1328. 22. Akın Ö., Oruç Ö., A Prey Predator Model with Fuzzy Initial Values, Hacettepe Journal of Mathematics and Statistics, Volume, Vol.41, No.3, 2012, pp.387–395. 23. Akın Ö., Khaniyev T., Oruç Ö., Türkşen I.B., 2013, An algorithm for the solution of second order fuzzy initial value problems. Expert Systems with Applications, Vol.40, No3, pp.953–957. 24. Akın Ö., Khaniyev T., Oruç Ö., Türkşen I. B., Some possible fuzzy solutions for second order fuzzy initial value problems involving forcing terms, Appl. Comput. Math., Vol.13 No.2, 2014, pp.239-249. 25. Bede B., Stefanini L., Generalized differentiability of fuzzy-valued functions. Fuzzy Sets and Systems, 230, 2013, pp.119–141. 26. Buckley J.J., Feuring, T., Fuzzy initial value problem for N th-order linear differential equations. Fuzzy Sets and Systems, 121, 2001, pp.247–255. 27. Stefanini L., Bede B., Generalized Hukuhara differentiability of interval- valued functions and interval differential equations, Nonlinear Analysis, Theory, Methods and Applications, Vol.71, No.3-4, 2009, pp.1311–1328. 28. Akın Ö., Khaniyev T., Oruç Ö., Türkşen, I. B., An indicator operator algorithm for solving a second order fuzzy initial value problem, 2015 (Submitted).

AN INDICATOR OPERATOR ALGORITHM FOR SOLVING A SECOND ORDER FUZZY INITIAL VALUE PROBLEM Ömer Akin1, Tahir Khaniyev2,3, Fikri Gokpinar4, Burhan Turksen2,5, Selami Bayeg1 1

TOBB University of Economics and Technology, Department of Mathematics, Ankara, Turkey TOBB University of Economics and Technology, Dep. of Industrial Engineering, Ankara, Turkey 3 Institute of Control Systems, Azerbaijan National Academy of Sciences, Baku, Azerbaijan 4 Gazi University, Departmant of Statistics, Ankara, Turkey 5 Toronto University, Toronto, Canada e-mail: [email protected]

2

The idea of the fuzzy number and fuzzy arithmetic were firstly introduced by Zadeh (Zadeh, 1965). Fuzzy differential term was conceptualized by Kandel and Byatt (Kandel, &Byatt 1987), and fuzzy differential equations were formulated by Seikkala (Seikkala, 1987) and Kaleva (Kaleva, 1987). In (Bede, Rudas, & Bencsik, 2007), the Hukuhara derivative was generalized and it was applied to the fuzzy initial value problem. First order linear nonhomogeneous ordinary differential equation was examined by Mondal et al. (Mondal, Banerjee, & Roy, 2013). Fuzzy initial value problems were extensively investigated by some other researchers such as (J J 190


Buckley, Feuring, & Hayashi, 2002; J. J. Buckley, Reilly, & Jowers, 2005; James J. Buckley & Feuring, 2000; Congxin & Shiji, 1998; Ding, Ma, & Kandel, 1997; Zarei, Kamyad, & Heydari, 2012). Buckley and Feuring (2001) have considered the initial value problem for order fuzzy differential equations. In Akın et al.(2013), a similar fuzzy initial value problem, which has fuzzy coefficients and fuzzy forcing functions as fuzzy numbers, has been solved according to the signs of the solution and its first and second order derivatives. However, as the α-cuts may cross over each other (Bede &Stefanini, 2013) it is difficult to find the α-cuts for the solution of the fuzzy initial value problem over the domain where crisp solution exists. Since this intersection may exist at any points in the domain, it restricts the researches to a limited domain to find the α-cuts .( Bede & Stefanini, 2013; James J. Buckley & Feuring, 2001; Stefanini & Bede, 2009). Hence to overcome this problem in finding the solution of second order fuzzy initial value problem, we formulate a new algorithm for finding an analytical form of α-cuts for the solution of the fuzzy initial value problem of the second order differential equation with the help of an indicator operator. This new algorithm allows us to find the solution of the fuzzy initial value problem without introducing any constraints on the domain where the crisp solution exists or without using interval conditions (Ömer Akın & Oruç, 2012; Ömer Akın & Oruç, 2014; Bede & Stefanini, 2013; James J. Buckley & Feuring, 2001; Stefanini & Bede, 2009). Briefly, in this work we propose a new algorithm to find the analytical form of α-cuts of the solution of the following second order nonhomogeneous fuzzy initial value problem with fuzzy initial values and fuzzy forcing coefficients y ( x)  a1 y ( x)  a 2 y( x) 

r

 b g ( x), i

i

(1)

i 1

y(0)   0 ;

y (0)  y1 .

(2)

Here a1 and a 2 are crisp constants and g i ( x), (i  1,..., r ) are continuous functions on the interval [0, ) . The initial conditions  0 ,  1 and forcing coefficients bi , (i  1,..., r ) are fuzzy numbers.

Firstly we apply Zadeh’s Extension Principle to the solution of our crisp initial value problem. Then, we apply our new algorithm to find the analytical form of α-cuts of the solution of the fuzzy initial value problem. Finally, we illustrate some examples by using this algorithm.

Keywords: fuzzy number, fuzzy arithmetic, fuzzy initial value problem, fuzzy forcing coefficients. AMS Subject Classification: 90A, 94D.

191


References 1. Akın Ö., Khaniyev T., Oruç Ö., Türkşen, I. B., Some possible fuzzy solutions for second order fuzzy initial value problems involving forcing terms, Appl. Comput. Math., Vol.13, No.2, 2014, pp.239-249. 2. Akın Ö., Khaniyev T., Oruç Ö., Türkşen I. B., An algorithm for the solution of second order fuzzy initial

value

problems,

Expert

Systems

with Applications,

40,

2013,

pp.953–957.

doi:10.1016/j.eswa.2012.05.052. 3. Akın Ö., Oruç Ö., A Prey Predator Model with Fuzzy Initial Values, 2012, Hacettepe Journal of 4. Mathematics and Statistics. Retrieved from http://dergipark.ulakbim.gov.tr/hujms/article/view/5000017263. 5. Beatson R.K., Bui, H. Q. (2007). Mollification formulas and implicit smoothing, Advances in Computational Mathematics, 27, pp.125–149. doi:10.1007/s10444-005-7512-3. 6. Bede B., Rudas I. J., Bencsik, A.L., First order linear fuzzy differential equations under generalized differentiability,

Information

Sciences,

Vol.177,

No.7,

2007,

pp.1648–1662,

doi:10.1016/j.ins.2006.08.021. 7. Bede B., Stefanini L., Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230, 2013, pp.119–141. doi:10.1016/j.fss.2012.10.003. 8. Buckley J. J., Feuring T., Fuzzy differential equations, Fuzzy Sets and Systems, Vol.110, No.1, 2000, pp.43–54. doi:10.1016/S0165-0114(98)00141-9. 9. Buckley J.J., Feuring T., Fuzzy initial value problem for N th-order linear differential equations, Fuzzy Sets and Systems, 121, (2001247–255. doi:10.1016/S0165-0114(00)00028-2 10. Buckley J.J., Feuring T., Hayashi Y., Linear systems of first order ordinary differential equations: fuzzy initial conditions, Soft Computing, 6, 2002, pp.415–421. doi:10.1007/s005000100155. 11. Buckley J. J., Reilly K.D., Jowers L.J., Simulating continuous fuzzy systems: I. Iranian Journal of Fuzzy Systems, 2, 2005, pp.1–17. doi:10.1016/j.ins.2006.03.005. 12. Congxin W., Shiji S., Existence theorem to the Cauchy problem of fuzzy differential equations under compactness-type conditions, Information Sciences, Vol.108, No.1-4, 1998, pp.123–134. doi:10.1016/S0020-0255(97)10064-0. 13. Ding Z., Ma M., Kandel A., Existence of the solutions of fuzzy differential equations with parameters, Information Sciences, Vol.99, No.3-4, 1997, pp.205–217 doi:10.1016/S00200255(96)00279-4. 14. Kaleva O., Fuzzy differential equations, Fuzzy Sets and Systems, Vol.24, No.3, 1987, pp.301–317. doi:10.1016/0165-0114(87)90029-7. 15. Mondal S.P., Banerjee S., Roy T.K., First Order Linear Homogeneous Ordinary Differential Equation in Fuzzy Environment, Vol.14, No.1, 2013, pp.16–26. 192


16. Seikkala S., On the fuzzy initial value problem, Fuzzy Sets and Systems, Vol.24, No.3, 1987, pp.319–330. doi:10.1016/0165-0114(87)90030-3. 17. Stefanini L., Bede B., Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis, Theory, Methods and Applications, Vol.71, No.3-4, 2009, pp.1311–1328. doi:10.1016/j.na.2008.12.005. 18. Zadeh L.A., Fuzzy sets, Information and Control. 1965 doi:10.1016/S0019-9958(65)90241-X. 19. Zarei H., Kamyad A.V., Heydari, A.A., Fuzzy modeling and control of HIV infection, Computational and Mathematical Methods in Medicine, 2012, doi:10.1155/2012/893474.

THE MODELING THE COMPLEX SYSTEMS OF THE OIL PRODUCTION: INTELLIGENT AND FUZZY CONTROL BY L.ZADEH'S FUZZY SETS THEORY M.I. Aliyev1, I.M. Aliyev1, E.A. Isaeva1, A.M. Aliyeva1, V.M. Baxishov2 1

Institute of Physics of National Academy of Sciences of Azerbaijan, Baku, Azerbaijan 2 Baku Bizness Center, Baku, Azerbaijan e-mail: [email protected]

It is well known that the statistics plays the great in our life. But statistics itself bases on the theory of probabilities (PT). Today PI is the universal theory. But 200 years ago PT was «the theory of casino games». And now, studying more deeply the L.Zadeh's fuzzy sets theory (FST) we can tell that triumph of this theory will be in near future. In sciences we study the fluctuations. In mathematics, they are being described by stochastic integral. But these integrals has many various definitions (by Ito, Stratanovich, Klimantovich). The problem, how must go process of decision choosing, is not solved in PT. Only in FST there is a decision - making process. By this point of view the modeling and control of oil production is very interesting. The oil field belongs to the class of the hierarchically arranged big and complex systems. Therefore there is a need for modeling, and in oil production there are known alternative modeling and integration of models. But these methods of decision-making allow to choose the best of a set of possible decisions only in the conditions of one concrete type of uncertainty. Using the L. Zadeh theory of fuzzy set we can consider all types of uncertainties at the same time and give a new method in modeling of systems of oil and gas production. In this case we will be able to use decision-making process in a new way and intellectually control and monitor the systems of oil and gas production.

193


It is known in the oil and gas deposit there are many kind of uncertainties. Here the developed quantitative methods of decision-making (such as the theory of games, the minimax theory, maximizing of expected usefulness, methods of maximum likelhood, the analysis "expenses - efficiency" and others) help to choose the best of a set of possible decisions only in the conditions of one concrete type of uncertainty. All these methods are based on the probability theory therefore here the decision-making process is impossible. The probability theory deals with the randomness, but the main source of uncertainty is the fuzziness. The L.Zadeh FST considers this kind of uncertainties and allows using the decision-making process. In this case we will be able to use decision-making process in a new way and and it is possible the intellectual control and monitoring of system of oil and gas production. We will use known in the oil production the variant modeling, stochastic modeling, evolution and integration of models. We will take models of the different classes, different types of difficulties, different levels of the description of the object, different means and technologies of their construction, interpretation and application. At multiple stochastic modeling, there are representative ensembles (set) of models of oil and gas deposit. The choice of the best variant of model occurs in the fuzzy environment. The statement «the chosen model is the best" is inexact. Such inaccuracy is expressed by a fuzzy set of all good models in which it is impossible to specify strict border between the elements which are belonging and not belonging to it. Therefore there is a membership function of x to a fuzzy set A. How decision-making process in fuzzy environment goes is considered by L.Zadeh and in his FST and we will apply its results to a case of oil and gas deposit in our paper. Thus Х={x} are the set of all versions of models. Then fuzzy set of the best models A in X is the set of pairs:   x,  A x ,

x  X where А(x) is membership function x to A .

The problem of estimation of membership functions  A (x) is the main in fuzzy sets theory and there are different methods. For example, to construct membership functions basing on the information of the expert. Also we will use the known in FST the fuzzy clustering, the base fuzzy algorithm of α –cut of average. After finding set of pairs x1 ,  A x1 ; x2 ,  A x2 ;.... we should decide another problem - problem of abstraction which plays central role in recognition of forms in FST. Let’s assume, that  A (x) are the known membership functions for all xX. The main elements of decision making process in FST are: a) set of alternatives X, in our case X is the set of variants of models x , 194


b) set of the restrictions G, with which must be accounted when choosing the necessary alternative. In our case the restrictions are the geological data about petroleum deposit – the scale, depth, quantity, quality and etc -and other data – the communications, urbanizations, system of pipelines and etc. c) a function of preference. Joint influence of fuzzy restrictions G can be presented by the crossing of G1 G2…Gn. A membership function in this case is:

G1G2 ...Gn  G1 x   G2 x  ...  Gn x  minG1 x ,...,Gn x

For example, there are 5 variants of models X = {1,2,3,4,5}. Let’s assume for only 3 of them the restrictions are such, as in the table: X Location

1

2

3

4

5

G

0,1

0,5

1,0

0,8

0,7

G

0,3

0,4

0,2

0,5

0,9

G

0,8

0,5

0,4

0,3

0,5

1

Quantity of oil

3

Quality of oil

4

In this case, the decision is the fuzzy set: D={1;0,1); (2; 0,4); (3;0,2); (4;0,3); (5; 0,5)}. Optimal decision is the precise subset DM of the fuzzy set D determined as : max  D x , x  K , x  K,  0

 D M x   

where K is the set of such X for which  D  max , every X from DM is the maximizing decision. Thus  D  0,5 is effective decision, x={5} is maximizing decision and variant about 5 will be the best. During formation of models there will be their various state (variants of models) Xt where t = 0,1 …. Here the entrance signals Ut (in our case - above mentioned restrictions G), UtU=1, 2 ... . U = 1, 2 take place. It is clear that the state Xt+1 depends from Xt and Ut and it is described by the equation of evolution:

Xt+1 =( Xt ,Ut) .

195


Let's assume at the certain moment of time the membership functions   i  and  U i  are given. In the Zadeh FST there find a sequence (U0, U1 ... Un-1 ), which maximizing the decision

D :  D U 0 ,U1 ,...U N 1    0 U 0   ... N 1 U N 1    N U N  Decision is represented as: U n   n xn  , where n is the accepted strategy, i.e. accepted rule of a choice of entrance signals e Un depending on realized Xn. After that the method of dynamic programming is applied to find Xn and maximizing (effective) decision. Keywords: L.Zade's FST, modeling, model of oil and gas deposit, fuzzy environment, decision-making, intellectual management, dynamic programming. AMS Subject Classification: 90A, 94D.

A NEW APPROACH TO NUMERICAL SOLUTION OF LINEAR FUZZY FREDHOLM INTEGRAL EQUATIONS USING ITERATIVE METHOD BASED ON FUZZY BERNSTEIN POLYNOMIALS Reza Ezzati1, Shokrollah Ziari2 , Seyed Mohsen Sadatrasoul1 1

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Iran e-mail: [email protected]

2

In this paper, we propose an iterative procedure based on fuzzy Bernstein approximation to solve linear fuzzy Fredholm integral equations of the second kind. Moreover, error estimation of the proposed method in terms of modulus of continuity is given. Finally, illustrative examples are included to demonstrate the accuracy and efficiency of the proposed method. Keywords: Fuzzy Fredholm Integral Equation; Modulus of continuity; Partial modulus of continuity; fuzzy Bernstein polynomials. AMS Subject Classification: 41-01, 45-XX.

196


SOLUTION METHOD FOR A FUZZY DELAY DIFERENTIAL EQUATION A.G. Fatullayev1, N.A. Gasilov2, Ş.E. Amrahov3 1,2 3

Baskent University, Ankara, Turkey Ankara University, Ankara, Turkey e-mail: [email protected]

In this paper, we consider a fuzzy delay differential equation (FDDE) [1-3] with fuzzy source function and fuzzy initial function. We assume these functions be in a special form, which we call triangular fuzzy function [4]. We present solution as a fuzzy set of real functions such that each real function satisfies the equation with some membership degree. We develop a method to find the fuzzy solution and we present the existence and uniqueness results for the solution of the considered FDDE. We also present an example to illustrate applicability of the proposed method. Keywords: fuzzy differential equation, fuzzy delay differential equation, fuzzy set, fuzzy function. AMS Subject Classification: 34K36, 34K28.

References 1. Azbelev N.V., Maksimov V.P., Rakhmatullina L.F., Introduction to the Theory of Functional Differential Equations and Applications, Hindawi Pub. Co., Cairo, 2007. 2. Hale J.K., Theory of Functional Differential Equations, Springer, New York, 1997. 3. Kolmanovskii V., Myshkis A., Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Acad. Pub., Dordrecht, 1999. 4. Gasilov N., Amrahov Ş.E., Fatullayev A.G., Solution of linear differential equations with fuzzy boundary values, Fuzzy Sets Syst., 257, 2014, pp.169-183.

SOLVING A NON-HOMOGENEOUS LINEAR SYSTEM OF INTERVAL DIFFERENTİAL EQUATIONS N.A. Gasilov1, Ş.E. Amrahov2, A.G. Fatullayev1 1

Baskent University, Turkey Ankara University, Turkey e-mail: [email protected] 2

Solving the optimal control problems lead to systems of set-valued (in particular, intervalvalued) differential equations. Besides, in most application problems, we don't know the values of input parameters exactly but we can determine intervals, where these values take place. In these

197


application problems the dynamic of the system is described by an interval-valued differential equation. In this study, we present a new approach to non-homogeneous system of interval differential equations. We consider linear differential equations with real coefficients but with interval initial values and with forcing terms, which are bunches of real functions. We assume each bunch to be linearly distributed between graphics of given two real functions. We look for solution not as a vector of interval-valued functions, as usual, but as a set (bunch) of real vector-functions. We develop a method to find the solution. We explain our approach and solution method with the help of an illustrative example. Keywords: interval differential equation, linear system of differential equations, set of functions. AMS Subject Classification: 34A26, 65L05, 68W99, 03E75.

THEORETICAL AND APPLIED ASPECTS OF FUZZY STRUCTURED SYSTEMS WITH FUZZY LOGIC OF DECISION-MAKING† K.S. Imanov1 1

Copyright Agency of Azerbaijan Republic, Baku, Azerbaijan

Many problems of systems analysis, decision making support in artificial intelligence and diagnostics, in organizational systems, and social structures lead to the necessity of the structural representation of the relevant systems and here along with the formulating decisions assumed also rearrangement of the structure, adequate to making decision. In most cases, such systems are weakly structured and poorly described, which leads to the use of the fuzzy systems and soft computing. A typical example of such systems is the copyright on the Internet, where interact such social groups as the Users (consumers), placed in a global network of intellectual production, Authors (creators), who created the creative results and transferred its intellectual property rights to third parties, the Right Holders (business- structure) acquired the exclusive rights to intellectual results and is a manufacturer of goods and services with intellectual property and finally Providers (information intermediaries) that provide access the above groups on the network. Guided by their own interests, which may not coincide, these social groups, entering into interaction in linguistic

This work was supported by the Science Development Fund under the President of Azerbaijan Republic - Grant № EIF-RITN-MQM-2 / IKT-2-2013-7 (13) -29/26/1 †

198


form, define important indicators, namely, the level of provision of copyright and openness (accessibility) on the Internet. Generalization of applied problems has led to the following algebraic models: - given an arbitrary set of elements x  X (elements, social groups, subdivisions, individuums, symptoms, conditions, situations, etc.); - the elements x  X are linked with different types of bonds c  C (information channels, communication indicators on managerial decisions, expert estimates, etc.); - in the front of system stands some set of problems (purposes)    , each of which forms a concrete connection between the elements x  X and thereby generates the certain structures

    X  X  C from the universal set of structures; - correspondence between the requirements of the problems    and implementing its structure is achieved by using the homomorphism α, given by an expert group of the system, or from the outside. Thus, the system under study is presented in the form of five basic algebraic models [1]:

S  X, C, M, Γ, α . Since the investigated systems are difficult, in these are involved people, as a rule, they are characterized by uncertainty of the description, subjectivism of conclusions and ideas, leading to their fuzzy character. In this regard, it was substantiated and introduced the algebraic model of structural-universal system with fuzzy individual cases S  X, C, V f  X, C , Γ, α

and the state of which is any fuzzy structure  V f  X, C  . Moreover, V f is a set of all possible fuzzy mappings of the direct product of X  X in the set of fuzzy structures  V f , each of which compares to the pair of elements of x, y  X 2 the element  x, y  C f  С , which is a subset of the set С   ( C f   1 ,..., k ck  c1





  , here i i  1, k ) is the membership function [2]. 

Are interpreted the fuzzy communication channels C f , are presented their linguistic specifications based on expert judgments in real application problems, and explore the impact of the action of problems 𝛾 on the dynamics of structural changes fuzzy system [3, 4]. In particular, for the applied problems "copyright - availability of the Internet", when Г = {𝛾1 , 𝛾2 , 𝛾3 } are 199


interpreting as 𝛾1 = free use of objects of copyright on the Internet; 𝛾2 = free use of objects on the Internet with the right to obtain compensation and 𝛾3= tightening of requirements for the use of objects, the order (sequence) of problems 𝛾1 , 𝛾2 , 𝛾3 are interpreting as the degree of their importance expert. The relation, pointing a pair of changing each other fuzzy structures 1 to  2 with membership functions 1 (С ) and 2 (С ) , is obtained: - structural losses:



 



 



12 (С)  1 (С)  1  1 (С)  1 (С)  1  2 (С) ;

- structural innovations:



21 (С)  2 (С)  1  1 (С)  2 (С )  1  2 (С) .

The important role of homomorphism, interpreted as a recommendation of the expert group of system, or as an a priori, outside the system representation led to the need of research in the case of fuzzy job. It was assumed that the job of the homomorphism in the form  :   H V f  from is a job   HomH V f  from  in H V f  all endomorphism of the set V f and the endomorphism 𝛿(𝛾) was considered as the choice of the set of structures which are necessary to solve the problem   . Fuzzy job of homomorphism is illustrated by the logic-linguistic rules of its describing. Because in practical applications, most systems have an ordered structure, was investigated class of fuzzy hierarchical systems based on the concept of fuzzy hierarchical structure [5]. Fuzzy hierarchical structure (hierarchy) 𝜑 In this case give for each pair of x, y  X 2 the fuzzy mapping

 x, y  С xyf  C , and satisfies the following conditions:  x, y  X ;Cxyf  Cxx ;  x, y ( x  y )  X ; Cxyf Cyxf   ;

 x, y, z  X ; Cxyf  Cyzf  Cxzf .

In particular, for the applied problems "copyright - the availability of the Internet", the numbering of the vertices and the direction of fuzzy connections graph - the hierarchy, corresponding to the fuzzy hierarchical structure is interpreted as a degree of importance of the Internet community actors (Users, Authors, Right Holders).

200


Description of fuzzy hierarchy led to increased of degree of fuzziness using fuzzy switching operations «  ~ » and fuzzy equality «≈» and thus introduced fuzzy defined hierarchy. The investigation of behavior of the fuzzy defined hierarchies based on the relationship between them and the corresponding fuzzy relations on the set X is the relations of the partial non-strict reflexive fuzzy order. For successive fuzzy structures in fuzzy hierarchical system [5] established the conditions under which the system is hierarchical [6]. For the applied problem "copyright - availability of the Internet", is built the play structure of three people (actors of the Internet community) with the possibilities to build coalitions (Users, Authors, Right Holders), which is reduced to the model of voting for one of the options, namely, a) - free use of copyright on the Internet; b) - free use of a right to receive compensation and c) the tightening of the requirements for the use of the objects. Taking into account the empirically revealed preference of actors is analyzed the sustainable solutions (of the strong coalition equilibriums) by the unanimous vote for one of the options and received the recommendations on priority of actions: option b) → variant c) → variant a). Are given the models of fuzzy logic conclusion in cases of job preferences of the parties in the form of modalities. To conclude recommendations are given regulatory developments and is offered an option of digital rights management [7, 8, 9, 10]. Keywords: fuzzy logic, system analysis, decision making, artificial intelligence. AMS Subject Classification: 90B06, 90B10, 90B20.

References 1. Birgkof G., Theory of Lattices, Moscow, Nauka, 1984. 2. Imanov K.S., Algebraic models of structural - blurry systems, Proceedings of IAM, Vol.3, No.1, 2014. 3. Zadeh L.A., Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy sets and systems, Vol.90, 1997. 4. Dubois D., Prade H., Possibility theorie, In R.Meyers, editor Ensyclopedic of Complexity and Systems Science. Springer, 2009. 5. Sadovskiy L.E. et.al., Issues of modeling of hierarchical systems, Rep. of USSR Academy of Sciences, Technical cybernetics, No.2, 1977. 6. Imanov K.S., On the ill-defined hierarchies, Rep. of ANAS, No.1-2, 1995. 7. Mulen E., Cooperative decision-making, Moscow, Mir, 1996. 201


8. Vatel I.A., Yereshko R.I., Mathematics of conflict and cooperation, Moscow, Nauka, 1973. 9. Imanov K.S., Management of intellectual property rights in digital networks, Baku, 2009. 10. Imanov K.S., Copyright and Internet: clash of interest and compromise search, Internet Governance Forum, Baku, 6-9 November, 2012, Intellectual property in speechs and presentations, II v., Baku, 2013.

MODELING OF FUZZY HIERARCHICAL STRUCTURES DESCRIBED BY GRAPHS OF PARTIAL ORDERING WITH THE HIGHEST VERTEX‡ K.S. Imanov1 1

Copyright Agency of the Republic of Azerbaijan

An important subclass of algebraic structures that generalize the idea of applied problems are hierarchical structures or Γ-hierarchy (hierarchy graphs), presented in the form of graphs with a partial ordering highest vertex. In [1] was analysed the fuzzy algebraic structures and discussed the applied questions of the problem arising from the main productions, including the problem of "copyright - the availability of the Internet", taking place in connection with the migration of copyright in the global network. In [2, 3] introduced into consideration the fuzzy hierarchical structures (Γ-fuzzy hierarchy) associated with the relation of fuzzy partially reflective order, strengthened illegibility on the basis of operations of the fuzzy inclusion and the fuzzy equality and was investigated the behavior of the defined hierarchies. In [4], was presented the results of computer modeling of hierarchical structures in the form of the graphs of partial ordering with the one maximum vertex and the algorithms of the constructive enumeration of these graphs. The present study is devoted to issues of computer modeling of fuzzy hierarchical structures described by graphs the partial ordering with the highest vertex. It is known that Γ-hierarchy has only basis, which is single rooted tree. On this basis for the enumeration Γ-hierarchies the initial is the algorithm of enumeration of the canonical (in some sense) trees recorded in the language convenient for computer representation, as such language is chosen the language of ρ-operation [5]. The language of ρ-operation (briefly – ρ-language) allows in a handy way to encode trees in the form of operating code and thus provides their monotonous generation on a computer. The formula of the tree in this language contains the operations – ρ-

This work was supported by the Science Development Fund under the President of Azerbaijan Republic - Grant № EIF-RITN-MQM-2 / IKT-2-2013-7 (13) -29/26/1 ‡

202


addition,

ρ-multiplying,

brackets,

integers,

symbol

ρ.

Wherein



the

formula



m  f m1 , m2 ,..., mk ,... is the operation code on 𝜌-language, and mi i  1, k - are arguments of

structural function that significantly more convenient of binary code proposed in [6]. Thus, for example, 𝑦 = (2𝜌) ∘ 𝜌  3𝜌 (here «» is the operation of 𝜌-addition, «∘» - the operation of 𝜌- multiplying, «( )» - brackets – is the operation code of the following tree):

and besides «elementary 𝜌-structure» - is the 𝜌-tree, which consists of two vertexes, connected by an edge (dangling vertex), «zero-structure» - is a е-tree, consisting of a single vertex; «operation е ∘ 𝜌 = 𝜌», and 𝜌-sum «» are compared totality of branches, suspended to the same vertex as to the root. Finally, for each ρ-structure are suspended the branches being either the tree mk , or ρstructure, or (conventionally) the null-structure е. The seniority of operations given by the condition: «∘» is older from «» and in the series of the operations of the structure expression this order can be modified by introducing «( )». Using the specified language and on the basis of the fact that for unmarked trees sufficient representative isomorphism class - canonical, the problem reduces to the identification and construction of the canonical representative, i.e. the eldest of each isomorphism class (older tree is a tree with the eldest lexicographical sequence branches), as well as for constructive transfer canonical trees in lexicographical order. It is assumed that for the isomorphic tree generates a permutation with repetitions and demanding numbering symbol errors reshuffle left to right in numerical order [6]. In [4] an enumerating algorithm of the non-isomorphic root trees is built, each of which is based on the previous one and it is minimally superior in the lexicographical sense. Based on the obtained results, in this work for fuzzy hierarchies are built the enumerating algorithms of fuzzy trees, which provide the computer generation of fuzzy hierarchy graphs. At the same time it is assumed that the fuzzy orgraph (in the Berge sense) there is a pair 𝐺 = (𝐸, Γ), and for a finite set of vertices 𝐸 as the universe, the mapping Γ compares to each vertex Х ∈ 𝐸 the fuzzy subset Γ (x),

203


̃, assigns to each vertex Х ∈ 𝐸 the fuzzy subset all arc G are focused. Introducing the mapping Γ ̃(Х) and of the universe 𝐸 and given by the formula Γ ̃𝑋 (Х𝑗 ) = Γ𝑋 (Х𝑖 ) Γ 𝑖 𝑗

̃𝑋 , Γ𝑋 – are the membership functions of fuzzy sets Γ ̃(Х𝑖 ), Γ(Х𝑗 ), respectively). (where Γ 𝑖 𝑗 Under fuzzy ortree we understand the fuzzy orgraph 𝐺 = (𝐸, Γ), for which: ̃(Х0 ) = ∅ = root of the tree 𝐺; 1. ∃! Х0 ∈ 𝐸: Γ ̃(Х𝑖 )] consists of a single vertex, other than Х𝑖 , i.e. for 2. ∀ Х𝑖 ≠ Х0 , Х𝑖 ∈ 𝐸: the carrier 𝑆[Γ each vertex, except the root, there is only one arc, included with this vertex; 3. If (Х𝑖1 , … , Х𝑖𝑟 ) – is the fuzzy path in 𝐺, т.е. ∀(Х𝑖𝑘 , Х𝑖𝑘+1 ), Х𝑖𝑘+1 ∈ 𝑆[Γ(Х𝑖𝑘 )], (𝑘 = 1, … , 𝑟 − 1), then Х𝑖 ∉ 𝑆[Γ(Х𝑖𝑟 )] – is the absence of contours. For the given full fuzzy orgraf 𝐺 = (𝐸, Γ), if 𝑚𝑖𝑗 = Γ𝑋𝑖 (Х𝑗 ), (𝑖, 𝑗) = 1, 2, … , 𝑛, the matrix 𝑀 = ‖𝑚𝑖𝑗 ‖ is the adjacency matrix with the main diagonal elements equal to one. Any subgraph of fuzzy orgraph 𝐺, satisfying the conditions 1-3 of the fuzzy ortree induced from 𝐺, the set of which will be 𝐺̃𝑘 . It is shown that if any tree is uniquely determined by the structural formula, for the fuzzy tree further order of the vertices indicates the levels from left to right, with the weight of the arcs are determined from the adjacency matrix of the graph 𝐺. For example, the fuzzy tree 𝑋1

0,1 𝑋2 0,5 𝑋6

0,8 0,5

1 𝑋3

𝑋5 0,3

0,3 𝑋4 𝑋 7 0,2

𝑋8

𝑋9

uniquely represented as follows: {ρ2  2ρ  (ρ2  ρ) ∘ ρ / 𝑋1 ; 𝑋2 , 𝑋3 , 𝑋4 , 𝑋5 ; 𝑋6 , 𝑋7 , 𝑋8 , 𝑋9 } here ρ – is the elementary structure , «∘» - sign of ρ-multiplying, «» - sign of ρ–sum; after the vertical line indicates the order of the vertices of the levels from left to right, and the levels are separated by the sign « ; »; weight of arcs are determined from the adjacency matrix of the graph, from which is induced this fuzzy ortree.

