Optimal control of delay systems with differential and algebraic dynamic constraints

Boris S. Mordukhovich; Lianwen Wang

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 11, Issue: 2, page 285-309
  • ISSN: 1292-8119

Abstract

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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 W-1,2. 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.

How to cite

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Mordukhovich, Boris S., and Wang, Lianwen. "Optimal control of delay systems with differential and algebraic dynamic constraints." ESAIM: Control, Optimisation and Calculus of Variations 11.2 (2010): 285-309. <http://eudml.org/doc/90766>.

@article{Mordukhovich2010,
abstract = { 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 W-1,2. 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. },
author = {Mordukhovich, Boris S., Wang, Lianwen},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Optimal control; variational analysis; functional-differential inclusions of neutral type; differential and algebraic dynamic constraints; discrete approximations; generalized differentiation; necessary optimality conditions.; optimal control; functional-differential inclusions of neutral type; generalized differentiation; necessary optimality conditions},
language = {eng},
month = {3},
number = {2},
pages = {285-309},
publisher = {EDP Sciences},
title = {Optimal control of delay systems with differential and algebraic dynamic constraints},
url = {http://eudml.org/doc/90766},
volume = {11},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 11
IS - 2
SP - 285
EP - 309
AB - 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 W-1,2. 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.
LA - eng
KW - Optimal control; variational analysis; functional-differential inclusions of neutral type; differential and algebraic dynamic constraints; discrete approximations; generalized differentiation; necessary optimality conditions.; optimal control; functional-differential inclusions of neutral type; generalized differentiation; necessary optimality conditions
UR - http://eudml.org/doc/90766
ER -

References

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