U.P.B. Sci. Bull., Series A, Vol. 76, Iss. 1, 2014
ISSN 1223-7027
PDES OF HAMILTON-PFAFF TYPE VIA MULTI-TIME OPTIMIZATION PROBLEMS ˘1 Savin Treant ¸a
In this paper, PDEs of Hamilton-Pfaff type are derived in the sense of exterior differential via a multi-time optimal control problem. Using the control Hamiltonian 1-form associated with our basic optimal control problem and the corresponding adjoint distributions, we obtain PDEs of Hamilton-Pfaff type. Keywords: Hamilton-Pfaff PDEs; control Hamiltonian; Lagrangian 1-form; EulerLagrange exterior PDEs; adjoint distributions. MSC2010: 70H05, 49J20, 34H05, 93C20, 65K10, 70S05.
1. Introduction Nowadays, the multi-time optimal control problems and multi-time optimization are intensively studied (see [10], [14]-[17]), both from theoretical and applied reasonings. The cost functionals of mechanical work type become very important in applications due to their physical meaning. In this direction (see [10]), let consider the following multi-time optimal control problem, formulated using as cost functional a path independent curvilinear integral with distribution-type constraints: ( ) Z β max J (u(·)) = Xβ t, x(t), xα1 (t), . . . , xα1 α2 ...αs−1 (t), u(t) dt (1.1) u(·)
Γt0 ,t1
subject to dxiα1 α2 ...αs−1 (t) = Xβi t, x(t), xα1 (t), . . . , xα1 α2 ...αs−1 (t), u(t) dtβ u(t) ∈ U, ∀t ∈ Ωt0 ,t1 ;
x(tξ ) = xξ , xα1 ...αj (tξ ) = x ˜α1 ...αj ξ
(1.2) (1.3)
i = 1, n, αζ ∈ {1, . . . , m}, ζ, j = 1, s − 1, ξ = 0, 1. The mathematical data used are: t = (tα ) ∈ Ωt0 ,t1 (see Ωt0 ,t1 ⊂ Rm + , the 1 m hyperparallelepiped with the diagonal opposite points t0 = t0 , . . . , t0 and t1 = 1 t11 , . . . , tm 1 ) is a multi-parameter of evolution or a multi-time; Γt0 ,t1 is a C -class curve joining the points t0 and t1 ; x(t) = (xi (t)), i = 1, n, is a C s+1 -class function, called state vector ; u(t) = (ua (t)), a = 1, k, is a continuous control vector ; β the running cost Xβ t, x(t), xα1 (t), . . . , xα1 α2 ...αs−1 (t), u(t) dt is a non-autonomous Lagrangian 1-form; the equations in (1.2) are distribution-type equations; the func tions Xβi t, x(t), xα1 (t), . . . , xα1 α2 ...αs−1 (t), u(t) are of C 1 -class; we accept the no∂x ∂ s−1 x tations xα1 (t) := α1 (t), . . . , xα1 ...αs−1 (t) := α1 (t), αj ∈ {1, 2, . . . , m}, ∂t ∂t . . . ∂tαs−1 1 Teaching Assistant, Department of Mathematics-Informatics, Faculty of Applied Sciences, University ”Politehnica” of Bucharest, Romania, E-mail: savin
[email protected]
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˘ Savin Treant ¸a
j = 1, s − 1. We assume summation over the repeated indices. For more details, see references [9], [10]. Consider u ˆ(t) ∈ IntU an interior optimal solution which determines the optimal evolution x(t) in (1.1), subject to (1.2) and (1.3). In a very recent work (see [10]), we proved that there exist the C 1 -class co-state 1-forms, pr = (pir ) , r = 1, s, defined on Ωt0 ,t1 , such that the relations ˆ(t), p(t) , pj1 (t1 ) = 0 (1.4) dpj1 (t) = −Hxj t, x(t), . . . , xα1 ...αs−1 (t), u ˆ(t), p(t) , pj2 (t1 ) = 0 dpj2 (t) = −Hxjα t, x(t), . . . , xα1 ...αs−1 (t), u 1
dpjs (t) = −Hxjα
1 ...αs−1
.. . ˆ(t), p(t) , t, x(t), . . . , xα1 ...αs−1 (t), u
pjs (t1 ) = 0