204


It is shown that in the case of fuzzy trees, as they are marked with (vertex labels), the number of trees that are isomorphic to the present, will have more than in the case with the unmarked trees. Are fixed as follows rules for the selection of the canonical representative of the isomorphism class of fuzzy trees (order of precedence): 1) the application of the rules for determining of precedence for ordinary trees [4]; 2) if these rules are not enough to select canonic, then compare the leftmost path of trees on weights and choose a tree with more weight as a canonic; 3) If the leftmost path have the same weights, then go directly to the right branches, and again compare their weights, etc.; 4) if there are trees coinciding on all parameters of the rules 1) -3), then any one of them is selected as the canonic. It is shown that to all representatives of any class of isomorphism corresponds one and the same hierarchical structure, i.e., There is a one-to-one correspondence between the canonical representatives and hierarchical structures. An algorithm was proposed for enumerating all fuzzy trees isomorphic to this, including the transfer of ordinary trees [4], the account marked vertices, followed by selection according to the rules 1)-4) of the canonical representative. For this is constructed all canonical unmarked trees with the number of ribs from 0 to 𝑀 = 𝑛 − 1, on system recurrently formulas: Fi0  e  nm F m  k f m   m  1, M  k ki i 1  n m  m, n m  n m F m k  2, k k k k -1 m  1 m m m m   f ki  e, f ki  f ki Fk -1 i  1, m k  2, k m





   

n km

here  - is the sign of ρ–sum of n mk structures; k m - the number of m -ribs non-isomorphic i 1

structures; Fkm - k -th in series m -ribs canonical tree; f kim - 𝑖-th pretornal tree branch Fkm ; n mk the number of the pretornal branches of the tree Fkm ; e - the graph, consisting of one vertex. The dependences n mk  n mk Fkm-1 , f kim  f kim Fkm-1  understood as the application of the algorithm for constructing directly of the older pretornal branch from the set, suggested in [1].

205


Then, in each of the resulting trees we do all sorts of marking on the vertices and thus obtain the whole set 𝐺̃𝑘 . At the final stage, making the transitive closure of each ortree and adding loops at the vertices of a non-zero degree of the outcome, we get the graph of a partial ordering with the highest vertex, which is a representation of the hierarchical structure. The paper presents the combinatorial considerations: is shown that the marked tree with the 𝑛 vertices 𝑋1 , … , 𝑋𝑛 has

   X  !  isomorphs trees, where n



i

   X i  - is the number of

i 1

the arcs, leaving from the vertex 𝑋𝑖 . For example, for 𝑚 = 4 the above algorithm builds the following trees (see, fig.). For each of them the first vertex can be marked with 𝑛 methods, the second - (𝑛 − 1) methods, ..., fifth - (𝑛 − 4) methods. Total from each 𝑚-ribs ortree (𝑚 = 4) , we obtain

𝑛!

𝑛(𝑛 − 1)(𝑛 − 2)(𝑛 − 3)(𝑛 − 4) = (𝑛−5)!, i.e. in the general case

𝑛! (𝑛−𝑚−1)!

𝑛 ) various 𝑚+1

=(

fuzzy ortree. Recall that the weights of the arcs are determined from the adjacency matrix of the graph, from which is induced these trees. Then the number of all 𝑚-ribs fuzzy ortrees is equal to 𝑛 ( ) ∙ 𝑘𝑚 . 𝑚+1 Thus, 𝐺̃ consists of

n 1

 n 

  m  1  k

m 0





m

  fuzzy ortrees. Then using the above rules 1)-4) in 

each class of isomorphism is defined the canonic tree. The totality of all these canonics gives us 𝐺̃𝑘 . The total number of fuzzy ortree in 𝐺̃𝑘 is equal to   n       n 1 k m   m  1 k    m1   m  0 k 1       i !   ,  k    i 1

 

206


 n   is the number of the fuzzy ortrees, obtained from Fkm by all sorts of marks of vertices here   m 1 k  n   is same  k  1, k m , and (in fact   m 1 k

 m 1      i !  - the number of the trees, isomorphs to Fkm  i 1 k

 

, if considering that the vertices are marked (  i - the number of arcs leaving from 𝑖–th vertex). In particular, for the given figure and 𝑛 = 5, 𝑘𝑚 = 4, we have  n     5 ! / 0 !  120 i  1,4  m  1 i

 

 

 5      i !   1;  i 1 1

 

 5      i !   2;  i 1 2

 5      i !   6;  i 1 3

 

 5      i !   24  i 1 4

120/1 + 120/2 + 120/6 + 120/24 = 205, i.e. the number of the canonic 4-ribs fuzzy ortrees is equal to 205 (provided that 𝑛 = 5). Keywords: Γ-fuzzy hierarchy, fuzzy algebraic structures, fuzzy ortree. AMS Subject Classification: 90B06, 90B10, 90B20.

References 1. Imanov K.S., Algebraic models of structural-fuzzy systems, Proceedings of IAM, Vol.3, No.1, 2014. 2. Imanov K.S., On the ill-defined hierarchies, Reports of ANAS, No.1-2, 1995. 3. Imanov K.S., On the computer modeling of hierarchical structures generated by the trees, Reports of ANAS, No.2-3, 1989. 4. Klimov A.N., The algebraic representations of structures described by the graph-trees, A and T., No.7, 1979. 5. Under. ed. of Faradzhev A.I., Algebraic Researchs in Combinatorics, Nauka, 1978.

207


FEATURE SELECTION METHOD FOR RECOGNITION OF HAND-PRINTED CHARACTERS Elviz Ismayilov1 1


There are a lot of both theoretical and practical difficulties associated with a huge variety of possible ways to write individual hand-printed characters, which don’t allow the finally solution of the recognition problem of hand-printed characters [1]. Many different approaches and algorithms have been offered for hand-printed character and digit recognition. Different algorithms have been developed using feature extraction methods, classifier algorithms, artificial neural networks, fuzzy logic techniques, et al., but there are many advantages of pattern recognition by features extraction: simplicity of implementation; fast training process; stability to form distortion et al. The large number of feature extraction methods reported in the literature, but to define optimal set of features for given class of symbols or images is very important [2]. In this paper was proposed method for optimal features selection process for hand-printed recognition task. Assume that, F = { f1, f2 ,..., fn } Ì Rn is the set of all features. Algorithm for definition of more appropriate vector of features consists of following steps: 

Step 1: numbers from 1 to n converted into binary numeration, we get following vectors:

bi = {bi1, bi2 ,..., bi3 },i =1, M, M = 2n-1 , here if bi j =1, j =1, n then jth feature aggregates in recognition, else, e.g. bi j = 0, j =1, n thenfj does not participate in recognition process; 

Step 2: Training and recognition is carried out for each vector of features and are calculated following indicators: minimized values of error function Ei ,i =1, M, , number of iterations

Iti ,i =1, M, time periods required for recognition Ti ,i =1, M, and percentage of recognizing symbols Pi ,i =1, M, ; 

Step 3: At last, must be selected the main indicator for recognition, for example the percentage of recognized symbols, then the optimal vector of features defined considering * to the highest value of correct recognition percentage, if P = max Pi , then the vector of i=1,M

features considered to the highest percentage is the optimal vector of features for given task. 208


For the best understanding of proposed method an example for recognition of geometrical figures is described below: Example: assume that, we have two features for recognition of geometrical figures (triangle, square, rectangle, circle, et al.): number of angles f1 and number of sides f2. Step1 : m=3, vectors of features are defined: b1=(0; 1); b2= (1; 0), b3=(1;1) it means that, number of angles participates in the second and the third recognition process and is not on the first recognition process; Step 2: as main indicator we take correct recognition percentage and calculate Pi ,i =1, 2,3 ; Step 3: in our example P1=92 %; P2=89%; P3=93,5%, P* = 93,5, i*=3 then optimal vector for given task is the b3=(1,1), (f1; f2) is the optimal set of features; For experiments, we get Azerbaijani hand-printed characters and recognition method based on ANFIS [3]. After preprocessing and skeletization of hand-printed characters, for their classification have been extracted a lot of feature classes. Using algorithm described above have found eight the most appropriate features with recognition percentage 94,3 % : 

F1:F6 – numbers of intersection points of character with lines carried out in rectangle where located recognized character;



F7 – diameter of the largest closed area of symbol;



F8 – location of the largest closed area of symbol [4].

These features are fuzzy sets with different number of terms and they are input variables of neuro – fuzzy network constructed for Azerbaijani hand-printed character recognition. The main advantage of proposed method is that, this method can be apply to recognition of all type of pattern recognition tasks: hand-printed character recognition of different alphabets, digits, geometrical figures et al. Also in this method number of aggregated features defines during experiments, it may be a one feature or maximal number of available features. Keywords: features extraction, hand-printed character, pattern recognition, optimal features, ANFIS. AMS Subject Classification: 68T10.

References 1. Aliyeva N., Ismayilov E., Analysis of effect of different feature classes in learning systems. Proceedings of 24th Mini EURO Conference on Continuous Optimization and Information-based technologies in the financial sector, Izmir, Turkey, June 23-26, 2010. pp. 270-274.

209


2. Trier O.D., Jain A.K., Taxt T., Feature Extraction Methods for Character Recognition – A survey, Pattern Recognition, Vol. 29, No. 4, 1996, pp. 641-662. 3. Ismayilov E., Bakhishoff U., Hand-printed character/digit recognition by ANFIS system, Journal of Contemporary Applied Mathematics, Vol.3, No. 2, 2014, pp. 17-24. 4. Ismayilov E., Ismayilova N. Fuzzy Features Extraction for Hand-Printed character/digit Recognition System, INISTA 2014 (Italy), pp 249-253.

FORMULATING AND OBTAINING T-BEST APPROXIMATION IN A FUZZY NORMED SPACEUSING SOME OPTIMIZATION METHODS Karim Ivaz1, Ali Beiranvand1 1

Faculty of Mathematical Science, University of Tabriz, Tabriz, Iran e-mail: [email protected]

L.A. Zadeh introduced fuzzy set theory in1965. Later, many researchers have applied this theoryto the well-known results in the classical set theory.Therefore it has become an area of active research fornearly forty years. Here some example of theapplications of fuzzy logic inengineering are given, such as, applications inthe population dynamics, the quantum physics, the control of chaos, the computer programming, and in bifurcation of nonlinear dynamical systems. One of the most important problems in fuzzy topology is to obtain an appropriate concept of fuzzy metricspace. This problem has been investigated by many authors from different points of view, see for example [1]. In particular, George and Veeramani [2] have introduced and studied a notion of fuzzy metric space with the help of continuous t-norms. Veeramani [4] in2001 introduced the concept of t-best approximation in fuzzy metric spaces and Vaezpour and Karimi [3] introduced the concept of t-best approximation in fuzzy normed spaces. However, there is no systematic way to obtain the t-best approximation in a given fuzzy normed space. In this paper the problem of finding t-best approximation will be considered as a minimization problem. To do this,t-best approximation will first be formulated as constrained maximization problem, then using penalty method it will be transformed into a unconstrained maximization problem. To solve the resulted problem, using inexact steepest descent algorithm, we reformulate the mentioned problem a minimization problem. At the end two numerical examples are given. The numerical results shows that the inexact steepest descent algorithm is an efficient and powerful method to attack this type of problems.

210


Keywords: t-best approximation, fuzzy normed space, steepest descent. AMS Subject Classification:46Bxx, 90C30, 49M37.

References 1. Bag. T., Samanta S.K., Finite dimensional fuzzy normedlinear spaces, J. Fuzzy Math., Vol.11, No.3, 2003, pp.678–705. 2. George A., Veeramani P., On some result in fuzzy metricspace, Fuzzy Sets and Systems, Vol.64, 1994, pp.395–399. 3. Karimi F., Vaezpour S.M., t-Best approximation in fuzzy normed spaces, Iranian Journal of Fuzzy Systems, Vol.5, No.2, 2008, pp.93–99. 4. Veeramani P., Best approximation in fuzzy metric spaces , J.Fuzzy Math., Vol.9, No.1, 2001, pp.75–80.

THE EXISTENCE THEOREM FOR FUZZY DIFFERENTIAL EQUATION M.M. Mutallimov1, A.A. Murtuzayeva1, B.M. Gasimov2, A.Kh. Abdullayev2 1

Institute of Applied Mathematics, Baku State University, Baku, Azerbaijan 2 Azerbaijan University of Economics, Baku, Azerbaijan e-mail: [email protected]

A fuzzy set A is characterized by a generalized characteristic function  A (.) , called membership function, defined on a universe X , which assumes values in [0,1] . For any

 [0,1] denote by

A  x  X :  A ( x)  a the  - cut of A . Let  A (.) is an upper

semicontinuous function and

sup p( A)  x  X :  A ( x)  a is bounded set of X . A fuzzy

 set is a fuzzy number if X  R and for any  [0,1] , the  -cut A is convex and the height

of A , that is, sup  A ( x) has to be equal to one. This fuzzy number usually is called convex xX

normal fuzzy number. Let's define by F the class of convex normal fuzzy numbers. Then for any a  F the set of

 -cut of fuzzy number a the interval a  [ La ( ), Ra ( )] ,  [0,1] , is defined. Let

a  F , b F and a  [ La ( ), Ra ( )] , b  [ Lb ( ), Rb ( )] . Then  -cut of fuzzy number

ab

and

ka, k  0 , define as

a  b  [ La ( )  Lb ( ), Ra ( )  Rb ( )]

ka  [kLa ( ), kRa ( )] , respectively. 211

and


Note that F is not a linear space (the operation of subtraction is not defined in F ). We consider the set of pairs (a, b)  F  F and define the operation of addition, multiplication and equivalency as

(a1 , a2 )  (b1 , b2 )  (a1  b1 , a2  b2 ), k  (a, b)  (ka, kb), k  0, (1)

(1)  (a, b)  (b, a), (a1 , a2 )  (b1 , b2 )  a1  b2  a2  b1 .

As zero element of this space is taken the pair (0,0) , i.e. the set of elements (a, a), a  F . From last relation (1) we get (a, a)  (0,0) . For any x  (a, b) ,  x  (b, a) .It is clear that

x  ( x)  (a, b)  (b, a)  (a  b, a  b)  (0,0) The set of all pairs (a, b)  F  F forms a structure of a linear space. Let

x  (a1 , a2 )  F  F , y  (b1 , b2 )  F  F .   Then ai  [ Lai ( ), Rai ( )] , bi  [ Lbi ( ), Rbi ( )] ,  [0,1].

For any x, y  F  F define the scalar product as 1

1 x y [(La1 ( )  La2 ( ))(Lb1 ( )  Lb2 ( ))  20



(2)

 ( Ra1 ( )  Ra2 ( ))(Rb1 ( )  Rb2 ( ))]d It may be shown that this definition satisfies all requirements of the scalar product. We denote this space by LF . Norm in this space is defined as 1

x

2

1  [(La1 ( )  La2 ( ))2 ( Ra1 ( )  Ra2 ( ))2 ]d 20



(3)

We define distance between two fuzzy numbers a  F and b F as

 (a, b)  x  y

(4)

where x  (a,0), y  (b,0) . Now, let’s consider fuzzy function f (t )  F for each t  t0 ,t1  and define a derivative of the function f (t ) . For any  [0,1] ,

f (t )  [ L f (t ) ( ), R f (t ) ( )] ,  [0,1] is called  -cut of the function f (t ) . 212

(5)


Definition. Let there exists such  (t )  F ,  (t )  F , t  t0 ,t1  , that

( f (t  t ,0)  ( f (t ),0)  ( (t ), (t )) t 0 t lim

Then the pair

(6)

( (t ), (t ))  F  F is called a derivative of the function f (t ) at the point

t  t0 ,t1  . This definition may be written in the following form

( f (t  t ,0)  ( f (t ),0)  ( (t ),  (t )) t 0 t lim

(7)

where ( (t ),  (t )) are  -cut for the functions  (t ) ,  (t ) . It is shown that, if L f (t ) ( ), R f (t ) ( ) is continuous differentiable relatively t , then f (t ) is differentiable. Each function f (t ) may be considered as an element ( f (t ),0) from F  F . Then

( f1 (t )  f 2 (t ))  f1(t )  f 2(t )

(8)

Now, let f (t ) be a pair of fuzzy functions, i.e. f (t )  ( f1 (t ), f 2 (t )),t  (t0 , t1 ). From relation f (t )  ( f1 (t ),0)  (0, f 2 (t ))  ( f1 (t ),0)  ( f 2 (t ),0) we see, that the derivative of the function f (t ) also is a pair from F  F . For any    (t )  F  F , which  (t )  F  F , consider the scalar product f (t )   (t ) defined by the formula (2). It can be shown that T



T

f ( )   ( )d  f ( )   ( ) t 

t

T

 f ( )   ( )d , t, T  (t0 , t1 )

(9)

t

Let D R n be a given bounded domain with smooth boundary S and fuzzy function

u  u( y)  F  F depends on the parameter y  ( y1 , y2 ,..., yn )  D , i.e. u  u( y), y  D . We’ll write

u  C (D) , if the function u ( y ) continues on y in D . Analogically we can

1 define U  C ( D) . Consider the boundary problem

u   f ( y), y  D ,

u( )  g ( ),   S.

(10) (11)

Let f ( y)  ( f1 ( y), f 2 ( y))  F  F , y  D, g ( )  ( g1 ( ), g 2 ( ))  F  F ,  D.

In the difference of traditional problems, here solution of the problem (6), (7) is fuzzy function

213


u  u( y)  F or pair of the fuzzy function u( y)  (u1 ( y),u2 ( y))  F  F. For the of simplicity, this type functions we’ll call fuzzy function. Equation (10) and boundary condition (11) we understand as equality pair of the domains. Theorem 1. Let f i  C1 ( D)  C ( D ) and g i  C (S ),i  1,2. Then there exists unequal solution u( y)  (u1 ( y),u2 ( y))  F  F , y  D of the problem (10), (11). Theorem

2.

Let

for

any

yD

and

 S

be

fuzzy

function

and

f  C 1 ( D)  C ( D ), g  C (S ). Then there exists unequal fuzzy function u  u( y)  F solution of

the problem (10), (11). Keywords: fuzzy number, scalar product, fuzzy function, boundary problem. AMS Subject Classification: 03E72, 34A07.

INDICATORS OF SUSTAINABLE DEVELOPMENT STRATEGY AZERBAIJAN REPUBLIC FOR THE INVESTIGATION OF FUZZY METHODS§ S.M. Salimov1, A.M. Maharramov2 1

Azerbaijan National Academy of Sciences, Institute of Economy, Baku, Azerbaijan 2 Baku State University, Baku, Azerbaijan e-mail: [email protected]

The impact of global economic processes towards sustainable development of the economy of Azerbaijan, in the formation for sustainable development strategy of the country is reflected significantly. Consideration of this problem, in solar and windy energy network and investigation of the use independent regime and the use of the same indicators for application of the Energy industry in Azerbaijan Republic is the main set goal. Posed objective are as follows for achievement this goal: develop that can be applied a system of indicators species alternative energy strategies for sustainable development of Azerbaijan in the desired scope; analyzing and assessments the current fuzzy system of the country in terms of the sustainable development strategy. Sustainable development of the economy and the impact of international economic processes, the country's sustainable development strategy, which shows itself in the formation

§

The work is supported by the joint grant of ANAS and SOCAR №26, 2014-2015.

214


significantly. Considering this issue, in order to ensure the sustainable development strategy for the implementation of international experience suggests that the main goal of the results. To achieve this goal, the proposed issues: the state of indicators of sustainable development in the countries of the world, prepared for the United Nations "Sustainable development strategy methodology" in the analysis; the economic indicators of the World Economic Forum "Global Competitiveness Index," the analysis; Heritage Foundation's "Index of Economic Freedom" in the analysis; indicators of environmental, political, and environmental correlations center at Yale University and Columbia University, jointly prepared by the experts of the "Environmental Sustainability Index" and "Environmental Situation Index" in the analysis; social indicators of the United Nations Development Programme (UNDP) prepared by the "Human Development Index" in the analysis; Department of Social and Economic Organization of the United Nations (UNDESA), prepared by the "knowledge-based society Index" of data analysis methodologies, as well as the legitimacy of the generalized fuzzy methods. The idea of the project is to explore new ways of strategy for sustainable development, assess the results of the examination of the country's sustainable development strategy and international experience to add to the model to develop new proposals. Indicators of sustainable development strategy of the Republic of Azerbaijan in the investigation of fuzzy methods, pressing problems of the modern era. The theme of sustainable development in the world's scientific literature the last 30 years occupies the central position, the analysis and evaluation of the concept of sustainable development economists, philosophers, sociologists, energy and ecology have to research as the fundamental problem of the modern period. In 1987, after the publication of Bruntland Report ("Our Common Future") by the United Nations, some national and international organizations developed the indicators of some aspects complex for sustainable development strategy. Untied Nation Organization held a conference of the “Rio+20 sustainable development strategy” in 20-22 June, 2012 in Rio de Jeanery. The final document of the conference was held under the slogan of the "Our desired future" was accented the need of the effective use of alternative energy in the sustainable development strategy of each country. On the recommendation of United Nation Organization, countries, international countries and non-governmental organizations should agree on national, international and regional levels of indicators of the sustainable development strategy. 215


According to the methodology of the Sustainable Development Strategy prepared by the UN for the world countries is divided into four aspects: social, economic, environmental and institutional aspects. According with Measurements of the main aspects of the indicators for sustainable development concept respectively themes and parameters of the themes are divided into 14 themes and topics by 30 parameters. According to the classification of some of the measurement parameters are related to more than one and more themes parameters. The themes and themes of parameters was prepared relevant conceptual structure of the UNO for measurement of the energy indicators of the sustainable development strategy. Energy indicators of the sustainable development strategy that make up the core list, according to the classification of measuring 30 indicators divided into 3 groups of aspects (social, economic and environmental). They agree on the classification of those 7 respectively divided into 19 parameters. According with measurement classifications of the indicators are related to more than one and themes and parameters. According to the project in Azerbaijan applied connecting devices to the network, intended consideration of physical - technical opportunities in the use of solar energy and windy and selfapplication mode. Economic indicators of the World Economic Forum's "Global Competitiveness Index", a report of the evaluation of macro- and micro-economic analysis in addition to competitiveness (2013-2014 according to the years), which allows a comparative analysis methodology are published in 148 countries. The main parameters of the methodology is divided into 12 groups and 114 of these criteria. It should be noted that the republic in 2014-2015 according to the methodology of the ranked 38 among 144 countries, according to the years 2012-2013, the country was ranked as 46 th. Recent years in the study of economic processes in order to produce high results use the modern new theory and Soft Computing computer technology. An ethnic Azerbaijani American scientist L. Zadeh is the author of this theory. New technologies proposed by L. Zadeh: fuzzy logic, neural network, neuro-fuzzy systems and others were given the opportunity to solution of important real problems, which could not be solved before. The fuzzy theory was founded in 1965, the economy began to be implemented in the 70s of the last century. Lyutvi-Zade, in addition to the theoretical foundations

216


of a new science gave and showed its practical application and fuzzy logic theory became the basis of a new generation of intellectual control systems. Number of scientific papers devoted to the study of micro and macroeconomic theory based on Soft Computing increases significantly. There are fuzzy majority in the global scientific community and some interesting work relating to the application of soft calculation on economic issues. Among them we can show such as "Analysis of investment activity of mutual funds fuzzy majority" (K.Preyin), "Artificial Intelligence in financial and investment activities" (R.Trippinin) and others. In the 1999 macroeconomic research dedicated to measurements of the shadow economy scientists from New Zealand and David R.Draeseke Glees had special significance. Keywords: sustainable development, energy strategy, of neuron network, fuzzy logic, rules of inference, system of the fuzzy etc. AMS Subject Classification: 34B99, 03B52, 46F10.

GENERALIZED FUZZY ENTROPY OPTIMIZATION METHODS WITH APPLICATION ON WIND SPEED DATA Aladdin Shamilov1, Sevil Senturk1, Nihal Yilmaz1 1

Anadolu University, Faculty of Science, Department of Statistics, Eskisehir, Turkey e-mail: [email protected], [email protected], [email protected]

In the present study, we have formulated a Fuzzy Entropy Optimization Problem (FEOP) and proposed sufficient conditions for existence of its solution. Mentioned problem consists of maximizing Fuzzy Entropy Measure(FEM) with respect to membership functions with finite number of the fuzzy values subject to constraints generated by given moment functions. The existence of its solution is proved by virtue of convexity property of FEM, the implicit function theorem and Lagrange multipliers method. Moreover, Generalized Fuzzy Entropy Optimization Methods (GFEOM) in the form of MinMax(F)EntM and MaxMax(F)EntM are suggested on the basis of primary maximizing FEM for fixed moment vector function in order to obtain the special functional with MaxEnt values of FEM and secondary optimization for mentioned functional with respect to moment vector functions. Distributions obtained by mentioned methods are defined as (MinMax(F)Ent )mwhich is closest to a given membership function and (MaxMax(F)Ent )m which is furthest from a given 217


membership function. Furthermore, fuzzy data analysis is fulfilled by applying GFEOM. In application, wind speed data measured on December, 2005in Eskisehirisconsidered and the results are obtained by using MATLAB. The performances of distributions of (MinMax(F)Ent )m and (MaxMax(F)Ent )m are established by Chi-Square, Root Mean Square and information criteria. 1. Maximum value of Fuzzy Entropy Measure. The concept of fuzzy sets De Luca and Termini [4] suggested that corresponding to Shannon’s [3] probabilistic entropy the measure 𝐻(𝐴) of fuzzy entropy for fuzzy set 𝐴 [1, 2] containing finite number elements can be expressed by formula 𝐻(𝐴) = −[∑𝑛𝑖=0 𝜇𝐴 (𝑥𝑖 )𝑙𝑜𝑔𝜇𝐴 (𝑥𝑖 ) + (1 − 𝜇𝐴 (𝑥𝑖 )) 𝑙𝑜𝑔(1 − 𝜇𝐴 (𝑥𝑖 ))].

(1)

Fuzzy Entropy Optimization Problem (FEOP) consists of maximizing Fuzzy Entropy Measure (FEM) (1) with respect to membership functions 𝜇𝐴 (𝑥) with finite number of the fuzzy values 𝜇𝐴 (𝑥𝑖 ) , 𝑖 = 0,1,...,𝑛 subject to constraints ∑𝑛𝑖=0 𝜇𝐴 (𝑥𝑖 ) 𝑔𝑗 (𝑥𝑖 ) = 𝜇𝑗 , 𝑗 = 0,1,2, … , 𝑚 ,

(2)

where 𝑔0 (𝑥) ≡ 1; 𝜇𝑗 , 𝑗 = 0,1,2, … , 𝑚 are moment values of 𝜇𝐴 (𝑥𝑖 ) , 𝑖 = 0,1,...,𝑛 with respect to moment functions 𝑔𝑗 (𝑥) , 𝑗 = 0,1,2, … , 𝑚; 𝑚 < 𝑛 and 𝜆𝑗 (𝑗 = 0,1, … , 𝑚).

