Keywords: singular, system, difference equations, linear, discrete time system. 1 Introduction. Many authors have studied generalised discrete & continuous time systems, see [1-27], and their ..... [34] Ogata, K: Discrete Time Control Systems.
A generalised linear system of difference equations with infinite many solutions Fernando Ortega1 , Nicholas Apostolopoulos2 1
2
Universitat Autonoma de Barcelona, Spain National Technical University of Athens, Greece
Abstract: In this article we study a class of generalised linear systems of difference equations with given non-consistent initial conditions and infinite many solutions. We take into consideration the case that the coefficients are square constant matrices with the leading coefficient singular. We provide optimal solutions and numerical examples to justify our theory. Keywords: singular, system, difference equations, linear, discrete time system.
1
Introduction
Many authors have studied generalised discrete & continuous time systems, see [1-27], and their applications, see [28-37]. Many of these results have already been extended to systems of differential & difference equations with fractional operators, see [38-47]. We consider the generalised discrete time system of the form F Yk+1 = GYk + Vk ,
k = 1, 2, ...,
(1)
and the known initial conditions (IC) Y0 . r×m
m
(2) r
Where F, G ∈ R , Yk ∈ R , and Vk ∈ R . The matrices F , G can be non-square (r 6= m) or square (r = m) with F singular (detF =0). Generalised linear systems of difference equations with given initial conditions don’t always guarantee to have unique solution. In the case where there exist solutions and they are infinite, we require optimal solutions for the system. The aim of this paper is to generalise existing results regarding the literature. An explicit and easily testable formula is derived of an optimal solution for the system.
2
Preliminaries
Throughout the paper we will use in several parts matrix pencil theory to establish our results. A matrix pencil is a family of matrices sF −G, parametrised by a complex number s, see [48, 49]. Definition 2.1. Given F, G ∈ Rr×m and an arbitrary s ∈ C, the matrix pencil sF − G is called:
1
1. Regular when r = m and det(sF − G) 6= 0; 2. Singular when r 6= m or r = m and det(sF − G) ≡ 0. In this article we consider the system (1) with a regular pencil, where the class of sF − G is characterised by a uniquely defined element, known as the Weierstrass canonical form, see [48, 49], specified by the complete set of invariants of sF − G. This is the set of elementary divisors of type (s − aj )pj , called finite elementary divisors, where aj is a finite eigenvalue of algebraic multiplicity pj (1 ≤ j ≤ ν), and the set of elementary divisors of type sˆq = s1q , called infinite Pν elementary divisors, where q is the algebraic multiplicity of the infinite eigenvalue. j=1 pj = p and p + q = m. From the regularity of sF − G, there exist non-singular matrices P , Q ∈ Rm×m such that Ip 0p,q PFQ = , 0q,p Hq (3) Jp 0p,q P GQ = . 0q,p Iq Jp , Hq are appropriate matrices with Hq a nilpotent matrix with index q∗ , Jp a Jordan matrix and p + q = m. With 0q,p we denote the zero matrix of q × p. The matrix Q can be written as Q = Qp Qq . (4) Qp ∈ Rm×p and Qq ∈ Rm×q . The matrix P can be written as P1 P = . P2
(5)
P1 ∈ Rp×r and P2 ∈ Rq×r . The following results have been proved. Theorem 2.1. (See [1-28]) We consider the systems (1) with a regular pencil. Then, its solution exists and for k ≥ 0, is given by the formula Yk = Qp Jpk C + QDk .
(6)
" P # k−1 k−i−1 J P V 1 i i=0 p P Where Dk = and C ∈ Rp is a constant vector. The matrices Qp , q∗ −1 − i=0 Hqi P2 Vk+i Qq , P1 , P2 , Jp , Hq are defined by (3), (4), (5). Definition 2.2. Consider the system (1) with known IC of type (2). Then the IC are called consistent if there exists a solution for the system (1) which satisfies the given conditions. Proposition 2.1. The IC (2) of system (1) are consistent if and only if Y0 ∈ colspanQp + QD0 .
2
Proposition 2.2. Consider the system (1) with given IC. Then the solution for the initial value problem (1)–(2) is unique if and only if the IC are consistent. Then, the unique solution is given by the formula Yk = Qp Jpk Z0p + QDk . " P # k−1 k−i−1 J P V 1 i p i=0 P where Dk = and Z0p is the unique solution of the algebraic system q∗ −1 − i=0 Hqi P2 Vk+i Y0 = Qp Z0p + D0 .