∀t ∈ Ωt0 ,t1 ; (1.5) ˆ(t), p(t) = 0, ∀t ∈ Ωt0 ,t1 ; Hua t, x(t), . . . , xα1 ...αs−1 (t), u ∂H dxiα1 ...αr−1 (t) = (1.6) ˆ(t), p(t) , ∀t ∈ Ωt0 ,t1 t, x(t), . . . , xα1 ...αs−1 (t), u ∂pir x(t0 ) = x0 , xα1 ...αj (t0 ) = x ˜α1 ...αj 0 , j = 1, s − 1, i = 1, n, r = 1, s see dxiα1 ...α0 (t) := dxi (t) are fulfilled (see H as the associated control Hamiltonian 1-form). The aim of this paper is to introduce PDEs of Hamilton-Pfaff type (often used in Mechanics) using the simplified multi-time maximum principle (see (1.4), (1.5), (1.6)). For other different ideas connected to this subject, see [1]-[8], [12], [13], [18]. Section 1 motivates the study and provides, for a better coherence of this paper, the main ingredients used in previous works (see [10], [11], [14]-[17]). Section 2 includes the main result, while Section 3 contains the conclusions of this study. 2. Hamilton-Pfaff PDEs Let consider the following path independent curvilinear integral functional Z J (u(·)) = Xβ x(t), xα1 (t), . . . , xα1 α2 ...αs−1 (t), u(t) dtβ Γt0 ,t1
subject to dxiα1 α2 ...αs−1 (t) = uiβ (t)dtβ m t ∈ Ωt0 ,t1 ⊂ R+ ;
x(t0 ) = x0 , xα1 ...αj (t0 ) = x ˜α1 ...αj 0
ζ, j = 1, s − 1. Here, the running cost Xβ x(t), xα1 (t), . . . , xα1 α2 ...αs−1 (t), u(t) dtβ is a C 2 -class au tonomous Lagrangian 1-form, the control vector u(t) = uiβ (t) is a continuous vector m function and Γt0 ,t1 is a C 1 -class curve joining the points t0 and t1 from R+ . For solving the associated basic control problem, we need the control Hamiltonian 1-form H (v1 (t), v2 (t), . . . , vs (t), u(t), p1 (t), . . . , ps (t)) i = 1, n,
αζ ∈ {1, . . . , m},
i = Xβ (v1 (t), v2 (t), . . . , vs (t), u(t)) dtβ + pi1 (t)v1α (t)dtα1 1 i + . . . + pis−1 (t)vs−1α (t)dtαs−1 + pis (t)uiβ (t)dtβ , s−1
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where {v1 (t), . . . , vs (t)} are auxiliary variables, defined as follows: x(t) := v1 (t), (xα1 (t)) := v2 (t), · · · , xα1 α2 ...αs−1 (t) := vs (t), dvsi (t) = uiβ (t)dtβ , i = 1, n, β = 1, m, or, equivalently, (v1α1 (t)) = v2 (t),
(v2α2 (t)) = v3 (t), · · · , vs−1αs−1 (t) = vs (t)
i vsβ (t) = uiβ (t), i = 1, n, ∂vγ (t), γ = 1, s, η ∈ {1, . . . , m}), and the co-state (we have denoted vγη (t) := ∂tη 1-forms pγ (t) = piγ (t)dvγi , γ = 1, s, satisfying the following adjoint distributions:
∂Xβ (2.1) x(t), xα1 (t), . . . , xα1 ...αs−1 (t), u(t) dtβ j ∂x ∂Xβ dpj2 (t) = −pj1 (t)dtα1 − j x(t), xα1 (t), . . . , xα1 ...αs−1 (t), u(t) dtβ ∂xα1 .. . ∂X β dpjs (t) = −pjs−1 (t)dtαs−1 − j x(t), xα1 (t), . . . , xα1 ...αs−1 (t), u(t) dtβ . ∂xα1 ...αs−1 According to (1.4) and (1.5), we have Hxjα α ...α x(t), . . . , xα1 α2 ...αs−1 (t), u(t), p1 (t), . . . , ps (t) = −dpjη+1 (t) (2.2) dpj1 (t) = −
η
1 2
η = 0, s − 1,
Huj
β
t ∈ Ωt0 ,t1 ,
pjr (t1 ) = 0, r = 1, s x(t), xα1 (t), . . . , xα1 α2 ...αs−1 (t), u(t), p1 (t), . . . , ps (t) = 0,
∀t ∈ Ωt0 ,t1 .
By a direct computation, we get Hxi x(t), . . . , xα1 α2 ...αs−1 (t), u(t), p1 (t), . . . , ps (t) = Xβxi x(t), . . . , xα1 α2 ...αs−1 (t), u(t) dtβ ; x(t), . . . , xα1 α2 ...αs−1 (t), u(t), p1 (t), . . . , ps (t) = Xβxiα α ...α x(t), . . . , xα1 α2 ...αs−1 (t), u(t) dtβ + piη (t)dtαη , η = 1, s − 1; η 1 2 Hui x(t), . . . , xα1 α2 ...αs−1 (t), u(t), p1 (t), . . . , ps (t) β = Xβui x(t), . . . , xα1 α2 ...αs−1 (t), u(t) dtβ + pis (t)dtβ , Hxiα
1 α2 ...αη
β
or, equivalently, (see (2.2))
Xβxiα
1
Xβxi x(t), . . . , xα1 α2 ...αs−1 (t), u(t) dtβ + dpi1 (t) = 0 x(t), . . . , xα1 ...αs−1 (t), u(t) dtβ + piη (t)dtαη + dpiη+1 (t) = 0 ...α
(2.3)
η
Xβui
β
η = 1, s − 1 x(t), . . . , xα1 α2 ...αs−1 (t), u(t) dtβ + pis (t)dtβ = 0.