The problem of

maximizing fuzzy entropy function (1) has solution 𝜇𝐴 (𝑥𝑖 ) =

1

∑𝑚 𝜆 𝑔 (𝑥 ) 1+𝑒 𝑗=0 𝑗 𝑗 𝑖

, 𝑖 = 0,1, … , 𝑛

(3)

where 𝜆𝑗 , 𝑗 = 0,1, … , 𝑚 are Lagrange multipliers. We note that mentioned problem for Entropy Optimization Measure is suggested and solved in [5-7]. In application, one of the obtaining 𝐹𝑟𝑞𝑖

methods of membership values is used in the form 𝜇𝑖 = ∑𝑛

𝑖=0 𝐹𝑟𝑞𝑖

, 𝑖 = 0,1, … 𝑛[10].If Eq. (3) is

substituted in Eq.(1), the maximum value of FEM (1)is obtained: 𝑚

𝑚𝑎𝑥𝐻𝐴 = 𝑈(𝑔) =

− ∑𝑛𝑖=0

𝑙𝑛

∑ 𝜆 𝑔 (𝑥 ) 𝑒 𝑗=0 𝑗 𝑗 𝑖

∑𝑚 𝜆 𝑔 (𝑥 ) 1+𝑒 𝑗=0 𝑗 𝑗 𝑖

+ ∑𝑚 𝑗=0 𝜆𝑗 𝜇𝑗 .

(4)

2. Generalized Fuzzy Entropy Optimization Methods. Solving the MinMax(F)Ent and MaxMax(F)Ent problems require to find vector functions (𝑔0 , 𝑔(1) (𝑥)) and (𝑔0 , 𝑔(2) (𝑥)) , where 𝑔0 (𝑥) ≡ 1, 𝑔(1) ∈ 𝐾0,𝑚 ,

𝑔(2) ∈ 𝐾0,𝑚 minimizing and maximizing functional 𝑈(𝑔)defined by

(4). It should be noted that 𝑈(𝑔) reaches its minimum (maximum) value subject to constraints (2) generated by function 𝑔0 (𝑥) and all 𝑚 −dimensional vector functions 𝑔(𝑥), 𝑔 ∈ 𝐾0,𝑚 . In other words, minimum (maximum) value of

𝑈(𝑔) is least (greatest) value of values

𝑈(𝑔)

corresponding to𝑔(𝑥), 𝑔 ∈ 𝐾0,𝑚 . In other words, (MinMax(F)Ent)m ((MaxMax(F)Ent)m ) is 218


distribution giving minimum (maximum) value to functional 𝑈(𝑔) along of all distributions generated by (mr ) number of moment vector functions𝑔(𝑥), 𝑔 ∈ 𝐾0,𝑚 . Mentioned distributions can be denoted by (MinMax(F)Ent)m and (MaxMax(F)Ent)m . 3. Application of (𝐌𝐢𝐧𝐌𝐚𝐱(𝐅)𝐄𝐧𝐭)𝐦and (𝐌𝐚𝐱𝐌𝐚𝐱(𝐅)𝐄𝐧𝐭)𝐦Methods to Fuzzy Data. In the present study, the data set is taken from [8, 9] wind speed measured in Eskisehir. In this section, it is shown that the (MinMax(F)Ent)m and (MaxMax(F)Ent)m distributions obtained by Max(F)Ent theory is suitable for the assessment of the wind energy potential. In order to choose the best distribution function for wind speed data can be determined according to the lowest values RMSE, 𝜒 2 and FEM. In our investigation as moment functions 𝑔0 (𝑥) = 1, 𝑔1 (𝑥) = √𝑥 , 𝑔2 (𝑥) = 𝑙𝑛𝑥, 𝑔3 (𝑥) = 𝑙𝑛(1 + 𝑥) , 𝑔4 (𝑥) = 𝑙𝑛(1 + 𝑥 2 ) are chosen. Table 1. The obtained results for (MinMax(F)Ent)m and (MaxMax(F)Ent)m Distributions Distributions of 𝐻

MinMax(F)Ent and

Moment

Calculated value

Table value of

MaxMax(F)Ent

Constraints

(𝑀𝑎𝑥𝑀𝑎𝑥(𝐹)𝐸𝑛𝑡)1

1, √𝑥

(𝑀𝑖𝑛𝑀𝑎𝑥(𝐹)𝐸𝑛𝑡)1

RMSE

of Chi-Square

Chi-Square

3.3698

0.0023

2 𝜒17,𝛼 =27.587

0.0428

1, 𝑙𝑛(1 + 𝑥)

2.5656

0.0025

2 𝜒17,𝛼 = 27.587

0.0447

(𝑀𝑎𝑥𝑀𝑎𝑥(𝐹)𝐸𝑛𝑡)2

1, √𝑥 , 𝑙𝑛(1 + 𝑥)

2.5473

0.0024

2 𝜒16,𝛼 = 26.296

0.0436

(𝑀𝑖𝑛𝑀𝑎𝑥(𝐹)𝐸𝑛𝑡)2

1, √𝑥 , 𝑙𝑛𝑥

0.3744

0.0040

2 𝜒16,𝛼 = 26.296

0.0563

Figure 1. Histogram and obtained (MinMax(F)Ent)m and (MaxMax(F)Ent)m values for wind speed data

MinMax(F)Ent and MaxMax(F)Ent 0.16 Data set Histogram MinMax(F)Ent1 MinMax(F)Ent2 MaxMax(F)Ent1 MaxMax(F)Ent2

0.14

Membership values

0.12 0.1 0.08 0.06 0.04 0.02 0

0

2

4

6

8

10 m/s

219

12

14

16

18

20


In this study, it is shown that (MaxMax(F)Ent)1 and(MaxMax(F)Ent)2 distributions successfully represent fuzzy data.(MaxMax(F)Ent)1distribution is more suitable for fuzzy data than (MaxMax(F)Ent)2 distribution in the sense of RMSE criteria while (MaxMax(F)Ent)1 is more suitable than (MaxMax(F)Ent)2 distribution in the sense of fuzzy entropy measure. Furthermore, our investigation indicates that GFEOM in fuzzy data analysis yields reasonable results.

Keywords: fuzzy entropy measure, generalized fuzzy entropy optimization problem, MinMax(F)Ent, MaxMax(F)Ent distributions. AMS Subject Classification: 94A17, 62B86, 28E10, 62A86.

References 1. Zadeh L.A., Fuzzy sets, Information and Control, 8, 1965, pp.338-353. 2. Lee Kwang H., First Course on Fuzzy Theory and Applications, Springer, 2002. 3. Kapur J.N., Kesavan H.K., Entropy Optimization Principles with Applications, Academic Press, New York, 1992. 4. De Luca A., Termini S., A definition of non-probabilistic entropy in setting of fuzzy set theory, Information and Control, Vol.20, 1971, pp.301-312. 5. Shamilov A., Entropy, Information and Entropy Optimization, Turkey, 2009. 6. Shamilov, A., A development of entropy optimization methods, WSEAS Trans. Math., 5, 2006, pp.568–575. 7. Shamilov A., Generalized entropy optimization problems and the existence of their solutions, Phys. A: Stat. Mech. Appl., 382, 2007, pp.465–472. 8. Shamilov A., Generalized entropy optimization problems with finite moment function sets, Journal of Statistics and Management Systems, Vol.13, 2010, pp.595-603. 9. Shamilov A., Usta I., Kantar Y., The Distribution of minimizing maximum entropy: alternative to Weibull distribution for wind speed, Proceedings of the 9th WSEAS International Conference on Applied Mathematics, 2006, pp.605-610. 10. Shamilov A., Ozdemir S.,Yilmaz N., Generalized entropy optimization methods for survival data, ALT2014: 5th International Conference on Accelerated Life Testing and Degradation Models, Pau, France, 2014, pp.174-183. 11. Bharathi Devi, B., Sarma V.V.S., Estimation of fuzzy membership from histograms, Information Sciences, 1985, pp.43-59.

220


ECONOMIC AND MATHEMATIC MODEL IN A VIRTUAL BUSINESS BY APPLYING A FUZZY INSTRUMENT**†† R.U. Shykhlinskaya1, E.R. Shafizadeh1, Т.F. Murtuzaliyev2 1

Institute of Applied Mathematics, Baku State University 2 AFB Bank

Standard mathematical programming problem is usually formulated as a maximization (or minimization) problem of the prescribed function in a given set of admissible alternatives that are described by a system of equalities or inequalities. A set of admissible alternatives constitutes an array of every possible means of distributing resources that the expert is going to put in the given operation. While modelling a decision-making in such a form of real problem, the expert has only fuzzy description of parameters, through linguistic variables. Different forms of description of source information determine different formulation of problems of fuzzy mathematical programming. Problem definition. Let us construct an economic-mathematical model of maximizing profit for an Internet shop, applying a fuzzy instrument. For this purpose, we have to define main parameters that have impact on the profit in this environment. In the virtual environment, products are fixed. As these products are in a certain base, from where they are delivered to customers, there are no expenditures associated with the lease of premises, arranging of a showcase, salary etc. And this means there is maximization of profit. Let’s take into account that profit from the sale of i-product in the virtual business that depends on the variety of qualitative and quantitative indices – price of the product, Internet access service market, long-term change tendency of the potential market size, seasonal fluctuation of business activity, Internet providers’ service quality, price attractiveness, impact of the speed of access to Internet resources, sources of client inflow and outflow etc. that have fuzzy description. To take them into account in a classical model is either generally impossible or is possible with significant assumptions. Fuzzy description of these parameters in the model, i.e. a fuzzy model may turn out to be more adequate.

**

This work was supported by the Science Development Foundation under the President of the Republic of Azerbaijan- Grant N EİF-RİTN-MQM-2/İKT-2-2013-7(13)-29/06/1-M-24. ††


221


Taking all these statements into consideration, profit from the sale of the unit i-product 𝑏𝑖 (𝑖 = ̅̅̅̅̅ 1, 𝑛) may be introduced as a variable depending on fuzzy indices 𝑎̃𝑗 ,

𝑗 = ̅̅̅̅̅̅ 1, 𝑚,

i.e.𝑏𝑖 (𝑎 ̃, ̃, ̃ ̃𝑗 ,𝑗 = ̅̅̅̅̅̅ 1, 𝑚 affecting the profit from the sale 1 𝑎 2 …𝑎 𝑚 ). Let us notice that fuzzy indices 𝑎 of unit i–product are such indices as incomes of consumers, living standards of consumers, fashion, prices of products-substitutes or complementary products etc., and technical parameters that are listed above. As 𝑎̃𝑗 ,𝑗 = ̅̅̅̅̅̅ 1, 𝑚 are fuzzy subsets, they have their own membership function assigned by the expert. Thus, the task may be introduced in the following fuzzy form: n

 b (a~ , a~ i 1

i

i1

i2

,..., a~im ) xi  max

𝑎̃𝑗 ⊂ 𝐴𝑗 .𝑗 = ̅̅̅̅̅̅ 1, 𝑚

(1) (2)

In this model, the target function (1) is assigned in an implicit form. It may be defined for each specific case using relevant mathematical or fuzzy methods. Let us give an example of the problem of achieving vaguely set goal with vaguely described criteria [1]. An example. Let us build a model of maximizing profit for a certain Internet shop. In this case, let us determine main criteria 𝑎̃𝑗 ,𝑗 = ̅̅̅̅̅̅ 1, 𝑚, on which profit of the Internet shop depends. For a simplicity, let us assume that profit 𝑏𝑖 depends on the price of i-type of product(с̃), 𝑖 cost of the ̃ ) and procedural expenses (𝑃̃) that are fuzzy subsets and accept web page (𝐴̃), advertisement (𝑀 values from corresponding universal sets assigned by the expert. Then, fuzzy model (1)-(2) will be in the following form [2]: n

~ ~ ~

 b (c~ , A, M , P ) x i 1

i

i

i

 max

(3)

c~i  Ci ,

(4)

~ A  D1 ,

(5)

~ M  D2

(6)

~ P  D3 ,

(7)

Having analysed, we would receive the following formula for fuzzy function (3): ∑𝑛𝑖=1 𝑏̃𝑖 𝑥𝑖 = ∑𝑛𝑖=1(𝑐𝑖𝑠ℎ − Here, N is the number of products. 222

̃ + 𝑃̃ 𝐴̃+𝑀 𝑁

) 𝑥𝑖 → 𝑚𝑎𝑥

(8)


̃ , 𝑃̃, in turn, depend on certain qualitative and quantitative iundices.That Fuzzy indices 𝐴̃, 𝑀 is to say, each of the above-listed values of entry is, in its turn, is exit of a block expressed by fuzzy qualitative and quantitative indices. This model is reduced to the fuzzy optimization problem and is solved by the method of fuzzy logic conclusions. Key words: mathematical-economic modelling,

virtual business, membership function, fuzzy

rules of logical conclusion, pricing. AMS Subject Classification: 90A, 94D.

References 1. Shafizadeh E.R., Shykhlinskaya R.U., Application of fuzzy rules of logical conclusion to modelling of optimizing production and sectoral structure of agriculture for provision of food safety, Pressing issues of economy, Vol.1, No.103, 2010, pp.286-294. 2. Shelobayev S.I. Economic and mathematical methods and models, Unity, Moscow, 2005, 285p.

FUZZY MODEL OF PROFIT MAXIMIZATION IN ONLINE STORE‡‡ §§ R.Y. Shikhlinskaya1, N.D. Hajiyev2, F.A. Mirzayev3 1

Institute of Applied Mathematics, Baku State University, Azerbaijan, Baku Department of the Economy of Information and Technologies, Azerbaijan State Economic University 3 Baku State University, Baku, Azerbaijan

2

In the scientific literature are analyzed more practical issues of conducting e-commerce: problems of promoting goods to new markets and organizing the system of sales management, problems of the effectiveness of advertising on the network and etc. We construct a simple model of profit maximization in the online store. Example. Assume Internet shop undertook to sell a book of the A. publisher. Let books are published in 1000 copies for a month in a publishing house. Website of the online store is ready ‡‡


§§

This work was supported by the Science Development Foundation under the President of the Republic of Azerbaijan – Grant № EIF-2013-9(15)-46/44/5

223


and the publishing house offers books for the price of 2 azn. Internet store adds markup Cn and offers it to customers in the store's website. Every month it spends on advertising on the site support a certain amount k. Then store’s profit can be calculated as follows: 𝑃𝑟 = Cn × 𝑥 − 𝑘

(1)

Standard mathematical programming problem is usually formulated as the problem of maximizing (or minimizing) of a given function on a given set of admissible alternatives, which describes a system of equations or inequalities. The set of admissible alternatives is a collection of all possible ways to allocate resources, which the expert is going to invest in this operation. The profit maximization problem for described online store can be formalized as follows: Pr(Cn , x, k) ⟶ 𝑚𝑎𝑥

(2)

Cn ≤ C𝑟

(3)

𝑥 ≤ 1000,

(4)

𝑘 ≤ 𝐾,

(5)

Cn , x, k ≥ 0

(9)

Where Pr-store’s month profit, Cn - markup, C𝑟 – a positive number, depending on the established market price for the product, x-number of books sold for a month,recurrent costs for a month, K - the maximum possible amount to be spent. The fuzzy description of these parameters in the model may be more appropriate than in a sense arbitrarily taken a clear description. The aim of research. There are various descriptions of different fuzziness and fuzzy objectives. In such modeling of real decision-making tasks in the expert’s directionare only fuzzy descriptions of the parameters by means of linguistic variables. And so, let’s define a linguistic description of the profit maximization problem for an online store: 𝐺(Purpose): «maximize the markup» 𝐴1 (Limitation 1): «must be sold much more than half of the books (𝑥̃) (>>500)» 𝐴2 (Limitation 2): «Costs (k) must much less 1000 &» ( 0, 𝑖 = 1, … , 𝑛) has been given. We want to place these objects into bins of a given capacity C (𝐶 > 359


𝑤𝑖 , 𝑖 = 1, … , 𝑛) so that the total number of bins needed is minimized. This problem has many practical applications: Vehicles such as are pallets, containers, trailers, trucks, rail cars, ships and so on, are to be loaded with different items. The aim is to use as few vehicles as possible to carry the loads without exceeding the capacity of each vehicle. Another example is where tubes or cables are to be cut from quantities of standard length C. We want to use as few tubes or cables of standard length as possible to meet the demand. The same idea is used in metal working where steel sheets of different sizes must be cut from "master" sheets. Yet another example is in scheduling, where tasks of varying duration must be allocated using the least number of machines or processors [1, 2]. In this study, a new version of the Bin Packing Problem and a solution approach for this problem is proposed. Proposed packing problem is both relevant for many transportation and logistics planning problems, especially freight shipping, when a company has to ship orders to different customers. Some of these orders may be urgent the other may wait before shipped. Also, different orders can be placed into the same shipping box with a certain condition. For example, containers which are loaded to ship, can be closed, opened or both, designed for food, liquids or both. On the other hand, a company demand to place together for their different orders. In the proposed problem, given a set of items joined into different groups according to particular features and characterized by volume and profit and a set of bins with given volume which create constraints based on groups of objects and has its own constraint of upper bound of total volume of bins, and cost. Part of the items, which denoted compulsory, must be loaded, while a selection has to be made among the non-compulsory objects. Also problem has fuzzy relation between the objects reflecting the degree of the consistency of the whole packing. The fuzzy sets technique makes it possible to take into account the degree of the consistency in packing and thus to enhance its quality. The fuzzy relation between the objects impose certain constraints on the placement of objects 𝑅 = ||𝑅(𝑥𝑖 , 𝑥𝑗 )|| , 𝑖, 𝑗 = 1 … 𝑛 are symmetric and reflecting the degree of mutual attachment of elements 𝑥𝑖 and 𝑥𝑗 . All the elements of the indicated matrix take values from the closed interval [0,1]. The value 1 indicates that these objects must necessarily be placed together, while the value 0 indicates the complete freedom of action. Here, the degree of consistency of separation of elements of container 𝑆𝑗 from the elements that are outside the container 𝑆𝑗 with regard

for

the

information

on

their 360

mutual

attachment

is

𝐾(𝑆𝑗 ) = 1 −


max(𝑅(𝑥1 , 𝑥2 )|𝑥1 ∈ 𝑆𝑗 , 𝑥2 ∉ 𝑆𝑗 ) 𝑗 = 1, … , 𝑛 [3]. Assume that we have n set of objects,𝑋, which joined into different groups 𝑙 ∈ 𝐿 according to particular features and characterized by volume (𝑤) and profit (𝑝). Also we have a set of bins (𝑦) with given volume (𝑣) which create constraints based on groups of objects and has its own constraint of upper bound of total volume of bins(𝑉 ), and cost(𝑐). The bins are separated into types with different upper availability limits(𝑈𝑡 ). Part of the object, which denoted compulsory(𝐶), must be loaded, while a selection has to be made among the non-compulsory (𝑁𝐶) objects. The objects are to minimize difference between two total cost of the used bins and total profit of loaded objects which are non-compulsory (the profit of the compulsory objects is not included because it is compensated to a constant) and maximize degree of quality, i.e., the degree of consistency of the whole packing with regard for the relation indicated above [3, 4]. Let I denote the set of n items, with the volume (𝑤𝑖 ) and the profit (𝑝𝑖 ) of item 𝑖 ∈ 𝐼. 𝐼 𝐶 ⊆ 𝐼 the subset of items define absolutely loaded objects and 𝐼 𝑁𝐶 = 𝐼/𝐼 𝐶 the subset of items which may be chosen if profitable. Ley 𝐽 denote the set of available containers and let 𝑇 be the set of container types. For any bin 𝑗 ∈ 𝐽, let 𝜎(𝑗) ∈ 𝑇 be the type of bin 𝑗. Thus, a mathematical model of the proposed problem is as follows: 𝑀𝑎𝑥{𝑀𝑖𝑛 𝐾(𝑆𝑗 )]} 𝑗∈𝐽

𝑀𝑖𝑛{∑(𝑐𝑗 𝑦𝑗 − ∑ ∑ 𝑝𝑖 𝑥𝑖𝑗𝑙 )} 𝑗∈𝐽

𝑙∈𝐿 𝑖∈𝐼 𝑁𝐶

Subject to, ∑ 𝑤𝑖 𝑥𝑖𝑗𝑙 ≤ 𝑣𝑙𝑗 , 𝑗 ∈ 𝐽, 𝑙 ∈ 𝐿; 𝑖∈𝐼

∑ ∑ 𝑦𝑗 𝑣𝑗𝑙 ≤ 𝑉; 𝑗∈𝐽 𝑙∈𝐿

∑ 𝑥𝑖𝑗𝑙 ≤ 1 , 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽; 𝑙∈𝐿

∑ 𝑥𝑖𝑗𝑙 = 1 , 𝑖 ∈ 𝐼 𝐶 , 𝑙 ∈ 𝐿; 𝑗∈𝐽

∑ 𝑥𝑖𝑗𝑙 ≤ 1 , 𝑖 ∈ 𝐼 𝑁𝐶 , 𝑙 ∈ 𝐿; 𝑗∈𝐽

∑ 𝑦𝑗 ≤ 𝑈𝑡 , 𝑡 ∈ 𝑇; 𝑗∈𝐽:𝜎(𝑗)

361


𝑦𝑗 ∈ {0,1}, 𝑥𝑖𝑗𝑙 ∈ {0,1}, 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽, 𝑙 ∈ 𝐿. Keywords: Bin Packing Problem, fuzzy relation. AMS Subject Classification: 05B40.

References 1. Mohamadi N., Application of genetic algorithm for the bin packing problem with a new representation scheme, Mathematical Sciences, Vol.4, No.4, 2010.pp.253-266. 2. Berberler M.E., Nuriyev U.G., A new heuristic algorithm for the one- dimensional cutting stock problem, Appl. Comput. Math., Vol.9, No.1, 2010, pp.19-30. 3. Nasibov E., An algorithm of constructing an admissible solution to the bin packing problem with fuzzy constraints, International Journal of Computer and Systems Sciences, Vol.43, No.2, 2004, pp.205-212. 4. Nuriyeva F., Tezel B.T., Nasiboğlu E., A mathematical model of the multicriteria limited bin packing problem with fuzzy qualities, Proceedings of the “Caucasian Mathematics Conference CMCI”, 2014, pp. 145-146.

THE BAND COLLOCATION PROBLEM IN TELECOMMUNICATION NETWORKS Urfat Nuriyev1, Hakan Kutucu2, Mehmet Kurt3, Arif Gursoy1 1

Ege University, Department of Mathematics, Izmir, Turkey Karabuk University, Department of Computer Engineering, Karabuk, Turkey 3 Izmir University, Department of Mathematics & Computer Science, Izmir, Turkey e-mail: [email protected] 2

Introduction. In this paper, we introduce a new combinatorial optimization problem, the socalled Band Collocation Problem,is based on a telecommunication problem named the Bandpass Problem (BP)which is proposed by Babayev et al. [1].In the BP, we are given a binary matrix A of size

m  n and a positive integer B . This matrix corresponds to a flow of communication where each Figure 1: Sending point, m packages and n destinations on a communication network.

row and column represents a package and a

362


destination, respectively. If an entry (i, j ) of the matrix is 1, this means that the package i is sent to destination j . Such a communication flow is illustrated in Figure 1. The Bandpass problem consists of finding a permutation of rows of a binary matrix that maximizes the total number of non-overlapping B consecutive 1’s in all columns. B is called the bandpass number. Consecutive 1’s may lead toan opportunity to reduce the cost in telecommunication networks. For detailed information, the reader is referred to [1]. The standard Bandpass problem has a fixed Bandpass number B to be grouped packages in a communication matrix. Therefore, this problem is not practical in real applications. In this paper, we propose an improved version of the Bandbass Problem in which various bandpass numbers are allowed in the same column. The Band Collocation (BC) Problem. In fiber-optic communications, WavelengthDivision Multiplexing (WDM) technologycan carry information coded in a given m different wavelengths. An optical device named Add-Drop Multiplexer (ADM) facilitates flows on some wavelengths to exit the cable according to their paths. In each ADM, special cards control each wavelength; they may eitherpass through the ADM or dropped at their destination. If wavelengths carrying information are consecutive, then there is an opportunity of packing wavelengths which pass the same specific ADM into bandpasses. This is the origin the BP. However, recent changes in ADM technology allow an ADM to drop a wavelength even if it doesn’t carry any information. This removes the obligation that wavelengths to be dropped at the same destination must be consecutive. Therefore, BP which is aimed to maximize the number of bandpasses (groups of consecutive ones)is not effective to reduce the cost. Moreover, various bandpass numbers at destinations represented by columns are required. Thus, we consider a new problem and its model having various bandpass numbers and their costs. The Mathematical Model of the BC Problem. We give a mathematical formulation of the BC problem as a binary integer nonlinear programming model.Let A  {aij } be an m  n matrix and aij be an element of the matrix A . Let B be a B  Band number. B is a power of two such as B0  20 , B1  21  2 , B2  22  4 , …, Bk  2k , where k  0,1, 2,..., t . Here, t  log 2 m .

However, t can be determined according to the capabilities of the hardwarein the network. Let Ck be the cost of forming a band of length Bk. For a band Bk, there can be less than k consecutive 1’s in a column. Let us define decision variables as follows: 1, if row i is relocated to position r xir   0, otherwise

i, r  1,..., m

363


1, if row i is the first row of a band Bk in column j yijk   i  1,..., m; j  1,..., n; k  1,..., t 0, otherwise 1, if aij is an element of a band Bk in column j zijk   0, otherwise

i  1,..., m; j  1,..., n; k  1,..., t

We can form the binary integer nonlinear programming model of the BC problem as follows: t m2k 1 n

ck yijk    k 0 i 1 j 1

Minimize

(1)

Subject to m

xir  1,  r 1

i  1,2,..., m

(2)

r  1, 2,..., m

(3)

k  0, 1, , t; j  1,, n; l  1,, m  2k  1

(4)

m

xir  1,  i 1

2k ylik 

l  2k 1 m

zijk xir ,   r l i 1

t m2k 1

  k 0 i 1 yljk



l  2k 1 t

yijp  1,   i l 1 p 0

m

2k yijk   aij ,

j  1,, n

(5)

i 1

k  0, 1, , t; j  1, , n; l  1, , m  2k 1 t

yijk  1,  k 0 t

zijk  aij ,  k 0 t

zijk  1,  k 0

(6)

i  1,..., m, j  1,..., n

(7)

i  1,..., m, j  1,..., n

(8)

i  1,..., m, j  1,..., n

(9)

xir 0,1 , yijk 0,1 , zijk 0,1 , i, r  1,..., m;

j  1,..., n;

k  0,1,..., t .

(10)

The constraints (2) express the fact that row i must be relocated into one new position r only, (3) express that only one row imust be relocated to each new position r,(4) guarantee to find the coordinates of bands Bk,(5) say that the total length of bands in column j can not be less than the number of 1’s in column j, (6) guarantee that no two bands may have a common element, (7) guarantee that any entry of the matrix belongs to a unique band Bk, (8) say that each non-zero entry of the matrix has to be an element of a band Bk. The binary integer nonlinear model (1)-(10) finds

364


an optimal permutation of rows to form bands of different lengths among all columns having the minimum total cost. Acknowledgement. This work is supported by the Scientific and Technological ResearchCouncil of Turkey-TUBITAK 3001 Project (Project No.:114F073)

Keywords: bandpass problem, band collocation problem, mathematical modeling, telecommunication. AMS Subject Classification: 97C30, 90C27,90C09, 90B18.

References 1. Babayev D.A., Bell G.I., Nuriyev U.G., The Bandpass Problem, Combinatorial Optimization and Library of Problems, J. Comb. Optim., 18, 2009, pp.151-172.

FEATURES OF SAS ENTERPRISE GUIDE FOR PROBABILISTIC MODELING SYSTEM, MACROECONOMIC ANALYSIS AND FORECASTING Tetyana Prosyankina-Zharova1, Oleksandr Terentiev2, Petro Bidyuk3, Aydin Gasanov4 1

European University, Kyiv, Ukraine National Technical University of Ukraine “KPI, Kyiv, Ukraine 3 National Technical University of Ukraine “KPI, Kyiv, Ukraine 4 International Scientific and Training Center of Information Technologies and Systems, Ukraine, Kyiv e-mail: [email protected], [email protected] [email protected], [email protected] 2

The use of SAS Enterprise Guide 6.1 [1] as a means for building probabilistic models and as optimum method of modelinggross domestic product in terms of the economic crisis and social threats is proposed. Introduction. Today in a complex socio-political and economic situation growing influence of external factors, presence of uncertainties and risks there exists a problem of anticipating potential threats in the humanitarian and social spheres and ways to overcome them aiming to provide food security and controllability of ecological situation [2]. All these problems, as reported in the NATO program “Science for Peace and Security”,are of high priority forthe countries that need to take into account threats to security, including Ukraine. That is why in the framework of the project NUKR.SFPP G4877 “Modeling and Mitigation of Social Disasters Caused by Catastrophes and Terrorism” the problems of scientific prediction of national economy

365


for the period to 2020 as one of the measures preventing growth of social tension in the country are disclosed. Problem. The problem of integrated system development for macroeconomic planning and forecasting through the application of modern mathematical tools and information technologies remains unresolved. Solution of this problem is constrained by inertia as the executive authorities and the lack of established system of scientifically grounded methodological approaches to the analysis and modeling of national economy and its constituents. Therefore, the substantiationof the choice of methods and tools for analysis and forecasting of national economy is an urgent task. The main material. Creating a system for forecasting development of national economy which will avoid the subjectivity of expert estimation, to process large volumes of indicators notonly quantitative but also qualitative ones in a short period of time is one of the priorities for reforming national economy. To solve the problem many mathematical and statistical forecasting models such as multiple regression models, neural network, models based on fuzzy logic, probabilistic model and so on are developed and used. However, the problem of development of common methods for constructing scenarios based on cause-effect relationships between factors which influence the economic dynamics and development of key macroeconomic indicators, including gross domestic product – the main indicator of economic development remains unsolved. The complexity of this problem is related to the problem of lack of sufficient total volume of statistical data to identify patterns, to establish an adequate structure and calculation of estimates of the model parameters. Because of confidentiality restrictions on certain socioeconomic indicators, statistical methodology of adaptation to international standards complicates the formation of sufficiently long time series of these sets of indicators. This leads to the need for greater use of modern intelligent assets able to work and to identify patterns of short-time sampling data, solving the problem known in statistics as “curse of dimensionality”. As the main analytical tool of datasuch a powerful analytical tool like SAS Enterprise Guide 6.1 was used, this made it possible to process large volumes of statistics. In general more than 10 thousand of time series ofannual section of regions of Ukraine and economic activities according to current classifier of economic activities by means of ETL procedures in the programming language SAS Base were processed. One of the methods that have proved it selfpositively in modeling gross domestic product, depending on the scenario is Bayesian network. Since the formation of the gross domestic product affects a significant number of factors that not only stimulate its growth, but slow it down also it was important to identify causal relationships 366


between factors. The study was performed in two stages. The first stage was built by Bayesian network topology that provides the information needed to identify causal relationships between variables and power relations between them. To build a Bayesian network the procedure HPBNET was used, with which one can build different types of networks: network naive Bayes, Bayesian network like tree,increased naive Bayesian network, the structure of parent-child and Markov blanket. HPBNET advantages of the procedure are that it provides a range of variables on the basis of independent tests, and automatically selects the best model. When building a Bayesian network 56 indicators of socio-economic development of Ukraine were used, with which 18 were used to build a network. Statistical characteristics of the constructed model: the total classification error (Misclassification Rate) – 22%, ROC-index (Roc Index) – 0.84. On the basis of the constructed Bayesian network topologythe most important variables affecting the target are determined. After that the multiple regressing with a forced inclusion in the model of indentified variables is built. Estimation of the model parameters is performed basing on the method of recursive least squares. Analyzing the results of prediction of the consequences of the financial crisis on the developed model, it should be noted that with high probability (51%) a decline in GDP of Ukraine at 18.5% in 2015 was predicted against the value of the index in 2014, which fully corresponds the previous data. However, given the complexity of the task and the limited statistical information, the use of complex methods that combine the tools of economic analysis, econometric techniques, systems analysis methods, process time series and data mining, etc. is an optimal variant of forecasting development of national economy during the crisis (Table.1). Table 1. Scheme of application of methods Step

Methodical approach

1. Collecting information, forming a model of diagnosis 2. Review of the status and dynamics of the research object, identity of the characteristic features

Empirical research methods, statistical method of observation, analysis and synthesis, system analysis Statistical analysis, data-mining, factor analysis, multivariate data analysis, principal components method, correlation and regression analysis, typological and structural grouping, RFM-analysis, cluster analysis

3. Causal analysis

Correlation and regression analysis, probabilistic modeling (Bayesian Network)

4. Development of scenarios of events

Expert assessments, morphological analysis, scenario modeling

5. Substantiation of management decisions

Data-mining, neural networks, econometric modeling, SWOT-analysis, cognitive modeling

6. Forecasting

Econometric analysis, data mining methods, probabilistic modeling

7. Analysis of results

Method of expert estimates, Delphi method, graphic

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The proposed scheme of the research provides an integrated analysis and forecasting, allows to analyze the extent and the character of impact of various factors on the development of the situation in the national economy, both on short and long term and to build appropriate models of acceptable quality.