3
Non-consistent initial conditions
We begin this Section with some already existing results. Corollary 3.1. The initial conditions (2) of the system (1) for Vk = 0r,1 are consistent if and only if Y0 ∈ colspanQp . We can now state the following Theorem. Theorem 3.1. We consider the system (1) with Vk = 0r,1 and known non-consistent initial conditions (Y0 ∈ / colspanQp ) of type (2). For the case that the pencil sF − G is regular, after a perturbation to the non-consistent initial conditions (2) accordingly
min Y0 − Yˆ0 , 2
or, equivalently,
Y0 − Qp (Q∗p Qp )−1 Y0 , 2 an optimal solution of the initial value problem (1)–(2) is given by Yˆk = Qp Jpk (Q∗p Qp )−1 Q∗p Y0 .
(7)
The matrices Qp , Jp are given by by (3), (4), (5). Proof. From Theorem 2.1 the solution of the system (1) exists and is given by (6). Note that where Dk = 0m,1 , since it is assumed Vk = 0r,1 . For given initial conditions of type (2), C is the solution of the linear system Qp C = Y0 .
(8)
Since the matrix Q is non-singular, the matrix Qp has linear independent columns, i.e. rankQp = p. The system has m linear equations with p unknowns and m > p, i.e. the system is overdetermined. To sum up, Qp is a tall matrix (more rows than columns) with linearly independent columns. Thus, since Y0 ∈ / colspanQp and rankQp = p, the system ˆ (8) has no solutions. Let Y0 be a vector such that Yˆ0 ∈ colspanQp and let Cˆ be the unique
3
solution of the system Qp Cˆ = Yˆ0 . Hence we want to solve the following optimisation problem
2
min Y0 − Yˆ0 2 s.t. Qp Cˆ = Yˆ0 , or, equivalently,
2
min Y0 − Qp Cˆ . 2
Where Cˆ is the optimal solution, in terms of least squares, of the linear system (8). Thus we seek a solution Cˆ by minimising the functional
2
ˆ = D1 (C)
Y0 − Qp Cˆ . 2
ˆ gives Expanding D1 (C) ˆ = (Y0 − Qp C) ˆ ∗ (Y0 − Qp C), ˆ D1 (C) or, equivalently, ˆ = Y0∗ Y0 − 2Y0∗ Qp Cˆ + (C) ˆ ∗ Q∗p Qp C, ˆ D1 (C) ˆ ∗ Q∗p Y0 . Furthermore because Y0∗ Qp Cˆ = (C) ∂ ˆ = −2Q∗p Y0 + 2Q∗p Qp C. ˆ D1 (C) ˆ ∂C Setting the derivative to zero,
∂ ˆ ˆ D1 (C) ∂C
= 0, we get
Q∗p Qp Cˆ = Q∗p Y0 . Since rankQp = p, the matrix Q∗p Qp is invertible and the solution is given by Cˆ = (Q∗p Qp )−1 Q∗p Y0 . Note that the optimization methods used in this proof can also be applied in several other similar systems that appear in different areas of Applied Mathematics, see [50-59]. The proof is completed.
Numerical example Example 3.1. Assume the system (1) with non-consistent initial conditions of type (2) and Vk = 05,1 . Let 0 0 −1 0 1 0 −1 −1 1 1 F = −1 −1 1 1 0 , 0 1 2 0 −2 0 0 0 0 0
4
1 G= 5
−5 0 −11 −1 −2 −2 11 2 −5 0
and
8 5 −3 14 11 −8 2 2 0 −14 −11 8 10 5 −5
Y0 =
1 1 1 1 1
.
Then det(sF − G)=s(s − 51 )(s − 25 ) and the pencil is regular. The three finite eigenvalues (p=3) of the pencil are 0, 51 , 25 and the Jordan matrix Jp has the form 0 0 0 Jp = 0 15 0 . 0 0 25 By calculating the eigenvectors of the finite eigenvalues we get the matrix 1 0 0 0 1 1 Qp = 0 0 0 . 1 0 1 0 0 1 From Proposition 3.1, the initial conditions are non-consistent and thus from Theorem 3.1 and , the optimal solution of the initial value problem (1) – (2) is given by Yˆk = Qp Jpk (QTp Qp )−1 QTp Y0 . Where
and
2 1 (QTp Qp )−1 = 1 3 −1 2 QTp Y0 = 1 3
1 −1 5 −2 , 2 −2
2 1 (QTp Qp )−1 QTp Y0 = 1 . 3 2
Hence
Yˆk =
1 3 · 5k
5
0 3 0 4 4
.
