This paper concerns constrained dynamic optimization problems governed by delay control systems whose dynamic constraints are described by both delay-differential inclusions and linear algebraic equations. This is a new class of optimal control systems that, on one hand, may be treated as a specific type of variational problems for neutral functional-differential inclusions while, on the other hand, is related to a special class of differential-algebraic systems with a general delay-differential inclusion and a linear constraint link between “slow” and “fast” variables. We pursue a twofold goal: to study variational stability for this class of control systems with respect to discrete approximations and to derive necessary optimality conditions for both delayed differential-algebraic systems under consideration and their finite-difference counterparts using modern tools of variational analysis and generalized differentiation. The authors are not familiar with any results in these directions for such systems even in the delay-free case. In the first part of the paper we establish the value convergence of discrete approximations as well as the strong convergence of optimal arcs in the classical Sobolev space . Then using discrete approximations as a vehicle, we derive necessary optimality conditions for the initial continuous-time systems in both Euler-Lagrange and hamiltonian forms via basic generalized differential constructions of variational analysis.
Mots-clés : optimal control, variational analysis, functional-differential inclusions of neutral type, differential and algebraic dynamic constraints, discrete approximations, generalized differentiation, necessary optimality conditions
@article{COCV_2005__11_2_285_0,
author = {Mordukhovich, Boris S. and Wang, Lianwen},
title = {Optimal control of delay systems with differential and algebraic dynamic constraints},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {285--309},
publisher = {EDP-Sciences},
volume = {11},
number = {2},
year = {2005},
doi = {10.1051/cocv:2005008},
mrnumber = {2141891},
zbl = {1081.49017},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv:2005008/}
}
TY - JOUR AU - Mordukhovich, Boris S. AU - Wang, Lianwen TI - Optimal control of delay systems with differential and algebraic dynamic constraints JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2005 SP - 285 EP - 309 VL - 11 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2005008/ DO - 10.1051/cocv:2005008 LA - en ID - COCV_2005__11_2_285_0 ER -
%0 Journal Article %A Mordukhovich, Boris S. %A Wang, Lianwen %T Optimal control of delay systems with differential and algebraic dynamic constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2005 %P 285-309 %V 11 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2005008/ %R 10.1051/cocv:2005008 %G en %F COCV_2005__11_2_285_0
Mordukhovich, Boris S.; Wang, Lianwen. Optimal control of delay systems with differential and algebraic dynamic constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 2, pp. 285-309. doi : 10.1051/cocv:2005008. http://archive.numdam.org/articles/10.1051/cocv:2005008/
[1] , and, Numerical Solution of Initial Value Problems in Differential-Algebraic Equations. North-Holland, New York (1989). | MR | Zbl
[2] and, Maximum principle for implicit control systems. Appl. Math. Optim. 40 (1999) 79-103. | Zbl
[3] and, Error estimates for discretized differential inclusions. Computing 41 (1989) 349-358. | Zbl
[4] , Differential Inclusions and Optimal Control. Kluwer, Dordrecht (1991). | MR | Zbl
[5] , Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40 (1976) 960-969. | Zbl
[6] , Approximation Methods in Problems of Optimization and Control. Nauka, Moscow (1988). | MR | Zbl
[7] , Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Amer. Math. Soc. 340 (1993) 1-35. | Zbl
[8] , Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions. SIAM J. Control Optim. 33 (1995) 882-915. | Zbl
[9] , and, An extended extremal principle with applications to multiobjective optimization. SIAM J. Optim. 14 (2003) 359-379. | Zbl
[10] and, Stability of discrete approximation and necessary optimality conditions for delay-differential inclusions. Ann. Oper. Res. 101 (2001) 149-170. | Zbl
[11] and, Optimal control of constrained delay-differential inclusions with multivalued initial condition. Control Cybernet. 32 (2003) 585-609. | Zbl
[12] and, Optimal control of neutral functional-differential inclusions. SIAM J. Control Optim. 43 (2004) 116-136. | MR | Zbl
[13] and, Optimal control of differential-algebraic inclusions, in Optimal Control, Stabilization, and Nonsmooth Analysis, M. de Queiroz et al., Eds., Lectures Notes in Control and Information Sciences, Springer-Verlag, Heidelberg 301 (2004) 73-83.
[14] and, Necessary conditions for optimal control problems involving nonlinear differential algebraic equations. J. Math. Anal. Appl. 212 (1997) 493-516. | Zbl
[15] ,, and, The mathematical modelling of transient systems using differential-algebraic equations. Comput. Chem. Engrg. 12 (1988) 449-454.
[16] , Equivalent subgradient versions of Hamiltonian and Euler-Lagrange conditions in variational analysis. SIAM J. Control Optim. 34 (1996) 1300-1314. | Zbl
[17] and, Variational Analysis. Springer-Verlag, Berlin (1998). | MR | Zbl
[18] , Introduction to the Theory of Differential Inclusions. American Mathematical Society, Providence, RI (2002). | MR | Zbl
[19] , Optimal Control. Birkhäuser, Boston (2000). | MR | Zbl
[20] , Optimal Control of Differential and Functional Equations. Academic Press, New York (1972). | MR | Zbl
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