Keywords: Bayesian networks, data-mining, forecasting, modeling, gross domestic product. AMS Subject Classification: 68U35.

References 1. SAS Enterprise Miner 13.2: Reference Help. SAS Documentation, SAS Institute Inc., Cary, NC, USA, 2014, 1740 p. 2. Pearl J., Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, San Francisco, Morgan Kaufmann, 1988, 552 p.

IMPULSIVE TWOPOINT BOUNDARY VALUE PROBLEMS FOR NONLINEAR qk -DIFFERENCE EQUATIONS Y.A. Sharifov 1 2 1

2

Baku State University, Baku, Azerbaijan Institut of Control Systems of ANAS, Baku, Azerbaijan e-mail: [email protected]

Impulsive differential equations have extensively been studied in the past two decades. In particular, initial and boundary value problems of impulsive fractional differential equations have attracted the attention of many researchers: for instance, see [1, 2] and references therein. We investigate the existence and uniqueness of solutions for a twopoint boundary value problem of nonlinear impulsive q k -difference equation in the this paper. We consider Dqk xt   f t , xt ,0  qk  1, t  J ,

xt k   I k xt k , k  1,2,...,m,

(1)

Ax0  BxT   C ,

where Dqk are q k -derivatives

k  0,1,2,...,m, f  C J  R n , R n , İ k  CR n , R n ,

   

J  0, T T  0,

 

0  t0  t1  ...  t m  t m1  T , J   J \ {t1 , t 2 ,...,t m } and xt k   x t k  x t k , where x t k

368

and xt k 


denote the right and the left limits of xt  at t  t k k  1,2,...,m, respectively, A, B  R nn , C  R n1 are given matrixes and det  A  B  0. Let us set J 0  0, t1 , J1  t1 , t 2 ,..., J m  t m ,T  . Introdduce the spase as follows:



 



x t k exist, k  1,2,...,m }

xt  PC J , R n  {x : J  R n : x  C J k , k  0,1,...,m, and

Obviosly, this is Banach space with norm x  sup xt . tJ

Let us recall some basic consepts of q k -calculus [3-6]. For 0  qk  1 and t  J k , we define the qk -derivatives of real valued continous function f as Dqk f t  

f t   f qk t  1  qk t k  , 1  qk t  tk 

Dqk f t k   lim Dqk f t . t tk

The qk -integral of a function is defined by Dq f t  

f t   f qt  , 1  q t

0 I q t  

t

f s d q s 



 t 1  q q f tq . 

n

n

n 0

0

Note that if t k  0 and qk  q , then Dk f  Dq f and

tk

I qk f  0 I q f , where Dq and 0 I q are the well-

known q-derivative and q-integral of the function f t  defined by Dq f t  

f t   f qt  , 1  q t

0 İ q f t  

t



f s d q s 

 t 1  q q f tq . 

n

n

n 0

0

Lemma. A function xt  PCJ , R n  is a solution of the impulsive twopoint boundary value problem (1) if and only if it is a solution of the following impulsive qk -integral equation: T

xt    A  B  C  K t ,  f  , x d qk   1

 0

m

 K t , t I xt  , i

k

i

i

i 0

1    A  B  A,0    t , 1`    A  B  B, t    T .

for t  J k  t k , t k 1 , k  0,1,2,...,m, where K t ,   

Theorem 1. Assume that there exist continuous functions at , bt  and nonnegative constant L such that f t , xt   at   bt  xt  İ k x   L, k  1,2,...,m.

Then boundary value problem has (1) at least one solution.

369


Theorem 2. Assume that there exist a function M t  C J , R   and positive constant N such that

f t , u   f t , v   M t  u  v ,

I k u   I k v   N u  v ,

for

t  J , u, v  R n

and

S MT  Nm   1, where S  max{  A  B  A ,  A  B  B , and M  sup M t . Then, boundary value 1

1

tJ

problem (1) has a unique solution.

Keywords:

qk -difference,

impulsive, two point boundary value problem, existence and uniqueness

solution. AMS Subject Classification: 39A13, 34A12.

References 1. Wang, G., Ahmad B., Zhang I., Impulsive anti-periodic boundary value problem for nonlinear differential equation of fractional order, Nonlinear Analysis: Theory, Methods, and Applications, Vol.74, No.3, 2011, pp.792-804,. 2. Ahmad B., Nieto J.J., Anti-periodic fractional boundary value problems,Computers and Mathematics with Applications, Vol.62, No.3, 2011, pp.1150-1156. 3. Jackson F.H., On q-difference equations, American Journal of Mathematics, Vol.32, 1910, .pp.305314. 4. Bangerezako G., q-Difference linear control systems, Journal of Difference Equations and Applications, Vol.17, No.9, 2011, pp.1229-1249. 5. Bangerezako G., Variational q-calculus, Journal of Mathematcal Analysis and Applications, Vol.289, No.2, 2004, pp.650-665. 6. Zhang L., Ahmad B., Wang G., Impulsive Antiperiodic Boundary Value Problems for Nonlinear

qk  Difference Equations, Abstract and Applied Analysis, Vol.2014, Article ID 165129, 2014. DEVELOPMENT OF THE TWO-DIMENSIONAL AXISYMMETRIC MODEL OF GAS LIFT N.М. Temirbekov1, A.K. Turarov1 1

D. Serikbayev East Kazakhstan State Technical University, Ust-Kamenogorsk, Kazakhstan e-mail: [email protected], [email protected]

In this paper, a model describing a real isothermal unsteady flow of fluid in the gas lift well in the form of the Navier-Stokes equations for a compressible gas and weakly compressible fluid 370


in a porous medium written in cylindrical coordinates is developed. This model more accurately describes the physical processes occurring in the oil reservoir when mining by this method. Studying the dynamics of gas and gas-liquid mixture in the gas lift well and carrying out the numerical solution of the equations make it possible to develop a general statement of the problem in a porous medius. Two-dimensional problem of the motion of gas and gas-liquid mixture is studied wherein annular space, seam roof and part of the production well is considered as the modeling domain. The proposed model takes into account the physical properties of each of the three environments such as porosity, permeability, and the characteristics of the oil. The review of the works [1-4] devoted to the modeling of oil displacement by a gas lift installation is conducted. In these studies, one dimensional model of the gas lift process is investigated, where the movement in the annular space and rising pipe and mechanism of gas lift installation is described by partial differential equations of hyperbolic type. In this paper, the initial and boundary conditions for the developed model are defined, the algorithm of numerical realization of the problem is suggested. Gas lift installation consists of two concentric tubes with lengths L and l and radii R and r0 , where L > l , R > r0 . Gas is injected through the entrance boundary of the annular space with

the velocity f1 . When contacting with a porous medium, gas moves up with formed gas-liquid mixture through the rising pipe. Based on the principle of operation of gas lift wells, the initial domain is divided into three subdomains: D1 - annular space of the well, D2 - porous medium, and D3 - rising pipe. In the area D ( D = D1  D2  D3 ), we consider the following system of differential equations describing the isothermal unsteady flow of gas-liquid mixture in a porous medium and oil: a

 1  (ru ) (rw)    = 0, t r  r z 

(1)

u 1   (ru 2 )  (ruw)  p 1     4 u 2 w 2 u           =     r   t r  r z  r r  r   3 r 3 z 3 r  

(2) 

   u w    4 u 2 u 2 w   r             ku, z   z z    3 r 3 r 3 z 

371


w 1   (ruw)  (rw 2 )  p 1     u w         =     r   t r  r z  z r  r   z r        4 w 2 u 2 u    g  kw,   r       z   3 z 3 r 3 r  

(3)

p = R0T ,

(4)

where t is time;  is density; u,w are velocity components along the coordinate axes r, z ; p is pressure; T is temperature;  is the coefficient of viscosity; g is acceleration of gravity; R0 is a gas constant. The coefficient a is selected as follows: 1, if r , z   D1 ,  a = 1, if r , z   D2 ,  , if r , z   D , 3 

where  is a small parameter. The viscosity of the gas and liquid mixture  r, z  and coefficient of permeability of a porous medium k r , z  are defined as follows:  1501   2  R  2   r , z  = ã   í  ã   1     , 3    d   

150  1     R  k r , z  =   , Re   3 d  2

2

where  ã is the gas viscosity,  í is the oil viscosity,  is a parameter describing porosity of the porous medium, R is the outer radius of the well, d is capillary diameter. The parameter  is selected as follows:  1, if r , z  D1 ,   = 0.5, if r , z  D2 , 0.7, if r , z  D . 3 

Equations (1) - (4) are solved in the domains of D1 , D2 and D3 with the following initial and boundary conditions:

at t  0 , 0  r  R , 0  z  L (initial condition), ur , z  = u 0 r , z  ,

wr , z  = w0 r , z  ,  r , z  =  0 r , z  ;

1. In the area D1 : at z = L , r0  r < R (condition of entry), wr , L = f1 r  , ur , L = 0 ,

 r, L =  r  ; at r = R , l < z < L (solid wall), wR, z  = uR, z  = 0 ; at r = r0 , l < z < L (borehole wall), wr0 , z  = ur0 , z  = 0 . 2. In the area D2 : at z = l , 0  r  R put gluing conditions u z =l =  z =l = 0 ; at z = 0 , 372


0 < r  R (sole): wr,0 = ur,0 = 0 ; at r = R , 0  z  l (solid wall): wR, z  = uR, z  = 0 ; at r = 0 , 0  z  l (axis of symmetry):

w 0, z  = 0 , u0, z  = 0 ,  0, z  = 0 . r r

3. In the area D3 : at r = 0 , l  z  L (axis of symmetry):

w 0, z  = 0 , u0, z  = 0 , r

 0, z  = 0 ; at r = r0 , l  z  L (borehole wall): wr0 , z  = ur0 , z  = 0 ; at z = L , 0  r  r0 r

(out of the well), wr , L = f 2 r  , ur , L = 0 ,

 = 0. z

For the numerical implementation of the difference problem, we use the difference scheme of the two-step method of Lax-Wendroff . The results of two-dimensional problem solving model are presented. The analysis of the results is conducted. Keywords: gas lift process, a two-dimensional model, gas-liquid mixture, numerical experiments. AMS Subject Classification: 76S05, 35Q35, 65M06.

References 1. Aliev F., Guliev A.P., Ilyasov M.Kh., Aliev N.A., An algorithm for solving the problem of determining the spatial gas lift process, Proceedings of the Institute of Applied Mathematics, Vol.2, No.1, 2013, pp.91-98. 2.

Aliev F.A., Ilyasov M.H., Dzhamalbekov M.A., Simulation of gas lift wells, Report of the NAS Azerb., No.4, 2008, pp.107-116.

3. Aliev F.A., Mutallimov M.M., The algorithm for solving the problem of building and managing software path for oil gas lift, Report of the NAS Azerb., Vol.55, No.5, 2009, pp.9-18. 4. Barashkin R.L., Samarin I.V., Development of models and algorithms of functioning as a gas lift wells object operational management system, I.M. Gubkin Russian State University of Oil and Gas – Moscow, 2011, 152p.

373


MODEL MATCHING H2 OUTPUT FEEDBACK CONTROLLER DESIGN USING LINEAR MATRIX INEQUALITY OPTIMIZATION METHOD FOR OBTAINING NEGATIVE STIFFNESS ON 4 POLE U-TYPE HYBRID ELECTROMAGNET Barış Can Yalçın1, Kadir Erkan1 1

Yildiz Technical University, Istanbul, Turkey e-mail: [email protected], [email protected]

4-Pole U-type hybrid electromagnetic levitation systems can levitate ferromagnetic materials not only using the force of electromagnetic parts but also using the force of permanent magnet parts [1-3]. However, the system shows highly non-linear behavior. Obtaining negative stiffness for vibration isolation purpose is a control problem that has to be optimized in terms of non-linearty. The objective of this paper is to present a model matching H2 output feedback controller using linear matrix inequality optimization method for obtaining negative stiffness on 4-Pole U-type hybrid electromagnet. The effectiveness of proposed control method is proven by simulation results. Usage of permanent magnets in the structure of electromagnets constructing to hybrid design brings very important advantages such as decreased design volume and a more compact design [4, 5]. Moreover, necessary force for levitation of ferromagnetic material can be supplied by the force of permanent magnet(s). However, system’s behavior needs to be stabilized. U-Type electromagnets have been popularly used in many industrial applications that include suspending ferromagnetic objects [6, 7]. However, controlling more than one degree of freedom cannot be possible. To get rid of this problem, 4-pole u-type electromagnet structure has been proposed. Hence, control capability in multi degree of freedom and full redundancy are obtained. There are many studies of realizing negative stiffness for hybrid electromagnets [8-10]. In this work, a model matching H2 output feedback controller using linear matrix inequality optimization method is proposed. The effect of voltage, which is a controller input, on model matching error is minimized by decreasing ||Gcl||2 as much as possible, as if it is a disturbance attenuation problem. The rest of paper is organized as follows; in the second section, definition of negative stiffness and its usage in the field of vibration isolation systems are given, in the third section, structure & dynamics of 4-pole u-type electromagnet is described, also model matching H2 output

374


feedback controller based on linear matrix inequality optimization method is designed. In the fourth section, simulation results are given. Keywords: vibration isolation, infinite stiffness, negative stiffness, 4 Pole U-Type Electromagnet, model matching, H2 Output Feedback Controller, optimal control. AMS Subject Classification: 49K15.

References 1. Sinha P. K., Electromagnetic Suspension Dynamics & Control, Peter Peregrinus Ltd. On behalf of the Institution of Electrical Engineers. 2. Gieras J. F., Linear Synchronous Motors, Transportation and Automation System, CRC Press. 3. Morishita M., Azukizawa T., Kanda S., Tamura N., Yokoyama T., A new maglev system for magnetically levitated carrier system, IEEE Trans. Vehicular Technology, Vol.38, No.4, 1989, pp.230-236,. 4. Yakushi Y, Koseki T., Sone S., 3 degree-of-freedom zero power magnetic levitation control by a 4-pole type electromagnet, International Power Electronics Conference, Tokyo, Japan, 2000, Vol.4, pp.2136-2141. 5. Liu J., Yakushi K., Koseki T., Sone S., 3 degree-of-freedom control of zero-power magnetic levitation for flexible transport system, The 16th International Conference on Magnetically Levitated Systems and Linear Drives, Rio de Janeiro, Brazil, 2000, pp.382-386. 6. Mizuno T., Design of zero-power controllers for magnetic suspension system by a transfer function approach, The Third International Symposium on Linear Drives for Industry Applications, Nagano, Japan, 2001, pp.36-41. 7. Pei-Jen W., Design and analysis of a maglev transportation system for clean room applications, The Third International Symposium on Linear Drives for Industry Applications, Nagano, Japan, 2001, pp.48-51. 8. Hoque M.E., Mizuno T., Ishino Y., Takasaki M., A 3-DOF modular vibration isolation system using zero-power magnetic suspension with adjustable negative stiffness, Advanced Motion Control, 2010 11th IEEE International Workshop, 2010, pp. 661-666. 9. Hoque M. E., Mizuno T., Ishino Y., Takasaki M., A six-axis hybrid vibration isolation system using active zero-power control supported by passive weight support mechanism, Journal of Sound and Vibration, 2010, Vol. 329. 10. Hoque M.E., Mizuno T., Ishino Y., Takasaki M., Development of a three-axis active vibration isolator using zero-power control, Mechatronics, IEEE/ASME Transactions on, 2006, Vol.11.

375


NUMERICAL SOLUTION OF INTEGRO-DIFFERENTIAL EQUATION SYSTEMS M. Zarebnia1 1

Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran e-mail: [email protected]

In this paper a numerical solution for system of linear Fredholm integro-differential equations by means of the sinc collocation method is considered. This approximation reduce the system of integro-differential equations to an explicit system of algebraic equations. The method is applied to a few test examples to illustrate the accuracy and the implementation of the method. Keywords: Fredholm integro-differential equation, sinc function, sinc collocation method, collocation. AMS Subject Classification: 65D05, 65Gxx.

376


APPLICATIONS

THE COMPARISON OF B-CONVEX, B-1-CONVEX AND CONVEX FUNCTIONS Gabil Adilov1, Ilknur Yesilce2 1

Akdeniz University, Antalya, Turkey 2 Mersin University, Mersin, Turkey e-mail: [email protected]; [email protected]

B-convex functions are studied in [1,3,4]. B-1-convex functions are examined in [2,4]. In this work, some concrete examples of B-convex, B-1-convex and convex functions are given and the relations between these function classes are analyzed by taking the examples into consideration. Keywords: abstract convexity, B-convex function, B-1-convex function, convex function. AMS Subject Classification: 26B25, 52A41.

References 1. Adilov G., Rubinov A., B-convex sets and functions, Numerical Functional Analysis and Optimization, Vol.27, No.3-4, 2006, pp.237-257. 2. Adilov G., Yesilce I., B-1-convex set and B-1-measurable maps, Numerical Functional Analysis and Optimization, Vol.33, No.2, 2012, pp.131-141. 3. Briec W., Horvath C. D., B-convexity, Optimization, 53, 2004, pp.103-127. 4. Kemali S., Yesilce, I., Adilov G., B-convexity, B-1-convexity and their comparison, Numerical Functional Analysis and Optimization, Vol.36, No.2, 2015, pp.133-146.

DRAFT LABVIEW (LABORATORY VIRTUAL IDE DEVICES) IN AZERBAIJAN TECHNICAL UNIVERSITY N.B. Akhmedov1, A. Humbatova1, S. Ismayilov1, N. Ramazanova1 1

Azerbaijan Technical University, Baku, Azerbaijan e-mail: [email protected]

The project is developed taking into account the experience of the leading universities in the world to introduce innovative technologies in science, education and production. The project’s 377


essence is the use of modern information technology and original research, government and educational platforms produced by the American company National Instrument, a global leader in innovative technologies, using which it is possible construct a continuous educational process of preparing future of talented engineers. Introduction of innovative technologies in LABVIEW Az.TU helps us: - Introduction of modern information technologies and techniques in scientific and educational process - To prepare highly qualified specialists in various disciplines - To realize the creative potential of young scientists, students and staff - Ensurement of the participation of creative youth at the international competitions, conferences, and exhibitions. What is LabVIEW? LabVIEW or Laboratory Virtual Instrument Engineering Workbench (Workbench laboratory virtual instrumentation) is a graphical programming environment, which is widely used in industry, education and research laboratories as a standard tool for data collection and instrument control. LabVIEW - a powerful and flexible software environment used for the data measurement and analysis. LabVIEW - multi-platform environment, and you can use it on computers running Windows, MacOS, Linux, Solaris and HP-UX. Creating your own program on LabVIEW, or virtual instrument (VI) is a fairly simple affair. LabVIEW concept is very different from the sequential nature of traditional programming languages, providing the user with an easy-to-use graphical environment, which includes all the tools needed for data collection, analysis and presentation of the results. Using graphical programming language LabVIEW, referred to as G (J), you can program your task of graphical block diagram. Being an excellent programming environment for countless applications in the field of science and technology, LabVIEW can help you solve different problems, spending much less time and effort compared with traditional programming code. LabVIEW is a programming environment with which you can create applications using a graphical representation of all elements of the algorithm. However, LabVIEW is much more than just an algorithmic language. This development environment and execution of applications designed for researchers - scientists and engineers, for which programming is just part of the job. Powerful graphical programming language LabVIEW allows hundreds of times to increase productivity. As with LabVIEW requires only a few hours, since the package is specially designed 378


for programming the different measurements, data analysis and presentation of results. LabVIEW has a flexible graphical interface and is easy to program, it is also great for process modeling, application creation of a general nature and just for learning modern programming. The measuring system created in LabVIEW, has more flexibility than the standard laboratory instruments, because it is you and not your hardware manufacturer, determines the function of the device created. Your computer is equipped with a measurement and control hardware (NI ELVIS II, PXI, RIO, NI my DAQ) and LabVIEW comprise a fully customizable virtual instrument for the tasks. With LabVIEW is permissible to create the desired type of virtual instrument at very low cost compared to conventional tools. If necessary, you can make changes to it in minutes. LabVIEW is designed for ease of programming your tasks. For this purpose, there is an extensive library of functions and ready-to-use routines that implement a large number of common programming tasks. In LabVIEW also includes special library of virtual instruments for data input /output with embedded hardware (dataacquisition - DAQ), to work with a shared channel (GeneralPurposesInterfaceBus - GPIB), device management via a serial RS-232 port, the software components for analysis, presentation and storing data interaction via a network and the Internet. You will find that applications built on LabVIEW, improve quality work in many spheres of human activity - both in process automation, as well as in education and many others. LabVIEW is used in the most diverse spheres of human activity. In keeping with its name, it was originally used in research laboratories. And now is the most popular software packages, both in basic science laboratories (eg, LawrenceLivermore, Argonne, Batelle, Sandia, JetPropulsionLaboratory, WhiteSands and OakRidge in the United States, CERN in Europe) and in the sectoral industrial laboratories. Advantages of LabVIEW: 1. A full programming language; 2. An intuitive graphical programming process; 3. Extensive data collection, data processing and analysis, instrument control, reporting and data exchange through network interfaces; 4. Driver Support over 2000 devices; 5. Compatibility with operating systems Windows2000 / NT / XP, Mac OS X, Linux and Solaris. LabVIEW supports a huge range of equipment from different manufacturers and has in it more component libraries: 379


- To connect external equipment to the most common interfaces and protocols (RS-232, GPIB 488, TCP / IP, etc.); - For remote control the experiment course; - For to control robots and machine vision systems; - For generation and digital signals processing; - The application of mathematical methods for processing data; - For data visualization and processing results (including 3D-model); - For the modeling of complex systems; - To store information in databases and reporting; - To communicate with other applications within the concept of COM / DCOM / OLE; - For the implementation of original research projects for the shortest possible time, thanks to tight integration with measurement systems and a large number of built-in functions for numerical processing tools and the graphical representation of data. Keywords: LABVIEW, innovative technologies, control hardware, device management. AMS Subject Classification: 90B10, 90C10, 90C27, 68M10.

EVALUATION OF FINANCIAL STABILITY IN INDUSTRIAL ENTERPRISES R.M. Aliyev1 1

Institute of Economics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan e-mail: [email protected]

Progress in industrial infrastructure of the Republic of Azerbaijan has a leading role in economic development. It is impossible to imagine any field of economy without industry. From this point of view, rapid development of industry plays an important role in meeting material and social demand of the people. Therefore, it should be always paid attention to rapid development and operational strengthening of the industry sector. Various measures are taken by our state in this direction which oriented to the development of industrial fields in general. Practice of developed countries shows that giving birth to the economic development program of any industrial enterprise and basing in this program to the economy reflecting real status of the industry are necessary preconditions. It should be mentioned that finance has an incomparable role within the development program of the industrial sector. Financial mechanism

380


is a necessary and core economic tool. It is an important integral part of an industrial enterprise. Finance plays an important role in implementation of economic policy of an industrial enterprise. Finance is recognized as necessary means of economic process in the life of industrial enterprise closely related to cash flow. Being movement of financial-monetary assets, it is interrelated with reproduction. I can be deduced that financial relations in industrial enterprises are closely associated with socio-economic relations that firstly are related to financial operations and in fact constituting them. The reality is that formation of financial stability proceeds operations from production till consumption. At the same time, in financial position essense and analysis method, financial position reflects cash payment indicator, and distributed and usedfinancial resourses. We may come to the following conclusions regarding concept, types and essense of financial stability. Financial position is an economic category reflecting structure of private and called-up capital, its management, payment ability, financial stability, investment attractiveness and freely development. Financial stability is the basic indicator of financial position of any enterprise. Current financial position of industrial enterprises can be reflected in the following 4 types: 1. Absolute financial stability. 2. Normal financial stability. 3. Financial instability. 4. Crisis. Indicator

Absolute

Normal

Financial

financial

financial

instability

stability

stability

WA–Private working WA> = 0

Crisis

WA< = 0

WA< 0

WA< 0

FR> = 0

FR< 0

FR< 0

TFR> = 0

TFR=< 0

TFR< 0

asset FR-Financial

FR> = 0

resourses TFR.-Total financial TFR> = 0 resourses

The following conclusions can be made thereof: 1) In absolute financial stability 381


S(F)={1; 1; 1} where WA> = 0; FR> = 0; TFR> = 0 2) In normal financial stability S(F)= {0; 1; 1} where WA< = 0; FR> = 0; TFR> = 0 3) In financial instability S(F)={0; 0; 1} where WA< 0 FR< 0 TFR= < 0 4) In crisis S (F)={0; 0; 0} where WA< 0 FR< 0 TFR< 0 The reality is that decrease in expenses is the optimal way in the case of crisis and financial instability.

Keywords: insdustrial enterprise, financial stability, crisis, financial position. AMS Subject Classification: 34A38.

References 1. Aliyev R.M. Economy and Audit, Monthly scientific review, No.3, Baku, Azerbaijan, 2007, 64p. 2. Kalbiyev Y.A., General concept and problems of taxation, Azerbaijan Tax News, No.1, 2008, 34p. 3. Finance and Accounting Review, Baku, 2002-2012.

A SEARCH FOR REPEATED POLYPEPTIDE CHAIN PATTERNS: ORGANIZATION AND USE OF PRIOR STRUCTURAL KNOWLEDGE V. Amikishiyev1,2, T. Mehdiyev2, J. Shabiyev2, G.N. Murshudov3 1

Azerbaijan State Pedagogical University, Baku, Azerbaijan 2 Genetic Resources Institute of ANAS, Baku, Azerbaijan 3 Structural Studies Division, MRC-LMB, Cambridge, UK, CB20QH e-mail: [email protected]

Knowledge of the three-dimensional structures of Biological Macromolecules is an essential ingredient of understanding fundamental biological processes at atomic and electronic level. Three dimensional structures also form solid bases for structure based drug discovery. The last two decades have witnessed rapid advances in all aspects of the experimental structural biology that resulted in massive increase of the number of three dimensional macromolecular structures. According to the Protein Data Bank (PDB) [1] more than 109000 macromolecular structures now are know. For comparison, in year 2000 this number was under

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14000. Every year more than 9000 new macromolecular structures are studied and deposited to the PDB. This number does not include those studied in pharmaceutical companies since they are rarely available. Such a dramatic increase of the number of three dimensional structures while answering many fundamental biological questions poses new challenges such as its analysis, organisation, storage and use. Successful solution of such problems will allow development of new approaches to such fundamental problems as protein folding, extraction of biologically relevant information from very noisy and limited X-ray crystallographic and/or single particle Electron Microscopic data. It is well known that most the proteins share similar patterns of polypeptide chains at various levels. For example although the structures of more than 109000 proteins are known, the number of different folds are only at low few thousands. At lower level it is known that proteins share similar short polypeptide chains – secondary structures. Most of the known modern classification to date has been carried out subjectively, i.e. by experienced structural biologists and bioinformatics. The amount of the data in the PDB allows designing of new statistical approaches to pattern extraction from the protein structures objectively. Such automatic pattern recognition and extraction will also allow automatic organisation of the extracted information and classification of invariant information for various purposes. Classified patterns and pattern classes then can be used to tackle wide variety of biological problems. One of the problems is derivation of biological information from noisy and limited experimental data where there is a need for reliable chemical and structural knowledge. Importance of the use of structural information as prior knowledge to enhance reliability of X-ray Macromolecular Crystallographic analysis has already demonstrated by various researchers and software developers [2]. In this report the first results of the extraction and organisation of a polypeptide fragment library will be demonstrated. Early results of the use of structural information to enhance reliability and robustness of X-ray Crystal Structure analysis will also be reported. Structural information play the role of one of the components of the prior knowledge that is used to stabilise X-ray Crystallographic data analysis. The main goal of this research is an automatic classification of short polypeptide patterns. Analysis and classification of polypeptide fragments is done in two stages:

383


1) Using procrustes distances between short polypeptide fragments and equivalence class identification algorithm divide all fragments into classes, verify validity of classes and reclassify if necessary; 2) Using clusterisation algorithms find classes of patterns, analyse each class and find variability within and between classes. The purpose of the first stage is to reduce the large amount of data contained in the raw protein structures reducing complexity of the problem. The purpose of the second stage is to produce "true" classes of patterns and analyse them further using methods of multivariate data analysis. As a proof of principle procrustes analysis [3] was applied to one of the structures given in the Introduction – A chain of 2Y20 [4].