The previous relations give us piη dtαη = −dpiη+1 −
∂Xβ dtβ , ∂xiα1 α2 ...αη
η = 1, s − 1
(2.4)
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dpi1 = −
∂Xβ β dt , ∂xi
pis dtβ = −
∂Xβ β dt . ∂uiβ
From the relations (2.4), we find dpiη ∧ dtαη = −d
!
∂Xβ
∧ dtβ ,
∂xiα1 α2 ...αη
∂Xβ ∂uiβ
dpis ∧ dtβ = −d
η = 1, s − 1
! ∧ dtβ .
Now, using (2.4) and the above equalities, we get ∂Xβ β ∂Xβ α1 − ∧ dtβ = 0 dt ∧ dt + d ∂xi ∂xiα1 ( s−2 X −pir dtαr − r=1
∂Xβ β dt ∧ dtαr+1 + d ∂xiα1 ...αr
" −pis−1 dt
αs−1
−
∂xiα1 ...αr+1
#
∂Xβ
β
∂xiα1 ...αs−1
dt
!
∂Xβ
β
∧ dt + d
∂Xβ ∂uiβ
) ∧ dtβ
=0
! ∧ dtβ = 0
or, equivalently,
∂Xλ α1 ∂ δβ − λ i ∂x ∂t
∂Xβ ∂xiα1
dtλ ∧ dtβ = 0
(Euler-Lagrange exterior equations), and " !# ∂Xβ ∂Xλ ∂ αr+1 αr pir δλ + i δβ − λ dtλ ∧ dtβ = 0, r = 1, s − 2 ∂xα1 ...αr ∂xiα1 ...αr+1 ∂t " α pis−1 δλ s−1
+
∂Xλ ∂xiα1 ...αs−1
!
∂ − λ ∂t
∂Xβ ∂uiβ
!# dtλ ∧ dtβ = 0.
Suppose that the critical point condition (see (2.3), the third relation) pis (t)dtα = −Xβuiα x(t), . . . , xα1 α2 ...αs−1 (t), u(t) dtβ admits a unique solution that satisfies uiβ (t)dtβ = uiβ x(t), . . . , xα1 ...αs−1 (t), p1 (t), . . . , ps (t) dtβ = dxiα1 ...αs−1 (t). Using a path independent curvilinear integral, we have Z xiα1 ...αs−1 (t) = xiα1 ...αs−1 (t0 ) + uiβ x(l), . . . , xα1 ...αs−1 (l), p1 (l), . . . , ps (l) dlβ , Γt0 t
where Γt0 t is a piecewise C 1 -class curve included in Γt0 t1 . Let remark that the previous critical point condition defines the co-state pis as a moment along the curve Γt0 t . Now, we establish the main result of our study.
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Theorem 2.1. (PDEs of Hamilton-Pfaff type) Under the above assumptions, consider the control Hamiltonian 1-form H x(t), . . . , xα1 ...αs−1 (t), u x(t), . . . , xα1 ...αs−1 (t), p1 (t), . . . , ps (t) , p1 (t), . . . , ps (t) . Then, the associated PDEs of Hamilton-Pfaff type are given by dxiα1 ...αr−1 =
∂H , ∂pir
j = 1, s − 1,
t ∈ Ωt0 ,t1 ,
i = 1, n,
x(t0 ) = x0 ,
r = 1, s,
xα1 ...αj (t0 ) = x ˜α1 ...αj 0
see dxiα1 ...α0 (t) := dxi (t)
and −
∂H = dpi1 , ∂xi
−
∂H ∂H = dpi2 , · · · , − i = dpis . i ∂xα1 ∂xα1 ...αs−1
Proof. Computing the partial derivative of H x(t), . . . , xα1 ...αs−1 (t), u x(t), . . . , xα1 ...αs−1 (t), p1 (t), . . . , ps (t) , p1 (t), . . . , ps (t) with respect to pir , r = 1, s, we get the first part of Hamilton-Pfaff equations ∂H x(t), . . . , xα1 ...αs−1 (t), u(t), p1 (t), . . . , ps (t) dxiα1 ...αr−1 (t) = ∂pir t ∈ Ωt0 ,t1 , x(t0 ) = x0 , xα1 ...αj (t0 ) = x ˜α1 ...αj 0 , j = 1, s − 1, i = 1, n, r = 1, s see dxiα1 ...α0 (t) := dxi (t) . Let illustrate two computations: ∂Xβ β ∂uiα ∂uiα α ∂H j dt = + dx + p dt = dxj is ∂pj1 ∂uiα ∂pj1 ∂pj1 ∂Xβ β ∂uiα ∂uiα α ∂H j α dt = + u dt + p dt = dxjα1 α2 ...αs−1 . is α ∂pjs ∂uiα ∂pjs ∂pjs Now, by a direct computation and using the relations in (2.1), we obtain the second part of Hamilton-Pfaff equations ∂uiβ β ∂Xβ β ∂Xβ β ∂uiα ∂H dt + dt − p dt = dpj1 − j =− is ∂x ∂xj ∂uiα ∂xj ∂xj " # ∂uiβ β ∂Xβ β ∂Xβ β ∂uiα ∂H α1 dt − p dt − p − j =− dt + dt = dpj2 j1 is ∂uiα ∂xα1 ∂xjα1 ∂xjα1 ∂xjα1 .. . −
∂H ∂xjα1 ...αs−1
"
∂Xβ β ∂uiα β =− dt + dt ∂uiα ∂xjα1 ...αs−1 ∂xjα1 ...αs−1 ∂Xβ
−pis and the proof is complete.