Example 3.2. Assume the system (1) with initial conditions of type (2). Let 1 1 F = , 1 1 1 1 −2 G= 5 −2 0 and
Y0 =
2 3
.
Then det(sF − G)=s + 54 and the pencil is regular. The finite eigenvalue (p=1) of the pencil is − 54 and the Jordan matrix Jp has the form 4 Jp = − . 5 By calculating the eigenvectors of the finite eigenvalues we get the matrix 2 Qp = . 3 It is easy to observe that Y0 ∈ colspanQp . From Proposition 3.1, the initial conditions are consistent and thus from (6) the unique solution of the initial value problem (1) – (2) is given by Yk = Qp Jpk C. or, equivalently, 4k 2 C. 5k 3 Where C is the unique solution of the linear system 2 2 = C 3 3 Yk = −
and thus C = 1, i.e. Yk = −
4k 5k
2 3
.
Example 3.3. We assume the system (1) as in example 3.2 but with different initial conditions. Let 2.00001 Y0 = . 2.99999
6
It is easy to observe that Y0 ∈ / colspanQp . This means that from Proposition 3.1 the initial conditions are non-consistent and thus from Theorem 3.1 and (3) an optimal solution of the initial value problem is Yˆk = Qp Jpk (QTp Qp )−1 QTp Y0 . Where (QTp Qp )−1 =
1 , 13
QTp Y0 = 12.99999 and (QTp Qp )−1 QTp Y0 = Hence 12.9999 · 4 Yˆk = − 13 · 5k
12.99999 . 13 k
2 3
.
It is worth to observe that theabove optimal solution of the system (1) with non-consistent 2.00001 initial conditions , is almost equal to the unique solution of the same system 2.99999 2 with consistent initial conditions . This is because the perturbation to the non3
consistent initial conditions (2) accordingly min Y0 − Yˆ0 =0.00018 is very small. 2
Conclusions In this article we focused and provided properties for optimal solutions of a linear generalized discrete time system whose coefficients are square constant matrices and its pencil is regular.
References [1] T. M. Apostol; Explicit formulas for solutions of the second order matrix differential equation Y 00 = AY , Amer. Math. Monthly 82 (1975), pp. 159-162. [2] Apostolopoulos, Nicholas, and Fernando Ortega. The stability of systems of difference equation with non-consistent initial conditions. Dynamics of continuous, discrete and impulsive systems Series A: Mathematical Analysis, Volume 25, Number 1, pp. 31–40 (2018). [3] Apostolopoulos, N., Ortega, F. and Kalogeropoulos, G., 2015. Causality of singular linear discrete time systems. arXiv preprint arXiv:1512.04740.