Figure 1: Early results of procrustes analysis and clusterisation. a) Contour map of interfragment distances. b) The result of agglomerative clustering – there seems to be three clusters two of them close two each other reflecting alpha helical nature of the protein

Figure 1 shows the results of simple complete linkage agglomerative clustering. Distances between fragments were calculated using procrustes analysis. Fragment lengths were chosen to be 7. Only backbone atoms – CA, C, N, O were used in calculations. This figure shows clear clusters for αhelices and irregular structures. Keywords: protein data bank, procrustes analysis, protein classification, structural patterns. AMS Subject Classification: 92-08.

384


References 1. Berman H.M., Westbrook J., Feng Z, Gilliland G., Bhat T.N., Weissig H., Shindyalov I.N. and Bourne P.E., Nucleic Acids Research, 28, 2000, pp.235-242. 2. Murshudov G.N., Skubak P., Lebedev A.A., Pannu N.S., Steiner R.A., Nicholls R.A., Winn M.D., Long F, and Vagin A.A., Acta Cryst D67., 2011, pp.355-367. 3. Murzin A.G., Brenner S.E., Hubbard T. and Chothia C., J. Mol. Biol. 247, 1995, pp.536-540. 4. Mardia K.V., Kent J.T. and Bibby J.M., Multivariate Analysis (Probability and Mathematical Statistics, 1980. 5.

R Development Core Team, R: A Language and Environment for Statistical Computing, 2010.

THE EQUATIONS OF HEAT CONDUCTIVITY FOR POLYTROPHIC GAS E.M. Akhundova1, A.A. Akhundov1 1

Institute of Control Systems of the Azerbaijan NAS, Baku, Azerbaijan e-mail: [email protected], [email protected]

1. Introduction. A physical model that maps the heating process of the one-dimensional compounds described in many textbooks on equations of mathematical physics (see note 1). Mathematical model of the heating process, developed based on this model, according to [1, p. 180], is presented as the next heat equation: 𝑐𝜌𝑇𝑡 = (𝑘0 𝑇𝑥 )𝑥 ,

(1.1)

where 𝑇 = 𝑇(𝑡, 𝑥) > 0 – temperature, 𝑡 > 0 – time, 𝑥 > 0 – spatial coordinate, с – specific mass heat capacity, 𝜌 is the density of the working body and 𝑘0 – coefficient of heat conductivity. Values с, 𝜌 and 𝑘0 general, not constant and, for example, as noted in [1, s. 181] depend on temperature and spatial coordinates. Here we follow [2, 3] consider this equation (1.1) in which two of these three quantities are functions of temperature. It is about an 𝑘0 and density 𝜌. The dependence 𝑘0 from temperature in many works properly has the power view: 𝑘0 = 𝑘𝑇 𝜎 , 𝜎 > 0, 𝑘 > 0.

(1.2)

However, the kind of density dependence, we still have to define. Below, you will accomplish this task for the case heat perfect polytrophic gas, was received by a nonlinear differential equation of heat conductivity of a special kind. Are some exact solutions of this equation by means of the theory of Lie groups [4].

385


2. Definition of type based on density from temperature. In polytrophic process the value с – constant by definition [5, с.23]. The polytrophic process of ideal gas at the same time meets the two equations: equations of state and polytrophic. The I equation is the following [5]: 𝑝 = 𝜌𝑅𝑇,

(2.1)

where 𝑝 – pressure, 𝑅 is the specific gas constant. The polytrophic equation for an ideal gas is [5, с.23]: 𝑝𝑣 𝑟 = 𝑎0 , 𝑎0 – constant,

(2.2)

where 𝑣 – specific volume of an ideal gas, 𝑟 – rate of polytrophic (dimensionless value). Given the ratio of 𝜌 = 𝑣 −1 (see note 2) polytrophic equation (2.2) will take the form: 𝑝 = 𝑎0 𝜌 𝑟 .

(2.3)

From (2.1) and (2.3) of the act is changed the density depending on the values of the temperature: 𝜌 = 𝑎𝑇𝑏(𝑟) ,

(2.4)

where 𝑏(𝑟) = 1⁄(𝑟 − 1), 𝑎 = (𝑅 ⁄𝑎0 )𝑏(𝑟) . The heat conductivity equation for an ideal gas, participating in the polytrophic onedimensional process provided 𝑟 ≠ 1 is obtained from (1.1), (1.2) and (2.4) and has the form 𝑐𝑎𝑇𝑏(𝑟) 𝑇𝑡 = 𝑘(𝑇 𝜎 𝑇𝑥 )𝑥 .

(2.5)

3. Exact solutions of the equation (2.5). Group symmetry equation (2.5) differ in three cases: (𝑏(𝑟) ≠ 𝜎)&(𝑏(𝑟) ≠ −(3𝜎 + 4)); 𝑏(𝑟) = −(3𝜎 + 4); 𝑏(𝑟) = 𝜎. Arbitrary constant of integration are marked with symbols 𝐶 𝑎𝑛𝑑 𝐶𝑛 , 𝑛 = ̅̅̅̅̅ 1, 3. 1. Case (𝑏(𝑟) ≠ 𝜎)&(𝑏(𝑟) ≠ −(3𝜎 + 4)): 𝑇(𝑡, 𝑥) = (𝐶1 𝑥 + 𝐶2 )1⁄(𝜎+1) .

(3.1)

Exact solution (3.1) invariant to shifts in time group. 𝑇(𝑡, 𝑥) = (𝐶 +

𝑐𝑎(𝑏(𝑟)−𝜎) 𝑘(𝑏(𝑟)+1)

1⁄(𝜎−𝑏(𝑟))

(𝑥 − 𝑡))

, 𝑏(𝑟) ≠ −1.

(3.2)

Exact solution (3.2) is invariant to a homogeneous sprains-compressions group on variables 𝑡, 𝑥. 𝑇(𝑡, 𝑥) = 𝑥 2/(𝜎−𝑏(𝑟)) (𝐶 +

2𝑘(𝑏(𝑟)+𝜎+2) 𝑐𝑎(𝑏(𝑟)−𝜎)

𝑡)

1⁄(𝑏(𝑟)−𝜎)

.

(3.3)

Exact solution (3.3) is invariant operator: 0.5(𝜎 − 𝑏(𝑟))𝑥𝜕𝑥 + 𝑇𝜕𝑇 . 2. Case 𝑏(𝑟) = −(3𝜎 + 4): 𝑇(𝑡, 𝑥) = (4(𝜎 + 1)𝑡 + 𝐶3 )−1⁄4(𝜎+1) (𝐶1 (𝐶2 ± 𝑥)2 − 386

𝑐𝑎(𝜎+1) 1⁄2(𝜎+1) 𝑘𝐶1

)

, 𝐶1 ≠ 0.

(3.4)


Exact solution (3.4) is invariant operator: (4(𝜎 + 1)𝑡 + 𝐶3 )𝜕𝑡 − 𝑇𝜕𝑇 . 𝑇(𝑡, 𝑥) = 𝑥

1⁄(𝜎+1)

(𝐶 −

4𝑐𝑎(𝜎+1)2 3𝑘

(𝑡 +

1⁄4(𝜎+1) 1 )) . (𝜎+1)𝑥

(3.5)

Exact solution (3.5) is invariant operator: 𝜕𝑡 + (𝜎 + 1)𝑥 2 𝜕𝑥 + 𝑥𝑇𝜕𝑇 . 𝑇(𝑡, 𝑥) = 𝑥

1⁄(𝜎+1)

(𝐶 −

4𝑐𝑎(𝜎+1)2 3𝑘

(𝑡 −

1⁄4(𝜎+1) 1 )) . (𝜎+1)𝑥

(3.6)

Exact solution (3.6) is invariant operator: −𝜕𝑡 + (𝜎 + 1)𝑥 2 𝜕𝑥 + 𝑥𝑇𝜕𝑇 . 𝑇(𝑡, 𝑥) = (1 + (𝜎 + 1)𝑥 2 )1⁄2(𝜎+1) (𝐶 −

4𝑘(𝜎+1) 𝑐𝑎

𝑡)

−1⁄4(𝜎+1)

.

(3.7)

Exact solution (3.7) is invariant operator: (1 + (𝜎 + 1)𝑥 2 )𝜕𝑥 + 𝑥𝑇𝜕𝑇 . 3. Case 𝑏(𝑟) = 𝜎: 𝑇(𝑡, 𝑥) = (𝐶2 −

𝑘𝐶1 𝑐𝑎

𝑐𝑎

𝑒− 𝑘

(𝑥−𝑡)

1⁄(𝜎+1)

)

, 𝐶1 > 0.

(3.8)

Exact solution (3.8) is an invariant of the infinitesimal operator: 𝜕𝑡 + 𝜕𝑥 . 1⁄(𝜎+1)

𝑇(𝑡, 𝑥) = 𝑒 𝑡 (𝐶1 𝑐ℎ (𝑥√𝑐𝑎(𝜎 + 1)⁄𝑘) + 𝐶2 𝑠ℎ (𝑥√𝑐𝑎(𝜎 + 1)⁄𝑘 ))

.

(3.9)

.

(3.10)

Exact solution (3.9) is an invariant of the infinitesimal operator: 𝜕𝑡 + 𝑇𝜕𝑇 . 𝑇(𝑡, 𝑥) = 𝑒 −𝑡 (𝐶1 𝑐ℎ (𝑥√𝑐𝑎(𝜎 + 1)⁄𝑘) + 𝐶2 𝑠ℎ (𝑥√𝑐𝑎(𝜎 + 1)⁄𝑘))

1⁄(𝜎+1)

Exact solution (3.10) is an invariant of the infinitesimal operator −𝜕𝑡 + 𝑇𝜕𝑇 . 𝑇(𝑡, 𝑥) = 𝐶 ∙ 𝑡 −1⁄2(𝜎+1) 𝑒

−

𝑐𝑎 𝑥 2 𝑡 −1 4𝑘(𝜎+1)

.

(3.11)

Exact solution (3.11) of the equation (2.5) is an invariant of the infinitesimal operators: 2𝑡𝜕𝑡 + 𝑥𝜕𝑥 − (𝜎 + 1)−1 𝑇𝜕𝑇 and − (2𝑘(𝜎 + 1)⁄𝑐𝑎)𝑡𝜕𝑥 + 𝑥𝑇𝜕𝑇 . Each of these operators is the leading operator of sub algebras in an optimal system are allowed by the equation (2.5). Note 1. As a physical model can be considered a long the insulated on the sides of a cylindrical tube with small diameter smelted, located horizontally to the ground and filled with rarefied gas Heating can be done with a corked tube ends. Note 2. Formula 𝜌 = 𝑣 −1 easily derived from known formula for density 𝜌 = 𝑚⁄𝑉 taking into account the apparent ratio 𝑉 = 𝑚𝑣, where 𝑚 – mass of the substance and 𝑉 – volume. Keywords: polytrophic gas, exact solution, invariant. AMS Subject Classification: 35K05.

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References 1. Tikhonov A.N., Samarskii A.A. Equations of Mathematical Physics, Moscow, Nauka, 1966, 724p. 2. Akhundov A.A. Mathematical model of heat conduction-diffusion process when there is no flow, Proceedings of ANAS, Vol. XXIV, No.2, 2004, pp.178-183. 3. Akhundov A.A.,

Akhundova E.M. Nonlinear differential equation of heat conductivity of

polytrophic gas and some results of its continuous group analysis (part I), Proceedings of ANAS, Vol. XXV, No.2, 2014, pp.120-132. 4. Zaitsev V.F. Introduction to Modern Group Analysis (part 2), Saint-Petersburg: Russian STATE them. A.I. Herzen, 1996, 40p. 5. Chechetkin A.A., Zanemonetz N.A., The Heating Technics, Moscow, V. Shkola, 1986, 344p.

ESTIMATION OF FINANCIAL RISKS IN CONDITIONS OF UNCERTAINTY P.I. Bidyuk1, O.M. Trofymchuk2, A. S. Gasanov3, S.H. Abdullayev4 National Technical University of Ukraine “KPI”, Ukraine Institute of Telecommunications and Global Information Sphere at NAS of Ukraine, Kiev, Ukraine 3 International Scientific and Educational Center for Information Technologies and Systems, Kiev,Ukraine 4 Institute for Information Technologies ANAS, Baku, Azerbaijan e-mail: [email protected], [email protected], [email protected], [email protected] 1

2

The problems of financial and other risks estimation and management are highly urgent as for today due to substantial losses that emerge as a result of underestimation of the risks or inappropriate countermeasures [1]. It is especially important task to develop new mathematical approaches and techniques that will enable correct estimation of risk situations, prediction of possible losses and making quality managerial decisions. There exist various approaches to solving the problem dependent on specific application area. For example, the problem could be considered separately for information technologies that are hired in financial institutions, for clients of these institutions, for the methods and techniques relevant to financial operations, for influence of market etc. The systemic approach to modeling and estimation of operational risks is based on system analysis ideas that refer to hierarchical data processing schemas, modern techniques for analyzing model structure and parameters, identification and taking into consideration possible uncertainties related to data and estimation algorithms. The approach also includes the possibilities for control of relevant computational processes by appropriate set of quality criteria so that to provide for the high quality of final result. It supposes performing of analysis of internal and external influence factors to various sides of 388


financial company activities including stochastic disturbances of different nature and types, application of statistical simulation techniques in the frames of decision support systems (DSS) constructed on the purpose. The most general approach is based on general system theory and more specific techniques could be found in operations research, modern system analysis theory, cybernetics science, intellectual data analysis and data mining, and many others. Among specific analysis tools are modern theories of estimation, model building, forecasting and optimal control. Correct application of the techniques mentioned provides for development of specialized DSS that are characterized by necessary functional orthogonality and completeness as well as inherent intellectual features. The DSS proposed has features necessary for identification and taking into consideration possible uncertainties of structural, statistical and parametric type. The uncertainties are encountered in the process of preliminary data processing, model constructing, and computing forecasts. It is possible to substantially reduce their influence on the quality of final result using such modern techniques as data normalization, imputation, and filtering together with application of appropriate model structure and parameter estimation techniques. Mathematical model structure estimation is based on thorough correlation analysis of data, analysis of other characteristics, such as mutual information of the time series employed, etc. To get high quality of parameter estimates it is recommended to hire more than one parameter estimation technique, to compare the results achieved with each method and (possibly) to combine the estimates to reach better final result. For example, the set of model parameters estimation techniques should include the following ones: OLS, RLS, ML, nonlinear LS, and Monte Carlo for Markov Chains (MCMC). The last method is popular for estimating parameters of nonlinear models describing dynamics of variance, logistic regression parameters and other nonlinear models. Processing statistical uncertainty. The most often met statistical uncertainties related to model development and estimation of forecasts are provoked by the following factors: measurement errors (noise) that is available practically in all cases of data collection independently on the data origin (including economy, finances and industrial control systems); stochastic external disturbances that usually negatively influence the process under study and shift the processes from desired mode (say, offshore capital transfer from some country, low quality of higher administration, unstable often changed laws, substantial corruption, local hybrid wars); missed measurements (observations) and outliers (there is necessary special literature on both subjects);

389


multicollinearity, that requires special data processing techniques to reduce the degree of mutual correlation between separate time series. The most often means used to fight the measurement noise and external stochastic disturbances are digital and optimal filters (among them Kalman filter is a widely used instrument). Digital filters (DF) help to select for subsequent processing the frequency band of interest by processing the time series data with linear structures that could be represented, for example, by autoregression (AR) or autoregression with moving average (ARMA) equations. Note, that functioning of DF doesn’t require the model of a process under investigation [2]. Today there is a number of Kalman filtering techniques designed for solving the problems of state and parameter estimation in special cases. Appropriately designed adaptive Kalman filter provides a possibility for covariances estimation for stochastic disturbances and measurement noise as well as for estimation of quality short-term forecasts. Optimal filter design requires model of the process (system) under study in state space form allowing taking into consideration two random processes: state disturbances and measurement errors (noise). In spite of the fact that classic optimal filtering problem statement requires normal distribution for these processes today there are techniques for considering other types of distributions. Another positive feature of the filter is in its possibility for estimating non-measurable components of a state vector. Such problem comes to being rather often in many areas of processes analysis. For example, in economy and finances there hidden processes connected with “disappearing” capital in off-shore zones, or volumes of “black” salaries paid by employers to their workers without paying necessary tax. Among intellectual data analysis (IAD) techniques hired for risk analysis, estimation and management are static and dynamic Bayesian networks, decision trees, nonlinear Bayesian regression, special statistical data processing techniques, multivariate distributions, neural networks and fuzzy logic. Also very promising is combination of various techniques in the frames of appropriate scenario based approach. A key feature of DSS is in its possibilities for appropriate control of intermediate and final results. First, the quality of data should be tested with the criteria disclosing degree of information contained in data. Here variance and entropy based criteria are suitable as well as the number of derivatives that could be computed with the polynomial approximating the time series under study. As far as some observations can be lost or skipped during their collection it is also advisable to apply appropriate data imputation techniques to solve the problem of incomplete data sets. Another set of criteria should be hired for detecting nonlinearities, disclose integrated and co-integrated 390


time series, identify heteroskedasticity etc. All these measures are directed towards improvement of a process model structure. After constructing candidate models it is necessary to test the model quality (adequacy) with appropriate set of statistics (say, determination coefficient, DurbinWatson and Bayes-Schwartz statistics, and others dependently on a model type). Quality of forecasts are also to be determined with a set of statistical parameters among which are mean squared error, mean absolute percentage error, Theil coefficient and many others. Thus, all computational procedures in the frames of DSS are controlled with appropriate statistics what enhances substantially quality of the final result. The forecasts quality could also be improved with application of various combining techniques, such as simple averaging, weighted averaging, and optimization procedures based combining. Keywords: financial risk, systemic approach, decision support system, statistical uncertainty. AMS Subject Classification: 60, 62 (34A38). References 1. McNeil A.J., Frey R., Embrechts P. Quantitative risk management, Princeton: Princeton University Press, 2005 554p. 2. Bidyuk P.I., Romanenko V.D., Timoshchuk O.L. Time series analysis Kyiv: Polytechnika, NTUU KPI, 2013, 600p.

CHARACTERIZATION OF DEFECTS USING INVERSE ANALYSIS WITH EMAT SIGNAL AND ELECTROMAGNETISM-LIKE MECHANISM H. Bougheda1, T. Hacib1, H. Acikgoz2, M. Chelabi1, Y. Le Bihan3 1

L2EI Laboratory, Faculty of Science and Technology, Univ. Jijel, Algeria 2 Engineering Faculty, Karatay University, Konya, Turkey 3 Laboratoire LGEP, Supelec, UMR 8507 CNRS, UPMC, Univ. Paris-Sud, Gif-sur-Yvette Cedex, France e-mail: [email protected]

This paper is concerned with the characterization methodologies of defects in conducting materials by an Electromagnetic Acoustic Transducer (EMAT) testing system. First the inspection simulator based on EMAT is developed for the purpose of constructing the virtual nondestructive testing environments. EMAT is a new technology that provides a non contact process of testing materials compared to ultrasonic testing technique. The finite element method (FEM) is used like a simulator model of EMAT forward analysis. Inverse analysis based on Electromagnetism-Like 391


Mechanism (EM-LM) approach is applied for defect parameters identification. The EM-LM is a metaheuristic method proposed for global optimization. I. Introduction. The Electromagnetic Acoustic Transducer (EMAT) is an important ultrasonic exciting and receiving instrument for NDT. The main advantage of an EMAT is that no couplant is needed. One major advantage of an EMAT is that different guided modes can be generated by simply changing the coil or magnet geometry [1]. In the receiving mode, the magnetic flux density of the receiver EMAT interacts with particle's velocity of displacement in conductive material to generate a current in the materials underneath the receiver EMAT. This current induces a voltage in the receiving coil. This work is based on simulation of the numerical model, which includes the calculation of the Lorentz force, displacement. The EM-LM technique is combined with the developed model to solve the inverse problem in order to deduce defects information. II. The Electromagnetic Model. EMAT is composed from two important parts, the coils and permanent magnet, the coils are fed by an alternating current source under high frequency, the presence of conductive material gives us an induced eddy current Je,. The permanent magnet is placed directly above the coils; it creates a strong static magnetic field BS, The interaction between the induced current and the magnetic fields launches a Lorentz force FL inside material.

Fig.1. 2D structure of EMAT transmitter and receiver

The problem is expressed in terms of the magnetic vector potential Az and scalar potential V, as shown in the following diffusion equation: A 1   ( (  Az  Br ))   z  v  (  Az )  V  Jex μ t

(1)

Where,, µ, Jex are conductivity, permeability, velocity and external current of the kth coil. The transient excitation current ik in the meander coil of T-EMAT (Fig. 2), is assumed as: i k (t )  I 0 * e  ( t  ) cos (2 f (t   ))

Where I0=30 A, α =5 e-10, τ =10 μs, and f is frequency 200 kHz.

392

(2)


7

30 Lorentz force density [N/m3]

4

input current [A]

20 10 0 -10 -20 -30

0

0.5

1 time(s)

1.5

-5

under coil 1 under coil 6

2

0

-2

-4

2

x 10

0

0.5

x 10

Fig. 2. Plot of input current in coil of T-EMAT

1 time (s)

1.5

2 -5

x 10

Fig. 3. Plot of Lorentz force density at the surface of the aluminum material under coil 1and 6

The results prove that the force generated at the surface of the material have the same density, but the opposite direction underneath coil 1 and 6 respectively. III. Acoustic Model. The acoustic field equation is stated in terms of a particle displacement vector u as follows: 2u μ    u  ( λ  2 μ).u  ρ   FL t

(3)

where,  is the mass volume density, µ and  are Lame constants and u is the mechanical displacement vector. Fig.4 shows plot of the displacement in the x direction inside the aluminum material for different defect depths; defect is introduced at the surface 20 mm away to the transmitter EMAT Fig.5. The amplitude of mechanical displacement decrease when the depth defect is deeper. -12

X displacement (m)

2

x 10

1 mm defect 2 mm defect 3 mm defect

1

0

-1

-2

0

0.5

1 time(s)

1.5

2 -5

x 10

Fig. 5. 2D EMAT with Defect depth variable

Fig.4. Plot of mechanical displacement underneath the EMAT receiver at different defect depths

This result proves that it is feasible to characterize defects from inverse analysis based on EMLM approach in the material.

393


VI. Inverse Problem Solving Using the EM-LM. The EM-LM is a heuristic method proposed by Birbil and Fang [2] for global optimization. Its origin comes from the attraction– repulsion mechanism of the electromagnetism theory of physics by considering potential solutions as electrically charged particles spread around the solution space. The heuristic EM-LM consists of four steps: initialization of the algorithm, calculation of the total force exerted on each particle, movement along the direction of the force, and application of a neighborhood search to exploit the local minima. V. Conclusion. Numerical model of an EMAT was analyzed which includes a defect in the conductive material. The result proved the EMAT model can be detected the depth of a defect from displacement signal. The validity of the proposed approach based inversion method EM-LM is being assessed by comparing the results obtained using the proposed method with those obtained from an experiment. Keywords: finite element method, electromagnetism-like mechanism, EMAT system, nondestructive testing environments. AMS Subject Classification: 76M10.

References 1. Luo W., Rose J.L., Guided wave thickness measurement with EMATs, Insight, Vol.45, No.11.2003. 2. Birbil S. I., Fang S.C., An electromagnetism-like mechanism for global optimization, Journal of Global Optimiz., Vol.25, 2003, pp.263-282.

REGARDING HYPERBOLIC PROTOTYPE SOLUTIONS FOR SOME NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS Hasan Bulut1, Haci Mehmet Baskonus2 1

Department of Mathematics, Faculty of Science, Firat University, Elazig, Turkey Department of Computer Engineering, Faculty of Engineering, Tunceli University, Tunceli, Turkey e-mail: [email protected], [email protected]

2

In this study, we have considered the modified exp  -Ω  ξ   -expansion function approach being newly submitted to literature for obtaining some new prototype analytical solutions to timeand space-fractional Burger equation [1-5, 8, 18] and time-and space-fractional mKdV equation. 394


Consecutively, we have attained some new prototype analytical solutions such as complex hyperbolic function solution, exponential function solution, trigonometric function solution and rational function solutions for these differential equations [6,7, 9-17]. After we checked up whether these prototype solutions have verified the nonlinear ordinary differential equation, we have drawn two and three dimensional surfaces for different values of parameters by the help of commercial wolfram computer programming language Mathematica 9.





Keywords: the modified exp -Ω  ξ  -expansion function approach, time-and space-fractional Burger equation, time-and space-fractional mKdV equation, complex hyperbolic function solution, exponential function solution, complex trigonometric function solution. AMS Subject Classification: 35AXX, 35DXX, 35FXX, 35EXX, 34MXX, 34FXX, 35MXX, 35QXX.

References 1. Miller K.S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, New York, Wiley, 1993. 2. Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, San Diego, Elsevier, 2006. 3. Podlubny I., Fractional Differential Equations, San Diego, Academic Press,1999. 4. Song L., Zhang H., Solving the fractional BBM-Burgers equation using the homotopy analysis method, Chaos Solitons Fractals, Vol.40, 2009, pp.1616-1622. 5. Sahadevan R., Bakkyaraj T., Invariant analysis of time fractional generalized Burgers and Korteweg–de Vries equations, Journal of Mathematical Analysis and Applications, Vol.393, No.2, 2012, pp. 341-347, 6. Abbasbandy S., Shirzadi A., Homotopy analysis method for multiple solutions of the fractional Sturm–Liouville problems, Numerical Algorithms, Vol.54, No.4, 2010, pp.521-532. 7. Jafari H., Momani S., Solving fractional diffusion and wave equations by modified homotopy perturbation method, Physics Letters A, Vol.370, No.5-6, 2007, pp.388–396. 8. Baskonus H.M., Bulut H., Pandir Y., On the Solution of Nonlinear Time-Fractional Generalized Burgers Equation by Homotopy Analysis Method and Modified Trial Equation Method, International Journal of Modeling and Optimization, Vol.4, No.4, 2014, pp.305-309. 9. Zhang X., Liu J., Wen J., Tang B., He Y., Analysis for one-dimensional time-fractional Tricomitype equations by LDG methods, Numerical Algorithms, Vol.63, No.1, 2013, pp.143-164. 10. Bulut H., Kilic B., Exact solutions for some fractional nonlinear partial differential equations via Kudryashov method, e-Journal of New World Sciences Academy, Vol.8, No.1, 2013, pp.24-31. 395


11. Jumarie G., Modified Riemann-Liouville derivative and fractional Taylor series of non differentiable functions further results, Computers and Mathematics with Applications, Vol.51, No.9-10, 2006, pp.1367-1376. 12. Jumarie G., Fractional Hamilton-Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost function, Journal of Computational and Applied Mathematics, Vol.23, No.1-2, 2007, pp. 215-228. 13. Jumarie G., Table of some basic fractional calculus formulae derived from a modified RiemannLiouville derivative for non-differentiable functions, Applied Mathematics Letters, Vol.22, No.3, 2009, pp.378-385. 14. He J. H., Elegan S. K., Li Z. B., Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Phys. Lett. A, 376, 2012, pp.257–259. 15. Elghareb T., Elagan S. K., Hamed Y.S., Sayed M., An exact solutions for the generalized fractional Kolmogrove–Petrovskii–Piskunov equation by using the generalized G G -expansion method, Int. J. Basic Appl. Sci., Vol.13, No.1, 2013, pp.19–22. 16. Güner Ö., Bekir A., Exact solutions of some fractional differential equations arising in mathematical biology, International Journal of Biomathematics, Vol.8, No.1, 2015, 17 p. 17. Safwan A., Fractional Transformation Method for Constructing Solitary Wave Solutions to Some Nonlinear Fractional Partial Differential Equations, Applied Mathematical Sciences, Vol.8, No.116, 2014, pp.5751–5762. 18. Sahadevan R., Bakkyaraj T., Invariant analysis of time fractional generalized Burgers and Korteweg-de Vries equations, Journal of Mathematical Analysis and Applications, Vol.393, No.2, 2012, pp.341–347.

AN APPLICATION OF THE NEW FUNCTION METHOD TO NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Hasan Bulut1, Tolga Akturk1, Yusuf Gurefe2 1

Department of Mathematics, Firat University, Elazig, Turkey Department of Econometrics, Usak University, Usak, Turkey e-mail: [email protected]

2

In nonlinear science, the investigation of various traveling wave solutions to (N+1)dimensional double sine-Gordon and (N+1)-dimensional double sinh-cosh-Gordon equations have been widely studied by many authors [1-3]. On the other hand, a variety of effective methods have been defined to construct the traveling wave solutions of nonlinear partial differential equations. 396


It is given the new function method as one of most important methods and its applications [4-8]. In this paper, we apply the new function method, based on sine, sinh functions, to (N+1)dimensional double sine-Gordon and (N+1)-dimensional double sinh-cosh-Gordon equations. Thus, some new Jacobi elliptic function solutions are obtained by the using of this method. The obtained results reveal that the new function method is powerful mathematical tool for solving the (N+1)-dimensional sine-Gordon and sinh-cosh-Gordon equations. Keywords: new function method, (N+1)-dimensional double sine-Gordon equation, double sinh-coshGordon equation, Jacobi elliptic function AMS Subject Classification: 35C07, 35Q99, 37K40.