∂uiβ ∂xjα1 ...αs−1
# − pjs−1 dtαs−1
dtβ = dpjs ,
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˘ Savin Treant ¸a
3. Conclusion In this paper, we introduced a study of a multi-time optimal control problem subject to distribution-type constraints. Using a geometrical language and variational calculus techniques (under simplified hypotheses), we formulated the main result of this paper (see Theorem 2.1), that is, the form of Hamilton-Pfaff PDEs. Acknowledgments. The results presented in this article were obtained with the support of Ministry of Labour, Family and Social Protection through Sectorial Operational Programme Human Resources Development 2007-2013, Contract no. POSDRU/107/1.5/S/76909. REFERENCES [1] J. Baillieul, P. E. Crouch, J. E. Marsden, Nonholonomic Mechanics and Control, SpringerVerlag NY, 2003. [2] V. Barbu, Necessary conditions for multiple integral problem in the calculus of variations, Math. Ann., 260, 175-189, 1982. [3] M. Doroftei, S. Treant¸a ˘, Higher order hyperbolic equations involving a finite set of derivations, Balkan J. Geom. Appl., 17, 2, 22-33, 2012. [4] L. C. Evans, An Introduction to Mathematical Optimal Control Theory, Lecture Notes, University of California, Department of Mathematics, Berkeley, 2008. [5] E. B. Lee, L. Markus, Foundations of Optimal Control Theory, Wiley, 1967. [6] S. Pickenhain, M. Wagner, Pontryagin Principle for State-Constrained Control Problems Governed by First-Order PDE System, J. Optim. Theory Appl., 107, 2, 297-330, 2000. [7] J. P. Raymond, Optimal Control of Partial Differential Equations, Universit´e Paul Sabatier, Internet, 2013. [8] H. Rund, Pontryagin functions for multiple integral control problems, J. Optim. Theory Appl., 18, 4, 511-520, 1976. [9] D. J. Saunders, The Geometry of Jet Bundles, London Math. Soc., Lecture Notes Series 142, Cambridge Univ. Press, Cambridge, 1989. [10] S. Treant¸a ˘, Geometric PDEs and Control Problems, PhD Thesis, University ”Politehnica” of Bucharest, 2013. [11] S. Treant¸a ˘, On multi-time Hamilton-Jacobi theory via second order Lagrangians, U.P.B. Sci. Bull., Series A: Appl. Math. Phys., accepted. [12] S. Treant¸a ˘, C. Udri¸ste, Optimal control problems with higher order ODEs constraints, Balkan J. Geom. Appl., 18, 1, 71-86, 2013. [13] S. Treant¸a ˘, C. Vˆ arsan, Linear higher order PDEs of Hamilton-Jacobi and parabolic type, Math. Reports, accepted. [14] C. Udri¸ste, I. T ¸ evy, Multi-Time Euler-Lagrange-Hamilton Theory, WSEAS Transactions on Mathematics, 6, 6, 701-709, 2007. [15] C. Udri¸ste, Nonholonomic approach of multitime maximum principle, Balkan J. Geom. Appl., 14, 2, 101-116, 2009. [16] C. Udri¸ste, Simplified multitime maximum principle, Balkan J. Geom. Appl., 14, 1, 102-119, 2009. [17] C. Udri¸ste, Multitime optimal control with second-order PDEs constraints, Atti Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat. Nat., 91, 1, art. no. A2, 10 pages, 2013. [18] M. Wagner, Pontryagin maximum principle for Dieudonne-Rashevsky type problems involving Lipcshitz functions, Optimization, 46, 165-184, 1999.