7
[4] Apostolopoulos, N., Ortega F., and Kalogeropoulos, G. A boundary value problem of a generalised linear discrete time system with no solutions and infinitely many solutions. arXiv preprint arXiv:1610.08277 (2016). [5] R. Ben Taher and M. Rachidi; Linear matrix differential equations of higher-order and applications, E. J. of Differential Eq., Vol. 2008 (2008), No. 95, pp. 1-12. [6] L. Dai, Impulsive modes and causality in singular systems, International Journal of Control, Vol 50, number 4 (1989). [7] I.K. Dassios, Homogeneous linear matrix difference equations of higher order: regular case, Bull. Greek Math. Soc. 56, 57-64 (2009). [8] I. Dassios, On a boundary value problem of a class of generalized linear discrete time systems, Advances in Difference Equations, Springer, 2011:51 (2011). [9] I.K. Dassios, On non-homogeneous linear generalized linear discrete time systems, Circuits systems and signal processing, Volume 31, Number 5, 1699-1712 (2012). [10] I. Dassios, On solutions and algebraic duality of generalized linear discrete time systems, Discrete Mathematics and Applications, Volume 22, No. 5-6, 665–682 (2012). [11] I. Dassios, On stability and state feedback stabilization of singular linear matrix difference equations, Advances in difference equations, 2012:75 (2012). [12] I. Dassios, On robust stability of autonomous singular linear matrix difference equations, Applied Mathematics and Computation, Volume 218, Issue 12, 6912–6920 (2012). [13] I.K. Dassios, G. Kalogeropoulos, On a non-homogeneous singular linear discrete time system with a singular matrix pencil , Circuits systems and signal processing, Volume 32, Issue 4, 1615–1635 (2013). [14] I. Dassios, G. Kalogeropoulos, On the relation between consistent and non consistent initial conditions of singular discrete time systems, Dynamics of continuous, discrete and impulsive systems Series A: Mathematical Analysis, Volume 20, Number 4a, pp. 447–458 (2013). [15] Dassios I., On a Boundary Value Problem of a Singular Discrete Time System with a Singular Pencil, Dynamics of continuous. Discrete and Impulsive Systems Series A: Mathematical Analysis, 22(3): 211-231 (2015). [16] I. K. Dassios, K. Szajowski, Bayesian optimal control for a non-autonomous stochastic discrete time system, Applied Mathematics and Computation , Volume 274, 556– 564 (2016). [17] I. K. Dassios, K. Szajowski, A non-autonomous stochastic discrete time system with uniform disturbances, IFIP Advances in Information and Communication Technology, Volume 494, 2016, Pages 220-229 (2016). [18] E. Grispos, S. Giotopoulos, G. Kalogeropoulos; On generalised linear discrete-time regular delay systems., J. Inst. Math. Comput. Sci., Math. Ser. 13, No.2, 179-187, (2000).
8
[19] E. Grispos, Singular generalised autonomous linear differential systems., Bull. Greek Math. Soc. 34, 25-43 (1992). [20] Kaczorek, T.; General response formula for two-dimensional linear systems with variable coefficients. IEEE Trans. Aurom. Control Ac-31, 278-283, (1986). [21] J. Klamka, J. Wyrwa, Controllability of second-order infinite-dimensional systems. Syst. Control Lett. 57, No. 5, 386–391 (2008). [22] J. Klamka, Controllability of dynamical systems, Matematyka Stosowana, 50, no.9, pp.57-75, (2008). [23] F. L. Lewis; A survey of linear singular systems, Circuits Syst. Signal Process. 5, 3-36, (1986). [24] F.L. Lewis; Recent work in singular systems, Proc. Int. Symp. Singular systems, pp. 20-24, Atlanta, GA, (1987). [25] F. Milano; I. Dassios, Small-Signal Stability Analysis for Non-Index 1 Hessenberg Form Systems of Delay Differential-Algebraic Equations, Circuits and Systems I: Regular Papers, IEEE Transactions on 63(9):1521-1530 (2016). [26] F. Milano; I. Dassios, Primal and Dual Generalized Eigenvalue Problems for Power Systems Small-Signal Stability Analysis, IEEE Transactions on Power Systems, Volume: 32, Issue 6, pp. 4626 – 4635 (2017). [27] L. Verde-Star; Operator identities and the solution of linear matrix difference and differential equations, Studies in Applied Mathematics 91 (1994), pp. 153-177. [28] I. Dassios, A. Zimbidis, The classical Samuelson’s model in a multi-country context under a delayed framework with interaction, Dynamics of continuous, discrete and impulsive systems Series B: Applications & Algorithms, Volume 21, Number 4-5b pp. 261–274 (2014). [29] I. Dassios, A. Zimbidis, C. Kontzalis. The Delay Effect in a Stochastic MultiplierAccelerator Model. Journal of Economic Structures 2014, 3:7. [30] I. Dassios, G. Kalogeropoulos, On the stability of equilibrium for a reformulated foreign trade model of three countries. Journal of Industrial Engineering International, Springer, Volume 10, Issue 3, pp. 1-9 (2014). 10:71. [31] I. Dassios, C. Kontzalis, G. Kalogeropoulos, A stability result on a reformulated samuelson economical model. Proceedings of the IASTED International Conference on Modelling, Identification and Control 2013, Pages 248-252. [32] I. Dassios, C. Kontzalis On the stability of equilibrium for a foreign trade model. Proceedings of the IASTED International Conference on Modelling, Identification and Control 2013, Pages 253-257. [33] I. Dassios, M. Devine. A macroeconomic mathematical model for the national income of a union of countries with interaction and trade. Journal of Economic Structures 2016, 5:18.