References 1. Li J.B., Exact traveling wave solutions and dynamical behavior for the (n + 1)-dimensional multiple sine-Gordon equation, Science in China Series A: Mathematics, Vol.50, No.2, 2007, pp.153-164. 2. Lee J., Sakhtivel R., Travelling wave solutions for (N+1)-dimensional nonlinear evolution equations, Pramana-Journal of Physics, Vol.75, No.4, 2010, pp.565-578. 3. Wang D.S., Yan Z., Li H., Some special types of solutions of a class of the (N+1)-dimensional nonlinear wave equation, Computers and Mathematics with Applications, Vol.56, No.6, 2008, pp.1569-1579. 4. Shen G., Sun Y., Xiong Y., New travelling-wave solutions for Dodd-Bullough equation, Journal of Applied Mathematics, Vol.2013, Article ID.364718, 2013, 5 pages. 5. Sun Y., New travelling wave solutions for Sine-Gordon equation, Journal of Applied Mathematics, Vol.2014, Article ID.841416, 2014, 4 pages. 6. Bulut H., Akturk T., Gurefe Y., Traveling wave solutions of the (N+1)-dimensional sin-cosineGordon equation, AIP Conference Proceedings, Vol.1637, No.1, 2014, pp.145-149. 7. Bulut H., Akturk T., Gurefe Y., An application of the new function method to the generalized double sinh-Gordon equation, AIP Conference Proceedings, Vol.1648, No.370014, 2015, 4 pages. 8. Akturk T., Determining the exact solutions of some nonlinear partial differential equations by trial equation methods, Firat University, PhD Thesis, 2015.

397


APPLICATION OF NEW INVESTIGATION METHODS OF DEEP-PUMPING WELLS WITH VISCOUS - PLASTIC OIL Javida Rizvan Damirova Azerbaijan State Oil Academy, Baku, Azerbaijan e-mail: [email protected]

Determination of dynamic level of fluid level in deep-pumping wells with abnormal viscous – plastic oil has been considered in the article. According to the accepted conditions let’s determine dynamic level of the fluid for abnormal oil (viscous – plastic oil). With this purpose let’s write velocity of viscous – plastic oil moving in vertical direction in the exploitation column.



 P1  P2  hz  O  2 2  0 R  r   R r  4 O  hz O





(1)

where P1 – is steady accepted well-bottom pressure; P2 – is pressure in the dynamic level;  O – is viscousity of oil;  O – is average specific weight of oil;

 0 – is initial oil stress; R - is internal radius of exploitation column; r - is current

radius of fluid flow; hz – is height of fluid movement in the well. Thus according to accepted symbols of abovementioned parameters we can write hz  H d ; P2  0 and P1  P .

Then we can get



 P  H d  O  2 2  0 R  r   R r  4 O  Hd O





(2)

Let’s consider boundary conditions of viscous – plastic oil movement on vertical in the pipes. at

r  r0 ;   0 ;

r0 - radius of fluid flow core is determined as r0  2H d 0 / P . 0 - velocity of flow core is determined by the following: 0 

 P  H d  O  2 2  0 R  r0   R  r0  4 O  Hd O





First of all let’s determine average velocity of gradient layer 398


  P  H d  O  2   1 1 dr   R  r02  0 R  r0  dr   R  r0 r R  r0 r  4  O  H d O  0 0 R

Q  av 



R









  P  H d  O   2 R 3  Rr0  r02  0 3R  r0  4 3 O  H d 2 O





(3)

Now let’s determine average velocity of the flow

 av 

Q R  r0   0 r0  av

R









P  Hd O 3   R 3  r03   0 R 2  r02 4 6  O  RH d 2 OR







(4)

From other side total average velocity of flow movement is

 av 

Q Q  2 F R

(5)

here Q - is productivity of the wells with viscous – plastic oil; F  R 2 - is cross section of the exploitation column.

If to consider balance of expressions (3) and (4) we receive



 

PR R 2  8H d3 03 / P 3 Hd  6Q O   O R R 3  8H d3 03 P 3  9 0  R R 2  4 H d2 02 / P 2







(6)

Then let’s determine distance from the wellhead to the height of dynamic level of viscous – plastic fluid (oil) H wellhead  H  H d

(7)

where H wellhead is distance from wellhead to the height of fluid dynamic level; H d - is dynamic level height; H - is well depth.

Conclusion. 1.Analytic formula for determining of dynamic level height of viscous – plastic oil in deeppumping wells has been received. 2. Knowing values of viscous – plastic oil dynamic level we can determine distance from wellhead to dynamic level. 3. Knowing values of dynamic level we can determine pressure in well bottom with abnormal viscous – plastic oil which allows to apply sound metering method for well investigation.

Keywords: pump exploitation, dynamic level, abnormal oil, deep pump, fluid shift stress. AMS Subject Classification: 00A69.

399


References 1. Charniy I.A. Unsteady Movement of Real Fluid in the Pipes, Edition 2, Moscow, Nedra, 1975, 296p. 2. Shelkachev V.N., Collected works in 2 volumes; Vol.1, Moscow, Nedra, 1990, 399p.

BUILDING DYNAMIC HOMOGRAPHS OF LARGE PROPER NAMES A. Fatholahzadeh1 1

UMI GT-CNRS, Centrale Sup elec 2, rue Edouard Belin, 57078, Metz, France e-mail: [email protected]

In information retrieval processing, it is important to store a huge number of compound strings or words, including the proper names. Storing those words along with values (i.e. information associated with words) occupies a large amount of spaces. This is infinite number of compound words created which become long strings. This paper presents a linear time method based on randomized binary search tree to map a language onto a set of values including the homograph outputs leading to more compact representation than the existing competitors (e.g. transducers). The method is described along with its effectiveness on large and real world data, including Azeri and French ones.

Keywords: compound strings, dynamic homographs, compound words. AMS Subject Classification: 00A69.

SECRET SHARING SCHEMES AND SYNDROME DECODING Selda Çalkavur1 1

Kocaeli University, Department of Mathematics, Kocaeli, Turkey e-mail: [email protected]

Secret sharing is an important topic in cryptography. Secret sharing schemes were introduced by Blakley and Shamir in 1979 [1]. Then several constructions were developed [2-8]. One of them is based on linear codes. Error correcting codes are used to correct errors when messages are transmitted through a noisy communication channel [9-12]. Syndrome decoding provides to correct errors in the code. 400


In this paper, we explore some relations between secret sharing schemes based on the binary linear code which is a single error correcting and syndrome decoding. Then we obtain some results using this relation. Keywords: Secret sharing scheme, threshold scheme, error correcting code, syndrome decoding. AMS Subject Classification: 14G50, 94A60, 94C30.

References 1. Blakley G. R., Safeguarding Cryptographic Keys, Proc. 1979 National Computer Conf., New York, June 1979, pp.313-317. 2. Brickell E. F., Some Ideal Secret Sharing Schemes, Advances in Cryptology-EUROCRYPT 89, Lecture Notes in Computer Science, Vol.434, 1990, pp.468-475. 3. Ding C., Kohel D., Ling S., Secret Sharing with a Class of Ternary Codes, Theor. Comp. Sci., Vol.246, 2000, pp.285-298. 4. Hill R., A First Course in Coding Theory, Oxford, Oxford University, 1986. 5. Karnin E. D., Greene, J. W., Hellman, M. E., On Secret Sharing Systems, IEEE Trans. Inf. Theory, Vol. IT-29, No.1, Jan. 1983, pp. 35-41. 6. Massey J. L., Minimal Codewords and Secret Sharing, Proc. 6th Joint Swedish-Russian Workshop on Information Theory, Mölle, Sweden, Aug. 1993, pp.276-279. 7. Mc Eliece R., Sarwate D., On Sharing Secrets and Reed-Solomon Codes, Communications of the ACM, Vol.24, 1981, pp.583-584. 8. Okada K., Kurosawa K., MDS Secret Sharing Scheme Secure Against Cheaters, IEEE Trans. Inf. Theory, Vol.46, No.3, pp.1078-1081. 9. Özadam H., Özbudak F., Saygı Z., Secret Sharing Schemes and Linear Codes, Information Security Cryptology Conference with International Participation, Proceedings, December 2007, pp.101-106. 10. Piepryzk J., Zhang X.M., Ideal Threshold Schemes from MDS Codes, Information Security and Cryptology-Proc. of ICISC 2002 (Lecture Notes in Computer Science), Berlin, Germany, SpringerVerlag, 2003, Vol. 2587, pp. 269-279. 11. Shamir A., How to Share a Secret, Commun, Assoc. Comp, Mach., Vol. 22, 1979, pp. 612-613. 12. Some Applications of Coding Theory, Cryptography, Codes and Ciphers: Cryptography and Coding IV, 1995, pp. 33-47.

401


TRAIN AND CHECK DATA SAMPLES OPTIMAL VOLUMES SELECTION METHOD A.S. Gasanov1, N. A. Murga2, S.H. Abdullayev3 1

International Scientific and Educational Center for Information Technologies and Systems, Kiev, Ukraine 2 National Technical University of Ukraine “KPI”, Kiev, Ukraine 3 Institute for Information Technologies ANAS, Baku, Azerbaijan e-mail: [email protected], [email protected], [email protected]

Problem statement. Having NNP prediction model defined by their parameters’ time series, to build a mechanism for selecting the number of the time series realizations for the models’ parameters adjustment and determining of optimal review and readjustment of the process models’ set. Used terms and definitions Definition 1. Adjust data sample is a set of process model’s variables realizations on which one’s parameters will be adjusted. Definition 2. Test data sample is a set of process model’s variables realizations on which one’s adequacy will be checked. Due to the fact that the test data adjustment will be used in the proposed method for selecting time intervals, it is necessary to give the next two definitions. Definition 3. Train data sample is a set of process model parameters realizations, on which the method parameters will be adjusted. Definition 4. Check data sample is a set process model parameters realizations, on which the method effectiveness will be checked. Note 1. Train and test data samples contain one adjust and test data sample. Note 2. Since the method is designed to run its application in real time, it is proposed that all the data samples are arranged chronologically and randomization is not performed. Designation 1. Let V  the volume of train data sample. Designation 2. Let T  the volume of adjust and the train data samples. Note 3. It is obvious that the amount of test data and train data samples is V  T . Volumes selection for models adjustment and application method basis is the values of quality models use criteria on these realizations of process models parameters. Therefore, it is necessary to give the next designations.

402


Designation 3. E  a   aggregated quality criteria of model using on adjust data sample, under condition that time series for this sample creation was given from the interval T   ; T  . Designation 4. S  , t   aggregated quality criteria of model using on a test data sample, under condition that time series for train data sample was given from the interval T  t; T  and the test one was given from the interval T  1 ; T    . Note 4. It is obvious that in most cases S   1, t   S , t   f   1 where f  some scalar-valued function. Concrete definition of the problem. Given the limit of aggregate quality criteria for the adjust and the test sample correspondently  1 and  2 , to find such numbers t and  which satisfy the following conditions: 1. T  t; T   The time delay involved in the formation of the sample to adjust the process model. 2. T  1 ; T     The period of time in which the process model efficiency is guaranteed. 3. E  t   1 4. S  , t    2 5 t should be the minimum possible but not less than the process model power s . 6  must be the maximum possible. Note 5. The desire to make the t as smallest as it is possible and   as biggest as it is possible means necessary to adjust the model to the most relevant realizations of the process parameters time series, while ensuring the its efficiency on as longest time interval as it is possible. Mathematical problem statement. Concrete definition of the task allow to do the next its mathematical statement: max  , min t , t  s  1,   1,

x1 E  t   1

x1, S  , t    2 ,



T i s 1



k i 1

t  T ,   V T

(1)

xl ,i  k, k  1... V  T  1, l   s  1...T ,

xi ,p  1, p  1... V  T  ,

xt ,  0; 1 , t  N ,

403

 N

(2) (3)


In this statement the following additional notation is proposed: xt ,  fact indicator of the model effectiveness on adjust data sample time series of length, under the condition that the process model was adjusted on the time series of the test sample with length t . Note 6. Since both E  t  and S  , t  are calculated after the process model application, then under point of stated task view, they are numbers and the problem (1)-(3) is a linear discrete optimization problem. Note 7. By using linear convolution, criteria (1) - (2) can be reduced to one criterion:  t V T   min   , V T  T

(4)

Method effectiveness check. The proposed method of time series volumes selection has a practical purpose and, as a result, checking its effectiveness is possible only in the field of experimental research. Values t and  are obtained as a result of the method application, are an investigation object when method is applied on check data sample. Let the time interval of the check sample is in the form 0, V   , where V   the amount of the check sample. Let T   the volume of the adjust data sample on the check one, then 0, T    the time interval of adjust sample on check one, and T   1, V    is the time interval of test data sample on check one. Next, fix t   t and     , and the values t and  will continue to change. The method quality criteria is S  t,   . Suppose S  t,  , T    that the value of S  t,   for the concrete T  . Thus, if as the experiment parameters select T  , t , and  it is possible to obtain the following hypersurface in the space shown in Fig. 1. Use of this hyperspace, or rather its projections on threedimensional spaces, allows to evaluate of method effectiveness visually. Fig. 1 Method quality evaluation space

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The formal criterion for the effectiveness of the proposed method is the following expression:

t, : S t,   S t ,   t is near t    is near   







(5)

Correspondence of these criteria to the goals of experimental investigations is followed from complete correspondence of stated during method development task. Practical recommendations for the method application. At first glance, the number of constraints (1) is excessive. However, if we consider that when adjusting the model, values t and

t  1 are not perceived as different (the same applies to  and   1 for model effectiveness limits), it is advisable to consider not all the values from the intervals t  s;T  and   1, V  T  , but the elements from these sets that are multiples of some values. This will both reduce the set of constraints (2) and narrow down the set of feasible solutions of (1)-(3). Keywords: selection method, train and check data, method effectiveness, practical recommendations. AMS Subject Classification: 60, 62 (34A38).

DEVELOPMENT OF STARTUP-ECOSYSTEM IN AZERBAIJAN Anar Jafarov1 1

Azerbaijan State University of Economics, Baku, Azerbaijan e-mail: [email protected]

Startup – this is a newly established company which developes and builds its business on the basis of new innovative ideas, or on basis of newly emerging technologies. Most modern meaning of "start-up" refers to this or that venture project. One of the main reason for the creation of successful development and continued existence start-ups are considered clumsiness and slowness of the large corporations who successfully use existing products and barely engaged with development and creation of new. Therefore, start-ups, because of their mobility in terms of realization of new ideas compete with large corporations. A good innovative idea is the main resource for the creation of start-up. A major factor in the success of this idea is its relevance, ie, degree of necessity for the consumer. Also the success of start-ups contributes to youth of startups (the average age of startups on Statistics - twenty-five years), as well as their enthusiasm for the idea and deed. Development of start-ups is important for Azerbaijani economy. The State provides

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necessary support and takes targeted steps on creation a favorable environment for the development of start-up movement in Azerbaijan. The State Development Fund of Information Technology, which is priority areas are defined by the Ministry of Communications and High Technologies of Azerbaijan aims to become the foundation of the development of start-ups in the country. Sitimulation of activities in the sector of information and communication technologies, the expansion of the application of innovation and applied research in this area are provided by the financial support of the Fund. The Fund provides financing in the form of grants on the innovative and applied scientifically- technical start-up projects. Many interesting startups was funded with great potential for further development and access to markets of other countries during the last year in Azerbaijan. There is an interesting, which has turned into a real business, among the projects that have received state financial support. At present, many countries with advanced economies receive income from the sale of intellectual production. Availability of innovative projects is a clear indicator of the country's development, which aims at augmenting their number, so that Azerbaijan could join the ranks of advanced IT-industry. In order to ensure sustainable development and improve the competitiveness of the economy, the expansion of innovative and high-tech industries which are based on modern scientific and technological achievements, conducting scientific research and the creation of modern facilities for the development of new technologies by the government, also was established Business Incubation Center "High Technology Park" (HTP) Ltd. The advantage of HTP that is created on Pirallahi island will accumulate the basic components required for a successful operation. It is noteworthy that HTP residents get exemption from the following taxes - VAT, income tax, land tax, property tax. In addition, from the VAT will be released equipment which is imported for use of research projects of HTP. Prescribes formation of the branch network of HTP throughout the republic. This concept covers the period of planned building of HTP hightech, which will take about two years. During this period, the task of the branches of the creating technopark will be performed in the enterprise of the country- «KUR» plant for the production of computer equipment of Mingachevir, «Billur» Plant of Ganja, where entrepreneurs or aspirants can implement their projects. So the order of the President recently established Mingachevir high-tech park, which will have the infrastructure, material and technical basis and management structures needed to conduct research and development work in order to prepare, develop or improve product innovation and 406


high technology application (commercialization )of their results in industrial, service and other areas. Production capacity of Mingachevir Technopark will be involved for the production of hightech products that already planned for this year. Among them - the deployment of production of computers Acer, which envisages to invest about 10 million manat. Speaking about the production, the production is meant not only the computer and other high-tech products. Azerbaijan considers three areas of development of the production sector. The first –this is a «Hardware» market (IT-finished products). Of course, to achieve progress in this area in a short period is difficult, but nevertheless, the state is interested in the fact that local companies are able to achieve the world level development. The second direction is the software market, which is already receiving adequate support from the state. It is about supporting the start-up movement at the level of the State Fund for development of information technologies. The third direction - the sphere of IT-services. The plan also provides strengthening the export potential of local companies. Today, the export potential of Azerbaijan in high-tech is very small and is approximately $ 52 million. The policy of Azerbaijan in this area is based on the rule which is called "golden triangle". The first - the development of human resources, the second - creating a favorable business environment, and the third - investment. The necessary steps have already been taken on all these fronts. Unfortunately, the relationship between business and science are at a low level, which is the biggest barrier for innovative development of the country. Another problem is the weak legal framework. The absence of the Law "On venture funds" have a negative effect on this sector in the country. In this case to support businesses and coordination of business and science is important. For this you need to create a favorable business environment in order to large business is interested in the development of innovative economy. Keywords: High-Tech Park, HTP, the ICT sector, "start-up", innovative entrepreneurship, ICT Fund. AMS Subject Classification: 91B02. References 1. Gasimov F. H., Najafov Z. M. Innovation: creation, distribution and prospects of development, Baku, Elm, 2009. 2. Ammosov Y.P., The venture capitalism: from the beginnings to the present, SPb .: RAVI, 2004. 3. Krylov E.I., Vlasov V.M., Ovodenko A.A., Analysis the effectiveness of innovations: Proc. Benefit, SPbSUAI. SPb., 2003.

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the investments and


4. The speech of the Minister of Communications and Information Technologies and academician Ali Abbasov at the First Congress of Azerbaijan scientists. 5. The National Strategy for the Development of Science in 2009-2015 in Azerbaijan Republic. "Baku, 10.04.2008. http://www.science.gov.az/az/index.php?id=1786 6. “Azerbaijan 2020: Look into the Future” Concept of Development”.

MORPHOMETRIC STUDY OF THE HORIZONTAL DISMEMBERMENT DENSITY OF MOUNTAIN GEOMORPHOSYSTEMS IN THE GREATER CAUCASUS (WITHIN AZERBAIJAN) TO OPTMIZE RESOURCE MANAGEMENT M.M. Mehbaliyev1 Baku State University, Baku, Azerbaijan e-mail: [email protected]

Introduction. The density of the horizontal dismemberment is one of the main morphometric parameters of slopes, which mainly reflects the characteristics of transverse and longitudinal dismemberment of morphostructures, repeatability of slopes with different morphometric parameters. The main factor in the formation of the thickness of the horizontal dismemberment of geomorphosystems is tectonic movements. An important role is also played by lithology, jointing and deposition of rocks, the nature of soil, the slope of surface, the nature and amount of rainfall, terrain elevation, slope exposure, the direction of watersheds, vegetation, human activities, and etc. Its study is of great scientific and theoretical and applied [6] value (Tab.1). Table.1. The effect of the density of slopes horizontal dismemberment on economic activity №

Type of economic activity Effect

1.

Agriculture

Leads to a fractional fields of crop rotation, a sharp increase in the number of furrows, a growth in the number of turns and stopovers, fuel consumption, poor performance of agricultural machinery, the intensity of erosion, etc.

2.

Engineering works

Increase in the cost of engineering works due to the large number of small man-made structures and extension of lines while roundabout. Occurrence of gully landslides and erosion. The additional work to consolidate and fill up gullies, the destruction of soil strength, etc.

3.

Recreation

Defines diversity and practicability of area, the possibility of transporting the holidaymakers, staff, creation of infrastructure related to recreation, etc.

408


4.

Winter tourism

Defines possibility for manoeuvrability of a skier, the length of the ski slopes, and practicability of territory. Strongly dissected relief creates certain difficulties in the organization of winter tourism. Small and steep slopes dominate in densely dissected areas. Avalanches are often observed here, which negatively affect the organization of winter tourism.

Researchers. Alizada E.K. [1], Anisimov V.I. [2], Chervyakov V.A. [9], Berlyant A.M. [3], Piriyev A.D. [7], Rajabli T.R. [8], and others were engaged in the study of the horizontal dismemberment density of mountain geomorphosystems slopes. They defined the indicators of horizontal dismemberment density within a square. We defined horizontal dismemberment density within the elementary slope that is of objective character, has a natural border, and easily stands out on a topographic map and on the terrain. Object of the study. The object of the study is the mountain geomorphosystems of the Greater Caucasus (within Azerbaijan) with a total area of 16427,75 sq km. Here, 3,960 elementary slopes with different morphometric parameters were marked based on a topographic map of 1:100 000 scale. Research methods and materials. The density of horizontal dismemberment of the each of the slopes was calculated by the well-known formula: K

L , S

(1)

where K is an indicator of horizontal dismemberment density (km./sq km), L - the length of the horizontal dismemberment elements (km),

S - slope area (sq km). There were used mathematical and statistical, comparative and descriptive (visual), conjugated, map, graphics and other methods as well. GIS technologies with great potential and prospects in the morphometric study of mountain geomorphosystems and mapping were widely used. The morphometric studies were carried out on the basis of the topographic map of 1:100 000 scale. Numerous thematic maps were also used [4,5]. The studies, results and discussion. To achieve this task, there was made a density map of slopes horizontal dismemberment throughout the study area by cartograms way on (Fig.1) a scale of 1:100 000 using GIS technology (ArcGIS10.2.1). There was estimated an amount, measured the area, calculated the cumulative number, growing area, the average area and density of the slopes on the map. The slopes were classified for horizontal dismemberment density on a 409


scale: 0-1 (slightly dissected), 1-2 (medium dissected), 2-3 (strongly dissected), 3< (much dissected). The results of these studies are summarized in the table (Tab.2). Histograms were made based the table. Slightly dissected slopes are predominant in the study area, differing for forest-cover, lower steepness and jointing. Such slopes are distributed mainly in the foothills of the study area. Generally, they are beneficial to economic activity. According to the number and area they are followed by medium dissected (1044; 26,36%, 4831,13 sq km; 29,41%), strongly dissected (104; 2,63%,407,11 sq km, 2,48%) and very dissected (14; 0,35%, 48,46 sq km; 0,29%) slopes. Slightly dissected (0,170) slopes are characterized by the highest density. The medium dissected (0,064), strongly dissected (0,006) and very dissected (0,001) slopes come next. Medium dissected slopes are characterized by the maximum average area. The most significant indicators of the value of horizontal dismemberment density are observed mainly in the southern arid part of the study territory, especially in the Gobustan-Absheron physicalgeographical area. In the middle mountain area, where much rain falls, the significant value of horizontal dismemberment is formed due to the river system, and in the alpine zone - due to tectonics. In the highlands, where there are rocky outcrops, hardly dissolvable rocks and in forested areas (vegetation prevents the formation of gullies and ravines and valley network), the small quantities of horizontal dismemberment density of relief are observed. Horizontal dismemberment density negatively impacts on economic activity. It leads to an increase in the cost of engineering work due to a large number of small man-made structures and extension of lines when they rounded, occurrence of gullies and erosion, additional work on the consolidation and filling of gullies, destruction of soil strength, and etc. It allows objective evaluating the traficability degree. Significant values of horizontal dismemberment density (significant values of other morphometric parameters correspond with them mainly) create conditions for the formation of powerful mud torrents, avalanches, and landslides. They cause great damage to the farm. Depending on the nature and morphometric features of geomorphosystems, there may arise reservoir, groundwater, axial, jumping, dry, wet, damp and other types of avalanches. In densely disjointed areas, the productivity of agricultural machinery is low, there are some difficulties in the construction of tourist and recreational facilities, choice of hiking trails, and etc.

410


Horizontal dismemberment density strengthens ecogeomorphological tension of slopes. Primary state of the slopes is destroyed, erosion process increases, avalanches, debris, and rock falls which hamper economic activity are occurred. Therefore, a detailed and comprehensive study and mapping of horizontal dismemberment density using the new technology is very promising. Conclusion. 1. Weakly disjointed slopes are those modal on the study area (K = 0-1 km/sq km, 2798; 70,66%, 11141,05 sq km; 67,82%). With the increase in the value of horizontal dismemberment density the number and area of the slopes decreases, the average size first increases and then decreases. Density of the slopes is also reduced. In the study area, the average area of the slopes is 4,15 sq km. Thus, the study area as a whole is favorable for economic activity. Keywords: horizontal dismemberment, morphostructures, geomorphosystems. AMS Subject Classification: 09B02.

References 1. Alizada E.K. Regularities of morphostructural differentiation of mountain structures of the eastern segment of the central part of Alpine-Himalayan seam zone (based the materials of satellite images interpretation). Abstract of a doctoral thesis, Baku, 2004, p.53. 2. Anisimov V.I. Basics of morphometric analysis of relief. Grozny: CIGU, 1987, p.91. 3. Berlyant A.M. Image of space: Map and information. Moscow: Misl, 1986, p. 240. 4. Azerbaijan geology. 6 volumes, V.II, Baku: Nafta-Press, 2005, p.277. 5. Azerbaijan geology. 6 volumes, V.IV, Baku: Nafta-Press, 2005, p.505. 6. Mehbaliyev M.M. Applied morphometry. Materials of IV International Scientific and Practical Conference. Youth and Science: Reality and Future, V.IV, Natural and applied sciences, Nevinnomyssk, 2001, pp.400-404. 7. Piriyev R.H. The methods of relief morphometric analysis (by example of Azerbaijan), Baku; Elm, 1986, 119 p. 8. Rajabli T.R. Morphometric study of Karabakh plain and the adjoining slopes of the Small Caucasus in agricultural aims. Abstract of a doctoral thesis, Baku, 1990, p.24. 9. Chervyakov V.A. Quantitative methods in geography. Barnaul: Altai State University. 1998, p.258.

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MINIMIZATION PRINCIPLE OF EIGENVALUES AND RAYLEIGH QUOTIENT OF A BOUNDARY VALUE-TRANSMISSION PROBLEM Oktay Sh. Mukhtarov1, Kadriye Aydemir1 Department of Mathematics, Gaziosmanpaşa University, Tokat, Turkey e-mail: [email protected], [email protected]

1

In this article we discuss equiconvergence of eigenfunction expansions for the Sturm– Liouville boundary value problem with transmission conditions on a finite interval. We derive an eigenfunction expansion theorem for the Green's function of the problem as well as a theorem of uniform convergence of a certain class of functions. By applying the obtained results we extend and generalize such important spectral properties as Parseval and Carleman equations, Rayleigh quotient and Rayleigh-Ritz formula (minimization principle) for the considered problem. Keywords: eigenfunction expansions, Sturm–Liouville problem, Rayleigh-Ritz formula. AMS Subject Classification: 34B24, 34B27, 34L10, 65M60

ON THE USE OF COMPUTER NETWORKS L.M. Nusretli1 1

Baku State University, Department of Applied Mathematics and Cybernetics, Baku, Azerbaijan e-mail: [email protected]

A computer network or data network is a telecommunications network which allows computers to exchange data. In computer networks, networked computing devices pass data to each other along network links (data connections). Data is transferred in the form of packets. The connections between nodes are established using either cable media or wireless media. The bestknown computer network is the Internet. Network computer devices that originate, route and terminate the data are called network nodes [1]. Nodes can include hosts such as personal computers, phones, servers as well as networking hardware. Two such devices are said to be networked together when one device is able to exchange information with the other device, whether or not they have a direct connection to each other. Computer networks differ in the transmission media used to carry their signals, the communications protocols to organize network traffic, the network's size, topology and

412


organizational intent. In most cases, communications protocols are layered on (i.e. work using) other more specific or more general communications protocols, except for the physical layer that directly deals with the transmission media. Computer networks support applications such as access to the World Wide Web, shared use of application and storage servers, printers, and fax machines, and use of email and instant messaging applications. A computer network, or simply a network, is a collection of computers and other hardware components interconnected by communication channels that allow sharing of resources and information. As of 2015 computer networks are the core of modern communication. Computers control all modern aspects of the public switched telephone network (PSTN). Telephony increasingly runs over the Internet Protocol, although not necessarily over the public Internet. The scope of communication has increased significantly in the past decade. This boom in communications would not have been possible without the progressively advancing computer network. Computer networks, and the technologies that make communication between networked computers possible, continue to drive the hardware, software, and peripherals industries. The expansion of related industries is mirrored by growth in the numbers and types of people using networks, from the researcher to the home user. Computer networking may be considered a branch of electrical engineering, telecommunications, computer science, information technology or computer engineering, since it relies upon the theoretical and practical application of the related disciplines. A computer network facilitates interpersonal communications allowing people to communicate efficiently and easily via email, instant messaging, chat rooms, telephone, video telephone calls, and video conferencing. Providing access to information on shared storage devices is an important feature of many networks. A network allows sharing of files, data, and other types of information giving authorized users the ability to access information stored on other computers on the network. A network allows sharing of network and computing resources. Users may access and use resources provided by devices on the network, such as printing a document on a shared network printer. Distributed computing uses computing resources across a network to accomplish tasks. A computer network may be used by computer Crackers to deploy computer viruses or computer worms on devices connected to the network, or to prevent these devices from accessing the network (denial of service). A complex computer network may be difficult to set up. It may be costly to set up an effective computer network in a large organization. 413


Computer communication links that do not support packets, such as traditional point-topoint telecommunication links, simply transmit data as a bit stream. However, most information in computer networks is carried in packets. A network packet is a formatted unit of data (a list of bits or bytes, usually a few tens of bytes to a few kilobytes long) carried by a packet-switched network. In packet networks, the data is formatted into packets that are sent through the network to their destination. Once the packets arrive they are reassembled into their original message. With packets, the bandwidth of the transmission medium can be better shared among users than if the network were circuit switched. When one user is not sending packets, the link can be filled with packets from others users, and so the cost can be shared, with relatively little interference, provided the link isn't overused. Packets consist of two kinds of data: control information and user data (also known as payload). The control information provides data the network needs to deliver the user data, for example: source and destination network addresses, error detection codes, and sequencing information. Typically, control information is found in packet headers and trailers, with payload data in between. Often the route a packet needs to take through a network is not immediately available. In that case the packet is queued and waits until a link is free. Keywords: computer networks, network organizations. AMS Subject Classification: 90B18.

THE CENTRAL LIMIT THEOREM FOR THE FAMILY OF THE FIRST MOMENTS OF REACHING THE LEVEL OF A RANDOM WALK, DESCRIBED THE FIRSTORDER AR1 AUTOREGRESSION PROCESS F.H. Rahimov1, S.A. Aliev1, A.D. Farhadova1 1

Institute of Mathematics and Mechanics, ANAS e-mail: [email protected]

Let  n ; n  1 be a sequence of independent identically distributed random variables defined on some probability space ,F, P  .

414


It is known that the sequence of random variables X n , n  0 defined on ,F, P  is called the first order AR1 autoregression process if the following recurrent relation holds true X n  0 X n 1  n , n  1 ,

where  0 is some fixed number and initial value of X 0 does not depend on  n . Autoregressive process describes a time series model, in which the value of the process in this time is linearly dependent on the previous values of the same process. Such models are widely used in application of the stochastic processes (see [1, 2, 3]). Let us consider the sum   

2

n

 X

k

 X k 1 

k 1

As a function of the parameter  (see [3]). Denote by n  n  X 0 ,..., X n  the estimate for  that minimizes   i.e.  n   min   .

(1)



Then we have   

n



X k2 2

k 1

n



X k X k 1   2

k 1

n

X

2 k 1

.

k 1

The estimation  n is a solution of the equation '    0 having the form n

n 

X

k X k 1

k 1 n

X

2 k 1

k 1

and satisfying (1), since  ''    0 for all  . In [3] is proved that under the condition  0  1 and EX 02   the central limit theorem for





the estimation  n is valid i.e. lim P n  n  0   x  x , where n 

 x  

1 2

x

e

 y2/ 2

dy, x  R   ,  ,  



Define Tn 

n

X k 1

n

k X k 1

and S n   X k21 . k 1

415

1 1   02

.


Note that some asymptotical properties of the sums Tn and S n are investigated in [2]. Particularly in [2] is shown that for  0  1 Tn a.s.  S 1 .s.  0 2 , n   and n a , n , n n 1  0 1   02 .s. where a means almost sure convergence over the measure P .

It is clear that n 

Tn Sn

.s. and  n a  0 by n   .

It should be noted that in recent years there have been several studies in which boundary problems for random walks are investigated, described by the first-order autoregression process. In this work the central limit theorem is proved for the family of stopping moments  a , a  0 of the form  a  inf n  1: n  a,

(2)

where  n  n n . Here inf    for the correct definition of  a . The family of the stopping moments (2) may be written in the following form 

  Tn S n  ,   a , n n  

 a  inf n  1 : ng 

where g x, y  

x T S  and g  n , n    n . y n n 

Note that the family of stopping moments type of  a underlie the theory of Markov reconstruction and play a big role in the applications of the theory of stochastic processes and statistical sequential analysis ([1-3]). This work was supported by the Science Development Foundation under the President of the Republic of Azerbaijan – Grant N EIF -2013-9(15)-46/13/1. Keywords: central limit theorem autoregression process, stochastic process, statistical sequential analysis. AMS Subject Classification: 60G50, 82B41.

416


References 1. Novikov A.A. Notes on the time distribution of the first reach and optimal stopping of the AR1 sequences, Probability Theory and Applications, Vol.53, No.3, 2008, pp. 458-471. 2. Melfi V.F. Nonlinear Markov renewal theory with statistical applications, The Annals of Probability, Vol.20, No.2, 1992, pp.753-771. 3. Pollard D. Convergence of Stochastic Processes, Springer, New-York, 1984.

INTEGRAL LIMIT THEOREMS FOR THE FIRST REACHING THE LEVEL OF THE RANDOM WALK DESCRIBED BY THE FIRST-ORDER AR1 AUTOREGRESSION PROCESS F.H. Rahimov1, T.E. Hashimov1, A.D. Farhadova1 1

Institute of Mathematics and Mechanics, ANAS e-mail: [email protected]

Let in the probability space ,F, P  the sequence of independent identically distributed random variables  n   n   and a random variable X 0  X 0  ,    be given and X 0 does not depend on  n , n  1 . As an autoregression process of the first order  AR1 we take the solution of the recurrent equation X n  X n1   n , n  1 ,

where  is some nonrandom constant. For the case   1 the process X n is usual random walking i.e. is a sum of independent identically distributed random variables  n , n  1 . Note that in applications of the theory of stochastic processes many mathematical models are described using autoregressive processes. For example, in the economic models the process

 AR1

describes the change in prices, the value X n of which in the time n depends on its last

development ( X n 1 ) and posed on it innovation  n is not related with the past (the random variables  n and X n 1 are independent). Consider the sequence of sums Sn 

n

X

2 k,

k 0

and introduce the family of the stopping moments 417

n0


 a  inf n : Sn  a,

a0.

(1)

The quantity  a is a moment of the first reach the level by the process Sn , n  0 and always may be taken inf    . Such a family of stopping moments type of (1) arises in the applications of the theory of stochastic processes, and statistical sequential analysis ([1-3]). In this work we prove an integral limit theorem for a family of stopping moments

 a of

the form (1), which is defined as a statement that under certain conditions there are normalizing constants Aa  and Ba   0 (depending on the parameter a ) and not degenerated random variable

 such that the convergence over the distribution  a  Aa  Ba 

d    by a  

take place. In [3] is shown that the estimate for the parameter  obtained by the method of least squares, that minimizes the sum 2

n

 X

k

 X k 1 

k 1

has a form n

n 



n

X k X k 1

k 1 n

X



X

k X k 1

k 1 n1

2 k 1

k 1

X

 2 k

Tn , S n1

k 0

n

where Tn   X k X k 1 . k 1

Moreover under the condition   1 and EX 02   , the estimation  n is asymptotically normal by parameters 0,1   2  , i.e. the convergence



d n  n      N 0,1   2



take place over the distribution by n   , where N 0, 2  denotes the random variable with normal distribution with parameters 0,  2 , EN 0, 2   0, DN 0, 2    2  . In [2] is shown that under the condition   1 and E X 0   the strengthened law of large 2

numbers takes place for Tn and S n , i.e. the following almost sure convergences are valid 418


Tn a.s.   n 1  2

and S n п. н 1  n 1  2

by n   .

.s. From these relations follows that  n a  by n   .

Theorem. Let En  0, Dn  1, 0    1 and E X 0   . Then the following convergence 2

over distribution is true





 a  1  2 a a

 1  2 d   N  0,  2 

   

by a   . This work was supported by the Science Development Foundation under the President of the Republic of Azerbaijan – Grant N EIF -2013-9(15)-46/13/1. Keywords: integral limit theorem, autoregression process, stochastic process, statistical sequential analysis. AMS Subject Classification: 60G50, 82B41.

References 1. Novikov A.A. Notes on the time distribution of the first reach and optimal stopping of the AR1 sequences, Probability Theory and Applications, Vol. 53, No.3, 2008, pp. 458-471. 2. Melfi V.F. Nonlinear Markov renewal theory with statistical applications, The Annals of Probability, Vol.20, No.2, 1992, pp.753-771. 3. Pollard D. Convergence of Stochastic Processes, Springer, New-York, 1984.

K∞ – ROBUST CONTROL OF OBJECT IS WITH TIME DELAY ON STATE Rustam Rustamov1 Azerbaijan Technical University, Baku, Azerbaijan e-mail: [email protected]

Abstract – based on the method of Lyapunov function the original system is transformed into a linear fetch system the right side of which includes undefined object model. Robust control strategy is to suppress the object model as a potential source of parasitic dynamics and leading 419


fetch system to the homogeneous equation. The solution of model problems using the Matlab/Simulink software package has permitted to reach a number of positive conclusions with important applied meanings. I. Introduction. The important is a construction of the control systems for objects with an internal delay or delay on state. The control problem becomes much more complicated when the delay is variable. This feature leads to the permanent change in the topology of the roots of the characteristic equation of the closed-loop system, and thus - to change the structure (type) of the object. Therefore, the controller must be invariant to the structural uncertainties of the object. The report for the construction (design) of such regulator the method of "robust equivalent control" proposed in [1,2] is used. II. Statement of problem. Nonlinear uncertain object is considered with a known structure described by differential equations with delay on state as y ( n) (t )  f ( y(t ), y(t  τ), t )  b( y(t ), y(t  τ), t )u  (t ),

t  [0, )

(1)

where y(t )  ( y(t ), y (t ), y ( n 1) (t ))T  ( x1(t ), x2 (t ),..., xn (t ))T  Rn  observable or measurable state vector; y  R  controlling output; u  R  control; f (), b()  0  non-linear unknown bounded functions; (t )  uncontrolled bounded outward influence. The delay τ(t )  (1 (t ), 2 (t ),..., n (t ))T , t  0 is assumed unknown and variable. Background of the process in the interval t  [τi ,0] is defined by the condition

yi (t )  i (t ), where

i

is the maximal value of the delay time, i (t )  initial function,

i  1,2,..., n. The tracking problem consists of the construction of such control u (t ), that after the completion of the transition component provides the output motion y (t ) of the object (1) on the etalon trajectory yd (t ) with a given accuracy | e(t ) | s , t  ts . Here e(t )  y d (t )  y(t ) is an tracking error, t s  given setting time for the transition component with error  s  (1  5)% . III. Solution of the problem. In [1] on the base of the Lyapunov function method for V= 2 ½ s the following robust equivalent control is obtained

uRe q  Ks  k (c1e  c2e  ...  e( k 1) ),

(2)

where K  0  is big enough number; ci  0, i  1,..., n  1, angle coefficients of the hyper plane

s  0. 420


When the coefficient of the controller gain K tends to infinity, the closed-loop uncertain system with the object (1) and control (2) is described by the equation of the hyper plane s  c1e  c2e  ...  e( k 1)  0,

s(0)  s0 .

The methodology of determination of the turning parameters ci , K is given in [1]. IV. Simulation example. Example. Generator of nonlinear oscillations by Van-der Pol. The equation of such generator in the state variables is as follows:

x1  x2 , x2  ( x12  1) x2 (t  2 (t ))  x1 (t  1 (t ))  bu  m(t ), y  x1 ;   [0 1 2 10 20], 1  1  sin(2t ),  2  1  cos(2t ), b  2  0.5 sin(2t )  0.5 cos(6t ), yd  1  sin(2t )  cos(4t ),   0.5randn(1,200). In fig.1 the characteristics of the system is given by c1  2, К=200, t s  3s,  s  2% and

  [0, 1, 2, 10, 20].

а)

в)

Figure 1. Dynamical characteristics of the robust system

Consentration of the chains {y(t)} and {e(t)} is high enough. After t s  3 c the object output {у(t)} is enough close to the etalon trajectory yd (t ) . V. Conclusion. The advantage of the proposed methodology is as follows: minimal information is used about the object - just a structural representation of the model. The exception is H∞- theory, which uses the exact nominal object model; observers of the uncertainties are not used that reduce the system performance; simplicity of the synthesis and the ability to build simple robust controllers that are in demand in industrial applications; independent settings сi, i=1,…, k1 and K;

421


The solution of the model example using Matlab/Simulink allowed us to conclude some positive results with important applications. Keywords: robust system control, Lyapunov function, time delay on state, Van der Pol generator. AMS Subject Classification: 93D09.

References 1. Rustamov G.A. Robust control systems with gained potential, Transactions of Tomsk Polytechnic University, Vol.324, No.5, 2014, pp.13-19. 2. Rustamov G.A. Construction of the invariant systems on the base of equivalent robust controls, East-European Journal of Advanced Technologies, No.2, 73, 2015, pp.50-55, doi:10.15587/1729– 4061.2015.37177.

INTERPOLATION APPROACH TO SEARCH HIDDEN RESULTS IN GPR DATA Refik Samet1, Ertuğ Çelik2, Erkan Şengönül1, Serhat Tural1, Merve Özkan1 1

2

Ankara University, Ankara, Turkey Ankara Earth Sciences, Ankara, Turkey [email protected]

One important problem of Ground Penetrating Radar (GPR) applications is 3-D visualization and interpretation of anomalies in GPR data. In order to contribute to solve this problem this work proposes the interpolation approach to search hidden results in GPR data. Proposed approach consists of three stages. At first stages, 1-D interpolation techniques are used to resample the sample values of traces of GPR profiles in GPR data. At second stages, 2-D interpolation techniques are used to resample the traces of GPR profiles in GPR data. Finally, at third stage, 3D interpolation techniques are used to resample the profiles in GPR data. Implementation results showed that thanks to the proposed approach the shape of anomalies has been enhanced and accuracy of anomalies has been increased significantly. Introduction: GPR is a geophysicaltechnique for collecting data about nearsurface/subsurfaceearth materials and abnormal objects. Geologistsuse GPR data to obtain a view of terrain underground. During GPR data processing the anomalies are searched, analyzed, visualized and interpreted.A lot ofprograms and libraries exist for this aim [1]. These programs and libraries show the anomalies as parabolas which are not suitable for understanding and

422


interpretation of the shape and type of anomalies.In this work the interpolation approach is proposed to increase an accuracy of anomalies and enhance the shape of anomalies in GPR data. Proposed Interpolation Approach: GPR data consist of 𝑁 profiles. Each profile consists of 𝑀 traces. Each trace consists of 𝐾 sample values (Fig.1).

(a)

(b)

(c)

Fig.1. (a) Profiles of GPR data; (b) Traces of profile; (c) Sample values of trace

Proposed interpolation approach includes three stages (Fig.2).

Fig.2. Stages of proposed interpolation approach

First stage is related to the sample values of traces of GPR profiles in GPR data. By using 1-D interpolation techniques[2], [3]the sample values of traces are resampled in𝑅𝑆𝑉 times. At the result, the original traces will be reproduced and they will consist of (𝑅𝑆𝑉 ∗ 𝐾 − 1)element, where 𝑅𝑆𝑉 is resampling rate (Fig.3).

1-D Interpolation Techniques

Fig.3. Resampling of the sample values of traces of GPR profiles in GPR data

Second stage is related tothe traces of GPR profiles in GPR data. By using 2-D interpolation techniques the traces of GPR profiles are resampled in 𝑅𝑇 times. At the result, thenew traces

423


between the reproduced original traces (obtained at the result of the first stage) will be producedand reproduced profiles will consist of (𝑅𝑇 ∗ 𝑀 − 1)trace, where 𝑅𝑇 is resampling rate (Fig.4).


Fig.4. Resampling of the traces of GPR profiles in GPR data

Third stage is related to GPR profiles in GPR data.By using 3-D interpolation techniques the profilesin GPR data are resampled in𝑅𝑃 times. At the result, the new profiles between the reproduced profiles (obtained at the result of the second stage) will be produced and reproduced GPR data will consist of(𝑅𝑃 ∗ 𝑁 − 1)profile, where 𝑅𝑃 is resampling rate (Fig.5).


Fig.5. Resampling of GPR profiles in GPR data

At the result of interpolation approach reproduced GPR data is obtained. In order to visualize and interpret the effects of the proposed interpolation approach the reproduced 2-D profiles are visualized in 3-D space and 3-D volume of reproduced GPR datais created. Implementation of proposed interpolation approach: GPR data with 𝐾 = 256,𝑀 = 150and 𝑁 = 5were used. Different interpolation techniques such as mean, median, linear, cubic, spline, nearest, etc. were implemented on reproduced GPR data. Also, different resampling rates at different stages of interpolation approach were tested. Obtained results showed that the accuracy and shape of anomalies are improved significantly by increasing of the resampling rate. Conclusion: This work has proposed an interpolation approach to search hidden results in GPR data. Different interpolation techniques have been implemented on different sets of GPR data. Obtained results have proved that the proposed interpolation approach has improved the

424


accuracy and shape of anomalies. These improvements were very useful to analyze, visualize, understand and interpret the anomalies in GPR data. Acknowledgment: This work has been funded by TÜBİTAK of Turkey under grant 5130012. Keywords: GPR data, anomalies, search hidden results, interpolation techniques, profile, trace. AMS Subject Classification: 00A66, 41A05, 65D05, 68U10, 94A08.

References 1. Samet R., Çelik E., Tural S.,Şengönül E., A new multipurpose easy and quick GPR data processing and visualization software, XIVth International Conference – Geoinformatics: Theoretical and Applied Aspects, 11-14 May, 2015, Kiev, Ukraine. 2. Samet R., Tural S., Ercan T., Two-way real-time meteorological data analysis and mapping information system, Appl. Comput. Math., Vol.13, No.3, 2014, pp.350-365. 3.

Lehmann T.M., Gönner C., Spitzer K., Survey: Interpolation Methods in Medical Image Processing, IEEE Transactions on Medical Imaging, Vol.18, No.11, 1999, pp.1049-1075.

ASSESSMENT OF SCIENTIFIC PRODUCTIVITY ON HIGHER EDUCATION IN AZERBAIJAN: CROSS-UNIVERSITIES ANALYSIS Sardar A. Shabanov1 1

Reasearch Institute for Economic Studies, Azerbaijan State University of Economics, Baku, Azerbaijan e-mail: [email protected]

The study is dedicated to the assessment of the scientific productivity of 52 higher education universities in Azerbaijan. A scientific article published in impact factor (IF) indexed journals or an international patent are accepted as a unit of scientific product [1] in the world. We will only take into consideration articles published in prestigious scientific journals as the number of international patents are few in Azerbaijan. It is known that these articles are indexed in Web of Science (scientific base WOS) which combines approximately 12000 scientific journals and runs under Thomson Reuters agency. The investigation covers 2012-2013 years. Web of Science [2] was investigated and as a result the table-1 given below for universities running in Azerbaijan was made.

425


Table 1. Counts of paper published in WOS for 2012-2013 UNIVERSITY

2012

2013

139

118

257

1

AZERBAIJAN TECHNICAL UNIVERSITY

24

18

42

2

AZERBAIJAN STATE OIL ACADEMY

23

10

33

3

AZERBAIJAN MEDICAL UNIVERSITY

13

7

20

4

8

4

12

5

10

2

12

5

KHAZAR UNIVERSITY

4

1

5

6

NAKHCHIVAN STATE UNIVERSITY

2

2

4

7

GANJA STATE UNIVERSITY

0

3

3

8

AZERBAIJAN STATE AGRARIAN UNIVERSITY

2

0

2

9

AZERBAIJAN TEACHER UNIVERSITY

1

1

2

9

BAKU STATE UNIVERSITY

AZERBAIJAN UNIVERSITY SUMGAIT STATE UNIVERSITY

SUMMARY

RANK

If based on information [3] provided by the science division of the Ministry of Education of the Azerbaijan Republic (ME of the AR) for a two year period between 2012 and 2013 the higher education institutions of Azerbaijan represented at WOS are given as below: BSU-300 articles, ATU- 92 articles and Nakhchevan Teachers’ Institute – 8 articles. Information on remaining universities was not provided to the science division of the ME of the AR. Certain incongruity of exaggeration for the favor of the universities exist between information that is given and information provided by WOS. By reconsidering the number of academic staff in universities [3] we can measure the annual sceintific productivity. Table 2. Scientific productivity of universities’s staff for 2012-2013 RANK UNIVERSITY 1

BAKU STATE UNIVERSITY

2 3 4 5 6 7 8 9

SUMMARY

STAFF

paper per capita for year

257

1 397,0

0,0920

AZERBAIJAN TECHNICAL UNIVERSITY

42

594,5

0,0353

AZERBAIJAN STATE OIL ACADEMY

33

674,0

0,0245

AZERBAIJAN UNIVERSITY

12

354,5

0,0169

AZERBAIJAN MEDICAL UNIVERSITY

20

1 219,5

0,0082

AZERBAIJAN TEACHERS UNIVERSITY

2

143,0

0,0070

NAKHCHIVAN STATE UNIVERSITY

4

469,0

0,0043

GANJA STATE UNIVERSITY

3

563,5

0,0027

AZERBAIJAN STATE AGRARIAN UNIVERSITY

2

532,0

0,0019

As seen by table-2 only 9 of the 52 universities have published articles in indexed (IF) journals and thus only those 9 universities were measured for their scientific productivity. This international indicator is measured as zero for all the remaining 43 universities. Let’s note that this 426


a situation specific for countries leaving through their transition stage. Yet, how to differentiate “international zeros?” In this case calculations should be made on not one but several indicators. For instance, another 9 indicators can be added as suggested in [4] and their weight coefficient is shown in table -3 based on expert-statistical approach. Table 3. WEIGHTING COEFFICIENTS OF MAIN FACTORS № 1 2 3 4 5 6 7 8 9 10

INDICATOR Monograph (stamped) Paper in journal with IF International patent Tutorial (stamped) Textbook (stamped) Paper in journal recommended Supreme Attestation Commission of CIS Paper in journal recommended Supreme Attestation Commission of AR Patent of AR Thesis of International Conference Thesis of Republic Conference

WEIGHTING COEFFICIENTS 69,23 59,62 52,31 36,92 18,46 13,46 6,08 3,62 2,03 1,01

Three additional indicators on universities running in Azerbaijan were aggregated into one when collected by the science division of the ME of the AR. In other words, a monograph (with stamp), an international patent, and a copyright certificate was taken as one indicator for 22 universities which exist and can be accessed. This is the reason that we took the personal weight coefficient for this conditionally aggregated indicator as 69.23 in order to measure the scientific productivity in these universities. Other indicators that were used are shown as bold writings in the table-3. The table below provides us with the calculation of a professors’ annual scientific productivity.

1

Table 4. Scientific Productivity per Capita for one year (table was formed by Sardar Shabanov) Main Staff+ Total Scientific Productivity per University 1/2 Half Stuff Score capita for one year 26901,98 Azerbaijan State Agrarian University 532,0 25,28

2

Baku State University

Rank

3

1 397,0

68550,47

24,53

84,5

4120,01

24,38

22564,31

18,09

Nahchivan Institute of Teachers

4

Baku Slavic University

623,5

5

Academy of Public Administration

125,0

4218,52

16,87

469,0

14876,42

15,86

594,5

15082,75

12,69

6 7 8 9 10 11

Nakhchivan State University Azerbaijan Technical University Azerbaijan State university of Languages Azerbaijan State Pedagogy University Science and Culture Centre Tefekkur University Azerbaijan State Technology University

18251,5 729,5

12,51 15099,17

644,0

11,72 2912,19

159,5

9,13 4315,35

242,5

427

8,90


12 13

14165,18

Azerbaijan State university of Economics

899,0

7,88 2209,49

Azerbaijan Teachers University Academy of Labour and Social Relations

143,0

15

Lenkoran State University

106,0

1601,37

7,55

16

Western University

197,5

2411,57

6,11

235,5

2794,43

5,93

112,0

1320,16

5,89

563,5

5901,09

5,24

674,0

6885,39

5,11

354,5

3153,4

4,45

1 219,5

9493,11

3,89

14

17 18 19 20 21 22

7,73

1277,32 83,0

Azerbaijan Institute of Turism Mingechevir Polytechnic Institute Ganja State University Azerbaijan State Oil Academy Azerbaijan University Azerbaijan Medical University

7,69

As seen by table-4, 13 universities with bold-italic signs were able to position themselves for labor productivity in a certain way due to artificially increased number of less valued indicators during 2012-2013 although these universities had not published any single article in IF journals. One can draw a conclusion that information submitted to the science division of the ME of the AR is not reliable. In order to obtain reliable results all statistics must be presented in a retrospectively recoverable format. For this, establishment of an information base registering scientific successes of all the educational institutions in Azerbaijan would be advised. Establishment and investigation of such an information base would lay the foundation for the scientific monitoring of universities in Azerbaijan. Keywords: scientific productivity, impact factor, rank. AMS Subject Classification: 62P20, 65C20, 97M40.

References 1. http://www.nsf.gov/statistics/seind12/pdf/seind12.pdf - National Science Board. 2012. Science and Engineering Indicators 2012. Arlington VA: National Science Foundation (NSB 12-01). 2. http://webofscience.com – the most prestigious indexed scientific data base 3. Ahmedov N.B. Universities Ranking of Azerbaijan Republic, Baku, Az.Tech.Uni., 2014. 113p. 4. Abbasov, A.M., Hajiev A.H., Shabanov S. A., Vision for scientific results of universities in Azerbaijan, App. of Inf. and Commun. Techn. (AICT), 7th Int. Conf., IEEE, 2013, pp.433-437.

428


SURVIVABLE NETWORK DESIGN PROBLEMS IN DISTRIBUTION SYSTEMS F.A. Sharifov1 M. Glushkov Institute of Cybernetics NAS of Ukraine, Kyiv, Ukraine e-mail: [email protected]

In any distribution systems, networks handle increasing traffic and include supporting critical services such as emergency response. The communication networks evolve towards all optical networks based on wavelength division multiplexing technology a simple fiber is capable of carrying hundreds of wavelength working in parallel each at 10Gb/s or higher data rate. To ensure flows rerouted after some link failures, sufficient excess capacity must be available on surviving links. On undirected graph 𝐺 = (𝑉, 𝐸), for a given nonnegative integers 𝑟𝑣 associated with each node 𝑣 ∈ 𝑉 and costs 𝑐𝑒 associated with an edge 𝑒 ∈ 𝐸, the first formulation of the survivable network design problem is to design a minimum cost network satisfying connectivity (survivability) requirements expressed by integers 𝑟𝑣 . The integer vector 𝑟 = (𝑟𝑣 ; 𝑣 ∈ 𝑉) is called the connectivity type vector.

Resulting network 𝐺∗ = (𝑉, 𝐸(𝐺∗ )) satisfies the node

survivability requirements if it contains at least 𝑟𝑣𝑤 = 𝑚𝑖𝑛{𝑟𝑣 , 𝑟𝑤 } (𝑟𝑣𝑤 = 𝑚𝑎𝑥{𝑟𝑣 , 𝑟𝑤 }) node disjoint path between each pair of distinct nodes 𝑣, 𝑤 ∈ 𝑉, where 𝐸(𝐺∗ ) ⊆ 𝐸. This case of the problem is called the minimum cost node- survivability network design problem. Similarly the network 𝐺∗ satisfies the edge survivability requirements if it contains at least 𝑟𝑣𝑤 edge disjoint path between each pair of distinct nodes 𝑣, 𝑤 ∈ 𝑉 [1]. These conditions ensure that some path between nodes 𝑣 and 𝑤 will survive a pre-specified level of link failures. This case of the problem is called the minimum cost edge- survivability network design problem, and we abbreviate it as SNDP. For many cases of the graph 𝐺 and costs 𝑐𝑒 , it seems that the latter problem cannot be reduced to the former since it considered to be harder than the first. Note that the both above mentioned minimum cost survivability network design problems are 𝑁𝑃- hard [1]. When the connectivity vector 𝑟 such that 𝑟𝑣 = 𝑘 for all 𝑣 ∈ 𝑉 and 𝑘 is some integer, the SNDP is called 𝑘 -connected network design problem.

For solving the SNDP different heuristics have been

developed by many authors. A natural basic models of different variants of SNDP as integer linear programming (LP) formulations are based on the well known Menger's Theorem. These models include exponential number of linear equations or inequalities. Main stream of theoretical research directed towards

429


obtaining exact solution to these problems relies on the so-called branch and cut approach [1], [2]. More precisely, starting an integer or mixed integer LP formulation, valid inequalities and/or facet defining inequalities are identified for strengthening the initial formulation, and possibly those corresponding problems generated in branch and bound methods process. However from the view of practice, to get an approximate solution to these integer linear programming problems, one needs to solve appropriate separable problems associated with the inequality constraints. Then any separable problem solver combined by a linear programming problem solver can be used to get near optimal feasible solution. Usually, in above mentioned problems, a solution to the separable problem, more exactly, for a fixed values of variables, either to find a violated inequality (if such exist) or to be confirmed that they satisfy all constraints, is finding by the effective minimum cut algorithms. By this approach some practice problems have been solved (see [2]). The general survivable network design problem SNDP(𝐻)

introduced in [3], with

survivability requirements expressed in terms of edges and nodes failures of any subgraph (say 𝐺0 ) of initial network 𝐺, which is isomorphic to another given graph 𝐻 = (𝑉(𝐻), 𝐸(𝐻)), that is 𝐺0 and 𝐻 are isomorphic. It can be shown that SNDP is a particular case of the SNDP(𝐻). The flow models of SNDP(𝐻)) is presented in [3],[4] have many applications not only in telecommunication systems, but also in studying crystal structures. For the case of the graph 𝐻 when it is a single edge, that is |𝑉(𝐻)| = 2 and 𝐸(𝐻) = 1, SNDP(𝐻) is the two connected Steiner problem having many applications in communication systems. Effective algorithms to compute lower and upper bounds for the optimal value of the objective function for this case of SNDP(𝐻) have been proposed in [3]. To get an exact algorithm, these algorithms integrated into branch and bound framework. All algorithms coded in C and a number of real -wolrd problems are solved by this algorithm. The case of SNDP(𝐻), when 𝐻 is two non-adjacent links [4], the problem has strong interest in the field of network connectivity and commute detection and studying crystal structures [4], [5]. In [6], the authors consider the packet recovery problem from dual-links failures in Internet Protocol (IP) networks, and present an integer linear program for the minimum monitoring cost problem for fast two link failures localization in a given optical network. Many real-world networks, U.S. National Network, for example, have topology satisfying such type survivability requirements that are a feasible solution to the problem. In addition to the communication network, this type of topologies arise in lattices of different crystal structures [5], where nodes represent atoms and each atom is connected to its neighbor atoms by an edge. When 𝐻 is the 2-edge 430


matching graph, besides to 3- edge connected graphs, the set of feasible solutions to the SNDP(𝐻) contains also near triangulated graphs. This may be motivated future research on this case of SNDP(𝐻) since one would use the mathematical formulation to clarify some deep results about the graph triangulation problems. Other special cases of the SNDP(𝐻) introduced in the [3], where some interesting theoretical results were proved for different cases of the graph 𝐻. Another new class of network design problem, namely when a given cut function, is to design network in which values of the function are capacities of cuts in resulting network. The problem which is a generalization of this special case has many applications in transportation, telecommunication systems and logistics. For example, in [7] the authors show that the problem of determining the capacity of airports relative to the corresponding structure of the route network and the forecasted passenger traffic between the airports can be reduced to another case of the general problem. Despite of 𝑁𝑃 hardness of SNDP(𝐻), special cases of SNDP(𝐻) which can be solved in polynomial time depending on some special connectivity type vector. Keywords: survivability, network, design, cut, isomorphic. AMS Subject Classification: 90B10, 90C10, 90C27, 68M10.