9
[34] Ogata, K: Discrete Time Control Systems. Prentice Hall, (1987) [35] W.J. Rugh; Linear system theory, Prentice Hall International (Uk), London (1996). [36] J.T. Sandefur; Discrete Dynamical Systems, Academic Press, (1990). [37] A. P. Schinnar, The Leontief dynamic generalized inverse. The Quarterly Journal of Economics 92.4 pp. 641-652 (1978). [38] D. Baleanu, K. Diethelm, E. Scalas, Fractional Calculus: Models and Numerical Methods, World Scientific (2012). [39] I.K. Dassios, Optimal solutions for non-consistent singular linear systems of fractional nabla difference equations, Circuits, Systems and Signal Processing, Springer, Volume 34, Issue 6, pp. 1769-1797 (2015). DOI 10.1007/s00034-014-9930-2 [40] I.K. Dassios, D. Baleanu, On a singular system of fractional nabla difference equations with boundary conditions, Boundary Value Problems, 2013:148 (2013). [41] I.K. Dassios, D.I. Baleanu. Duality of singular linear systems of fractional nabla difference equations. Applied Mathematical Modeling, Elsevier, Volume 39, Issue 14, pp. 4180-4195 (2015). DOI 10.1016/j.apm.2014.12.039 [42] I. Dassios, D. Baleanu, G. Kalogeropoulos, On non-homogeneous singular systems of fractional nabla difference equations, Applied Mathematics and Computation, Volume 227, 112–131 (2014). [43] I. Dassios, Geometric relation between two different types of initial conditions of singular systems of fractional nabla difference equations, Math. Meth. Appl. Sci., 2015, doi: 10.1002/mma.3771. [44] I. Dassios, Stability and robustness of singular systems of fractional nabla difference equations. Circuits, Systems and Signal Processing, Springer. Volume: 36, Issue 1, pp. 49 – 64 (2017). [45] Dassios, I. K. A practical formula of solutions for a family of linear non-autonomous fractional nabla difference equations. Journal of Computational and Applied Mathematics (2017). https://doi.org/10.1016/j.cam.2017.09.030. [46] T. Kaczorek, Application of the Drazin inverse to the analysis of descriptor fractional discrete-time linear systems with regular pencils. Int. J. Appl. Math. Comput. Sci 23.1, 2013: 29–33 (2014). [47] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering,p. xxiv+340. Academic Press, San Diego, Calif, USA (1999). [48] R. F. Gantmacher; The theory of matrices I, II, Chelsea, New York, (1959). [49] G. I. Kalogeropoulos; Matrix pencils and linear systems, Ph.D Thesis, City University, London, (1985). [50] Boutarfa, B., Dassios, I.K., 2017. A stability result for a network of two triple junctions on the plane. Mathematical Methods in the Applied Sciences, 40(17), pp.60766084.
10
[51] H.-W. Cheng and S. S.-T. Yau; More explicit formulas for the matrix exponential, Linear Algebra Appl. 262 (1997), pp. 131-163. [52] B.N. Datta; Numerical Linear Algebra and Applications, Cole Publishing Company, 1995. [53] L. Dai, Singular Control Systems, Lecture Notes in Control and information Sciences Edited by M.Thoma and A.Wyner (1988). [54] I. Dassios, A. P. Jivkov, A. Abu-Muharib, P. James: A mathematical model for plasticity and damage: A discrete calculus formulation. Journal of Computational and Applied Mathematics 312 (2017): 27 – 38. [55] I. Dassios, K. Fountoulakis, J. Gondzio. A preconditioner for a primal-dual newton conjugate gradients method for compressed sensing problems. SIAM J. Sci. Comput., 37 (2015), pp. A2783 – A2812 [56] I. Dassios, Stability of triple junctions on the plane, Bull. Greek Math. Soc. 54 (2007) 281 – 300. [57] I. Dassios, Stability of basic steady states of networks in bounded domains. Computers & Mathematics with Applications, 70(9), 2177 – 2196 (2015). [58] I. E. Leonard; The matrix exponential, SIAM Review Vol. 38, No. 3 (1996), pp. 507-512. [59] G.W. Steward and J.G. Sun; Matrix Perturbation Theory, Oxford University Press, (1990).
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