References 1. Kerivin H., Mahjoub A.R., On Survivable Network Polyhedra, Discrete Mathematics, 290, 2005, pp.183-210. 2. Gr"otschel M., Monma C.L., Stoer M. Polyhedral and computational investigations for designing communication networks with high survivability requirements, Operations Research, 43, 1995, pp.1012-1024. 3. Shor N. Z., Sharifov F.A. General problem of synthesis reliability network, Problems of information and cybernetics, 2, 3, 2006. pp.126-131 (in Russian). 4. Sharifov F.A., Kutucu H., A Network Design Problem with Two-Edge Matching Failures, RAIRO-Oper.Res. 49, 2, 2015, pp.297-312. 5. Wang L.W., Zhang L., Lu K., Vacancy-decomposition -induced lattice instability and its correlation with the kinetic stability limit of crystal, Philosophical Magazine Letters, 5, 2005, pp.213-219. 6. Wu B., Yeung K.l., Ho P.H. Monitoring cycle Design for fast link Failure location in all-optical networks. Journal of lightwave technology, 9, 2009, pp.1392-1401. 7. KristinaV.M., Sharifov F.A., Yun G.N. A problem of airport capacity definition, Aeronautica, Issue 5, 2013, pp.1-13 431


RELATIONSHIP BETWEEN COMPOSITE PIEZOELECTRIC PROPERTIES AND CRYSTAL CHEMICAL PARAMETERS OF PIEZOFILLER AND POLYMER MATRIX F.N. Tatardar1, M.A. Kurbanov1, Z.A.Dadashev1 1

Institute of Physics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan e-mail: [email protected]

Relationship between component electronegativities of polymer – piezoelectric compositional system and d33 piezomodule is found. An improvement mechanism of component piezoelectric properties with matrix and piezoparticle electronegativity increase is proposed. The basic reason of piezoelectric factor (dij, gij) buildip is shown to be the modification of the composite charge state which results in growth of stabilized charge values at the interface during composite electrothermopolarization. Keywords: composite, piezoelectric, polimer matrix. AMS Subject Classification: 7784LF.

References 1.

Mamedov G.A., Panich A.E., Gurbanov M.A., Sultanakhmedov I.S., Mekhtili A.A., Yahyayev F.F., Tatardar F.N., Piezoelectric composites with high stability of piezomodul to the influence of the mechanical and temperature fields, Physics of solids, 2010, Vol.52, No.6, pp.1067–1074.

MULTIFUNCTIONAL MOBILE ROBOT M. Tatur1, M. Zhartybayeva2, K. Iskakov2, T. Babayev3 A. Pashayev3, E.Sabziev4 1

Belorussian State University of Informatics and Radioelectronics, Minsk, Belarus 2 L.N. Gulilyov Eurasian National University, Astana, Kazakhstan 3 Institute of Control Systems of Azerbaijan National Academy of Sciences, Baku, Azerbaijan 4 Kiber Ltd Company, Baku, Azerbaijan e-mail: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]

Development and construction of mobile robotic systems develops in two directions: the first is based on the creation of unique (mechanized) platforms, the second - on the use of serial chassis or product as a whole. Examples of robotic systems on specialized chassis are: multi432


functional robots for security services QinetiQ (UK) [1], a mobile robot for fire extinguishing (Russian) [2]. Majority of autoconcerns create their own robotic concept car that can drive without a driver on the roads and even "keep" rules of the road. [3] As a rule, such machines, robots are created for testing new technologies, and separate options and then implants into the serial cars. For example, the automatic transmission already has entered in everyday life, cruise control, direction, engine start-stop, parktronic, recognition of road signs, prevention of frontal impact with an obstacle, etc., Which are integral functions of the integrated management system of the mobile robot. However, there are no domestic mobile robotic systems and small-scale serial production, available for mass use in agriculture, utilities and / or law enforcement agencies yet. In the present work announces an international project on creation of multi-functional mobile robot based on the chassis serial domestic mini-tractors and introduces the concept of controlling such a robot. The project was initiated in 2013 LLC "Intellectual processors" (Belarus) and included in the program of innovative development of the Republic of Belarus. In 2014 the project was joined to the Eurasian National University named after L.N. Gumilev (Astana), supported by Ministry of Education and Science of the Republic of Kazakhstan. In 2015, organizations of Azerbaijan (Kiber Ltd Co. and Institute of Control Systems of ANASciences) were joined to this project. Created products is positioned as a multi-robotic system for use in environments connected with the risk to the health and life of the driver and staff, for example, in emergency situations - in case of liquidation threat of explosion, poisoning, at suppression of fires; in agriculture - when spraying pesticides and other fields. Specificity of targeted use of robotic system will be defined by the specially established attachments, and appropriate software. The peculiarity of the project is to maximize the use of mechanical components for serial production, which would reduce the cost of the final product. As one of the variants of the mobile platform the chassis of mini tractor "Belarus-132" with petrol engine (HONDA GX390) was used. The power and pulling force, originally calculated on the plowing the soil, it is enough to ensure that the movement of the trolley up to 500 kg, or clearing blockages with regular attachments. Articulated frame chassis of mini tractors "Belarus -132N" provides exceptional maneuverability of the robot with the smallest turning radius - 2,5 meters, which can be a key factor in the application of complex in a crowded urban traffic streets, in the woods, tunnels and even in large rooms. The weight of robotic system is about 400 kg with the size of 120 × 120 × 180 cm. Physical Specifications allow to transport it in the cargo minibus with a medium wheelbase or on usual single-axle trailer. 3D model of the robotic system shown in Figure 1. 433


Fig.1 General view of the robotic system based on the chassis "Belarus 132"

In developing the concept of mobile robot necessary to solve several of fundamental questions, including: to define a strategy (basic methods) positioning and navigation of mobile robot; add limitations and detail control modes, particularly semiautomatic and / or automatic; identify a list of basic algorithms and link management regimes; realization of of algorithms by distributed computing modules. The robotic complex based on the following driving modes: remote control (operator) via radio channel at a distance of 500 meters and a wired (technology) channel - to 10 meters; control of semi-autonomous navigation through the on-board video. In the present project identified three main levels of computing modules: an external computer (laptop); on-board computer; multi-channel controller executive devices (mechatronics) and in addition, a remote control, which can be considered as an additional layer (module). For each module specification of functions were defined, commands to initialize them, the service (information) data and inter-module interfaces. A simplified scheme of system control, distributed by constructional computing modules is shown in Figure 2.

Fig.2. A block scheme of management system of the mobile robot

434


By results of already completed stages of the project cost of mobile robot is expected within 20-50 thousand dollars, depending on optional equipment and attachments. This becomes possible by maximizing the use of domestically produced components (both standard and custom), as well as applying the concept of multilevel governance. Keywords: mobile robot, a multi-level management system, autopilot, video processor. AMS Subject Classification: 03B52, 93C42, 93C85, 68T40. References 1. QinetiQ [electronic resource]. - Access: www.qinetiq.com. Date of access 04/27/2014. 2. Robot firefighter on combat duty atonement in the Trans-Baikal region [electronic resource] Ministry of Emergency Situations of Russia - Access: http://www.mchs.gov.ru/dop/info/smi/news/item/3824166/. Date of access 04/27/2014. 3. Unmanned vehicles [electronic resource] / Access http://avto-mir.info/bespilotnye-avtomobili. Date of access 05/08/2015.

DESIGN OF A SERVICE ORIENTED ARCHITECTURE FOR EFFICIENT RESOURCE ALLOCATION IN MASS CUSTOMIZATION Reza Vatankhah1, Ali Vatankhah1 1

Department of Mechanical Engineering, Eastern Mediterranean University, Famagusta, Cyprus e-mail: [email protected]

Mass customization relies to the company’s ability for offering products and services with characteristics merged with consumers’ personal requirements using flexible manufacturing system [1-3]. Mass customization has been followed up by many companies as a competitive strategy for increasing customs satisfaction [4,6]. Efficient resource allocation in a manufacturing system working under mass customization philosophy is vital for meeting consumers’ expectations in terms of deadlines and product features, and offer high scalability for the company in high working seasons. A service-oriented architecture is presented here aiming to address the resource allocation issues of the manufacturing enterprises working under mass customized philosophy [7,8]. A number of scheduling algorithms and advance reservation systems is incorporated to ensure efficient resource usage in the manufacturing system. An example of resource allocation process for a manufacturing system holds five stations is simulated for number of customized

435


product workflow scenarios and the most appropriate algorithms for offline and online scheduling are introduced. Keywords: flexible manufacturing system, resource allocation process. AMS Subject Classification: 47N70.

References 1. Barenji R.V., Barenji A.V., Hashemipour M., A multi-agent RFID-enabled distributed control system for a flexible manufacturing shop, The International Journal of Advanced Manufacturing Technology, Vol.71, 2014, pp.1773-1791. 2. Wang C., Ghenniwa H., Shen W., Real time distributed shop floor scheduling using an agent-based service-oriented architecture,

International Journal of Production Research, 2008, Vol.46,

pp.2433-2452. 3. Chowdary B.V., Muthineni S., Selection of a flexible machining centre through a knowledge based expert system, Global Journal of Flexible Systems Management, Vol.13, 2012, pp. 3-10. 4. Zhu J., Li X., Shen W., A divide and conquer-based greedy search for two-machine no-wait job shop problems with makespan minimisation, International Journal of Production Research, Vol. 50, 2012, pp.2692-2704. 5. Zhang W., Zhang S., Cai M., Huang J., A new manufacturing resource allocation method for supply chain optimization using extended genetic algorithm, The International Journal of Advanced Manufacturing Technology, Vol.53, 2011, pp.1247-1260. 6. Kahraman C., Beskese A., Kaya I., Selection among ERP outsourcing alternatives using a fuzzy multi-criteria decision making methodology, International Journal of Production Research, Vol.48, 2010, pp.547-566. 7. Zeballos L., Quiroga O., Henning G. P., A constraint programming model for the scheduling of flexible manufacturing systems with machine and tool limitations, Engineering Applications of Artificial Intelligence, Vol.23, 2010, pp.229-248. 8. Kaufmann M., Wagner D., Drawing graphs: methods and models, Vol.2025: Springer Science & Business Media, 2001.

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FUZZY-BASED ANALYSIS OF SOCIAL NETWORKS L.A. Zadeh1, A.M. Abbasov2, Sh. N. Shahbazova3 1

Department of Electrical Engineering and Computer Sciences, University of California, USA 2 Ministry of Communications and High Technologies of the Republic of Azerbaijan 3 Department of Information Technology&Programming, Azerbaijan Technical University e-mail: [email protected], [email protected], [email protected]

Social networks have gained a lot attention. They are perceived as a vast source of information about their users. Variety of different methods and techniques has been proposed to analyze these networks in order to extract valuable information about the users – things they do and like/dislike. A lot of effort is put into improvement of analytical methods in order to grasp a more accurate and detailed image of users. The paper presents a short survey of works that use fuzzy-based technologies for analysis of social networks. It also contains a few target areas of social network analysis that could benefit from applications of fuzzy sets and systems methods. 1. Introduction. In the light of these statements, we would like to state that techniques of processing social networks of users and groups should reflect such impression. These techniques should be based on a human-like methodology, and the theory of fuzzy sets and systems is highly suitable for such a purpose. Its ability to deal with ambiguous data and facts, its ability to describe things in a human manner, and its ability to handle imprecision and ambiguity make fuzzy sets one of the best tools for analysis of social network. In this paper we provide an overview of a number of known examples how fuzzy-based methods and approaches are utilized to analyze social data. This overview is far form exhaustive, but represents a glimpse on exceptional abilities of fuzzy sets and systems to provide a new dimension in data analysis. 2. Structure-based analysis of

social networks. Let us start, before focusing on

application of fuzziness to network analysis, with a formal definition of a social network. In general, a social network is composed of actors and relations (nodes and edges in terms of graph theory) [1]. A social network (SN) can be defined as: 𝑆𝑁 = (𝐴, 𝑅)

(1)

where A is a set of actors, and R is a relation built on Am, i.e., it is a m-ary relation on the domain A. In literature, SNs are considered as networks with binary relations: 𝑆𝑁 = (𝐴, 𝑅𝑏 ⊆ 𝐴 × 𝐴)

437

(2)


where Rb represents a binary relation. Such a relation on A can be represented as its characteristic function – called adjacency matrix: 𝑅𝑏 : 𝐴 × 𝐴 → {0,1}

(3)

This matrix is a very popular tool used for analysis of SNs. It contains information about connections that exist between actors. Such a matrix indicates if there are binary relations between actors or not. However, in a more realistic view such information does not fully reflect the reality– the connections between individuals have different levels of strength. Therefore, it seems the ability of fuzzy sets to represent a degree of relation between individual changes the depth of analysis and brings new more realistic results. In such a case, a binary fuzzy relation can be perceived as a generalization of a binary relation Rb. Its membership function can be: 𝜇(𝑅𝑏 ) = 𝜇𝑏 : 𝐴 × 𝐴 → [0,1]

(4)

A SN with such generalized relations is called a fuzzy social network. Such networks are considered in the rest of the paper. 3. M-ary Relations between individuals. Analysis of social relations between actors is one of the most significant and most obvious ways of analyzing SNs [2]. It is a portion of a broader discipline of network analysis that focuses on studying relationships between objects that belonging to the network. The authors of [1] and [3] investigate benefits of dealing with a fuzzy version of adjacency matrix, Eq. (4). In such a case, intensity of relations can be taken into account. The authors have proposed usage of fuzzy theory to build a fuzzy adjacency matrix. The advantages of such an approach are multifold: -

ability to express realistic levels of intensity of relationships;

-

extension of existing techniques for analyzing the adjacency matrix;

-

application of measures/indices specific for fuzzy sets, for example a measure of fuzziness;

-

finding more elusive information about relations between individuals, especially when it is very difficult to establish if they exist.

The work extends the notion of an adjacency matrix, and proposes usage of m-ary fuzzy relations. Such relations represent social relationships among m individuals when a group of m individuals is taken into account. They define an m-ary fuzzy membership: 𝜇: 𝐴𝑚 → [0,1] that characterizes the relations in the following way:

438

(5)


1 𝑖𝑓 𝑎1 , 𝑎2 , … , 𝑎𝑚 𝑎𝑟𝑒 𝑟𝑒𝑙𝑎𝑡𝑒𝑑 𝑡𝑜 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟 𝜇(𝑎1 , 𝑎2 , … , 𝑎𝑚 ) = {(0,1)𝑖𝑓 𝑎1 , 𝑎2 , … , 𝑎𝑚 𝑎𝑟𝑒 𝑟𝑒𝑙𝑎𝑡𝑒𝑑 𝑡𝑜 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟 𝑡𝑜 𝑠𝑜𝑚𝑒 𝑒𝑥𝑡𝑒𝑛𝑡 0 𝑖𝑓 𝑎1 , 𝑎2 , … , 𝑎𝑚 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑟𝑒𝑙𝑎𝑡𝑒𝑑 𝑡𝑜 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟

(7)

They define such a relation as an aggregation of binary fuzzy relations between pairs of individuals from the group m. 𝜇(𝑎1 , 𝑎2 , … , 𝑎𝑚 ) = 𝐴𝐺𝑅(𝜇𝑏 (𝑎1 , 𝑎2 ), 𝜇𝑏 (𝑎1 , 𝑎3 ), … , 𝜇𝑏 (𝑎𝑚−1 , 𝑎𝑚 ))

(8)

In order to estimate m-ary fuzzy relations they have adopted a well-known OWA operator [4,5]. The authors conclude that OWA operator is a valuable tool in estimating m-ary relations. Keywords: fuzziness to network analysis, aggregation of binary fuzzy relations, fuzzy adjacency matrix. AMS Subject Classification: 94D05, 94D99.

References 1. Hannemanand R., Riddle M., Introduction to Social Network Methods, University of California, Riverside, 2005. 2. Scott A.J., Social Network Analysis. A Handbook, London, Sage, 2000. 3. Kwakernaak H., Sivan R., Linear Optimal Control Systems, Wiley, Interscience, New York, 1972, 650 p. 4. Brunelli M., Fedrizzi M., Fedrizzi M., Fuzzy m-ary adjacency relations in social network analysis: Optimization and consensus evaluation, Information Fusion, Vol.17, pp.36-45, 2014. 5. Yager R.R., Ordered weighted averaging operators in multi-criteria decision making, IEEE Trans on Systems, Man and Cybernetics, Vol.18, 1988, pp.183-190. 6. Yager R.R., Kacprzyk J., The Ordered Weighted Averaging Operators: Theory and Application, Kluwer Academic Publisher, Boston, 1997.

NONLINEAR PREDICTIVE CONTROL TELEOPERATION FOR EAMA：EAST ARTICULATED MAINTENANCE ARM Jing Wu12, Huapeng Wu2, Yuntao Song1, Yong Cheng1, Tao Zhang1, Zhonghui Yang1 1

Institute of Plasma Physics Chinese Academy of Sciences, 350 Shushanhu Rd Hefei Anhui, China 2 Lappeenranta University of Technology, Skinnarilankatu 34 Lappeenranta, Finland e-mail: [email protected]

EAMA (EAST Articulated Maintenance Arm) is an articulated serial robot arm, works in experimental advanced superconductor to kamak for inspection and maintenance. Sometimes it 439


works like a telemanipulator with motion control device. A graphic interface is used on which EAMA is superimposed on the realoperating system in the scene of the remote site. In grasping application, the operator has only visual information about vacuum vessel and the task execution. The task in this paper presents the application of model predictive control (MPC) techniques to EAMA teleoperation system with time delays and proposes a modified MPC scheme with unscented Kalman filter which guarantees the stability in the presence of dynamic uncertainty. By exploiting a model of the remote environment and slave device and programming telesensors to help control of the task execution predictive display. Experiment results is graphically simulated in real-time, presented to illustrate the performance and effect of the algorithm in compensating time-delay for teleoperation. Keywords: nonlinear model, predictive control, unscented Kalman filter, state estimation, teleoperation, time-delay. AMS Subject Classification: 68T40.

440

AUTHORS INDEX

Abazari N. 31 Abbasov A.M. 437 Abbasov E.M. 283 Abbasov M.E. 32 Abbasova A.Kh. 302 Abdel-Aty M. 21 Abdullaev V.M. 227 Abdullayev A.Kh. 211 Abdullayev S.H. 388, 402 Abdullayeva N.G. 33 Abuassba K. 51 Acikgoz H. 391 Adilov G. 377 Afanasyev A.P. 285 Agamalieva L.F. 279 Agamaliyeva A.I. 34, 134 Agayeva N.A. 283 Aghayeva Ch. 35 Ahmadkhanlu A. 286 Ahmadova Zh.B. 134 Ahmedov M. A. 184 Aida-zade K.R. 36, 183, 227, 230, 286 Akbarimajd A. 38 Akhmedov A.M. 288 Akhmedov N.B. 377 Akhundov A.A. 385 Akhundov H.S. 160 Akhundova E.M. 385 Akin Ö. 186, 190 Akturk T. 396 Alguliyev R. 299 Aliev F.A. 41, 179, 233, 237, 240, 242, 293 Aliev N.A. 43, 233, 234, 290, 295 Aliev S.A. 414 Alieva H.H. 128 Aliguliyev R. 299 Aliyev А.М. 234 Aliyev I.M. 193 Aliyev I.Z. 126 Aliyev J.M. 245 Aliyev K. 327 Aliyev M.I. 193 Aliyev R.M. 380

Aliyeva A. 73, 75 Aliyeva A.J. 245 Aliyeva A.M. 193 Aliyeva G. 301 Aliyeva S.T. 134 Aliyeva V.D. 263, 266 Allahverdiyeva N.A. 73, 75 Amikishiyev V. 382 Amirova L.I. 75 Amrahov Ş.E. 197 Asadova J.A. 46 Asadova O.H. 290, 302 Ashpazzadeh E. 249, 271 Ashrafova Y.R. 36, 252 Asil V. 310 Askerov I.M. 49 Askerov T.M. 263 Atayev G.N. 350 Aydemir K. 412 Azizbayov E.I. 118 Babanly V.Yu. 234 Babayev R.M. 305 Babayev T. 432 Babayeva U.R. 330 Badreddine B. 53 Baxishov V.M. 193 Baigereyev D. 307 Bakkouche A. 253 Bashirov A.E. 51 Baskonus H.M. 394 Baş S. 310 Başar T. 21 Bayeg S. 190 Bayramova N.V. 311 Beiranvand A. 210 Bektaş Ç.A. 61 Berberler M.E. 356 Bidyuk P.I. 365, 388 Boudjehem Dj. 53 Bougheda H. 391 Boyd S. 22 Bulut H. 394, 396 Cavadov I. 327 441

Chelabi M. 391 Cheng Y. 439 Çalkavur S. 400 Çelik E. 422 Çiçek G. 56 Çopuroğlu E. 346 Dadashev Z.A. 432 Damirova J.R. 398 Demir S. 58 Dzhalladova I.A. 118 Ebadi S. 81, 316 Emelyanov S.V. 285 Ercan A. 63 Ercan S. 61 Erkan K. 374 Eshkuvatov Z.K. 257 Evtushenko Y. 22 Ezzati R. 196 Fallahi S. 38 Faracova Sh.A. 276 Farhadova A.D. 414, 417 Farsadi M. 169 Fatholahzadeh A. 400 Fatullayev A.G. 197 Feyziev F.G. 65 Floudas Ch.A. 25 Gabasov R. 67 Gadirova S.Sh. 70 Gamzaev Kh.M. 320 Garayeva E.A. 72 Gasanov A.S. 365, 388, 402 Gasanov K.K. 233 Gasilov N.A. 197 Gasimov B.M. 211 Gasimov G.G. 257, 322 Gasimov Y.S. 73, 75, 237 Gasimova K.G. 240, 279 Ghannadiasl A. 78, 81, 316 Ghasemi R. 83 Gokpinar F. 190 Guliev A.P. 233 Guliyev H.F. 84, 86, 89, 91 Guliyev M.F. 293 Guliyev S.Z. 230, 324 Gurefe Y. 396

Gursoy A. 362 Guseynov S.I. 94 Hacib T. 391 Hajiyev N.D. 223 Haqverdiyev A.A. 342 Hameed H.H. 257 Hamidov S.I. 259 Hasanaliyev V. 324 Hasanov A.B. 330 Hasanov K.K. 97, 100 Hasanova G.R. 179 Hasanova L.K. 100 Hasanova Sh.A. 330 Hashimov T.E. 417 Heris F.N. 169 Humbatova A. 377 Huseynov A.F. 335 Huseynov A.H. 350 Huseynov S.T. 333 Huseynova Kh.T. 97 Huseynova N.Sh. 103 Huseynzade S.O. 322 Huseynzade Sh.S. 184 Huseynzadeh G.A. 106 Ibrahimov V.R. 266 Imanov K.S. 198, 202 Imanov M.H. 107 Imanova M.N. 263 Imanova U.Z. 132 Imran M. 145 Irandoust-Pakchin S. 262 Isaeva E.A. 193 Iskakov K. 432 Iskenderov A.D. 108 Ismailov N.A. 49, 237, 240 Ismailova G.G. 84 Ismayilov E. 208 Ismayilov S. 377 Ismayilova N. 299 Ivaz K. 210 Jabbarova A.Ya. 111 Jafarov A. 405 Jahanshahi M. 339 Jamalbayov M.A. 293 Kacprzyk J. 26 Kao Ch. 314 442

Karchevsky A.L. 113 Kasimbeyli R. 115 Kazimov H.K. 302 Kerimova S.R. 94 Khaniyev T. 190 Khankishiev Z.F. 339 Khodabin M. 262 Khusainov D.Y. 118 Kirillova F.M. 67 Kizilates G. 150 Konstantinov M.M. 112 Korpinar T. 310 Kurbanov M.A. 432 Kurt M. 362 Kutucu H. 158, 362 Quliyeva A.M. 266 Lakestani M. 249, 271 Larin V.B. 41 Le Bihan Y. 391 Maharramov A.M. 214 Mahdavi I. 353 Mahmudov E.N. 56, 58 Mahmudov E.S. 342 Mahmudov N.I. 131 Mamadov A.V. 28 Mamadzadeh R.B. 28 Mamedhasanov E.H. 295, 311 Mamedov A.C. 126 Mamedov B.A. 346 Mamedov I.G. 123 Mamedova J. J. 160 Mamedova N.Q. 290 Mamedova Y.V. 347 Mammadov A.C. 128 Mammadov J.F. 350 Mammadov K.Sh. 142 Mammadov R.S. 141 Mammadova A.H. 142 Mammadova G.H. 65 Mansimov K.B. 33, 34, 72, 134, 135, 137 Mansouri N. 253 Mardanov M.J. 27, 121 Mastaliyev R.O. 135 Mehbaliyev M.M. 408 Mehdiyev A.A. 89

Mehdiyev T. 382 Mehdiyeva G.Yu. 263, 266 Melikov A. 138 Melikov T.K. 121 Mirzayev F.A. 223 Mirzeyeva K.A. 330 Mirzoyev S.S. 269 Mohammadalizadeh P. 169 Mohammadzadeh R. 249, 271 Molani M. 353 Mukhtarov O.Sh. 412 Mukhtarova N.S. 242 Murga N.A. 402 Murshudov G.N. 382 Murtuzaliyev Т.F. 221 Murtuzayeva A.A. 211 Mustafaev V.A. 184 Mustafayeva Y.Y. 354 Mutallimov M.M. 103, 132, 211, 295, 311 Nabi-zade F.N. 65 Namazov A.A. 237, 240 Namazov M.A. 295 Nasiboglu R. 359 Nasiboğlu E. 356 Nasibov E. 359 Nasibov V.Kh. 335 Nasir H.M. 145 Nasiyyati М.М. 149 Nik Long N.M.A. 257 Nuriyev U. 362 Nuriyeva F. 150 Nusretli L.M. 412 Odabas Berberler Z.N. 356 Orucova M.Sh. 103 Ozbay H. 153 Özkan M. 422 Panakhov E. 63, 154 Pashayev A. 432 Petkov P.H. 112 Pogorilyy S.D. 155 Polyak B. 158 Posypkin M. 22 Potebnia A.V. 155 Prosyankina-Zharova T. 365 Rahimov A.B. 286 Rahimov F.H. 414, 417 443

Rajabov M.F. 237, 242 Ramazanov A.B. 274 Ramazanova A.T. 86 Ramazanova N. 377 Rasulova Sh.M. 137 Rustamov A. 138 Rustamov R. 419 Sabaz F. 158 Sabziev E. 432 Sadatrasoul S.M. 196 Sadikhov Z.A. 350 Sadygov M.A. 160, 163 Safarova N.A. 242, 276 Salavatov T.Sh. 28 Salimov S.M. 214 Samet R. 422 Senturk S. 217 Seyedhashemi Dijvejin M. 78 Seyfullayeva Kh.I. 91 Shabanov S.A. 425 Shabiyev J. 382 Shafizadeh E.R. 179, 221 Shahbazova Sh. N. 437 Shahir F.M. 169 Shamilov A. 217 Sharifov F.A. 429 Sharifov Y.A. 368 Shikhlinskaya R.Y. 221, 223 Shirazi B. 353 Solé P. 29 Song Y. 439 Szczepanik E. 168 Şengönül E. 422 Tagiev R.M. 43 Tagıyev H.T. 89 Tagiyev M.M. 311 Tagiyev R.K. 108 Taib B.M. 257 Talibov S.G. 183 Tatardar F.N. 432 Tatur M. 432 Temirbekov N.М. 370 Terentiev O. 365 Teymurov R.A. 169 Tezel B.T. 359 Tret'yakov A. 168

Trofymchuk O.M. 388 Tural S. 422 Turarov A.K. 370 Turksen B. 190 Ulusoy I. 154 Varnamkhasti M.J. 225 Vatankhah A. 435 Vatankhah R. 435 Velieva N.I. 276, 279 Volkova Y.V. 320 Wu H. 439 Wu J. 439 Yagubov A.A. 172 Yagubov М.А. 172 Yalçın B.C. 374 Yang Zh. 439 Yayli Y. 31 Yesilce I. 377 Yilmaz N. 217 Yusifbeyli N.A. 335 Yusifov B.M. 126, 128 Yusubov Sh.Sh. 175 Zadeh L.A. 437 Zarebnia M. 376 Zeynalov J. 179 Zhang T. 439 Zhartybayeva M. 432 Ziari Sh. 196 Zulfugarova R.T. 295

444