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Optimal control of delay systems with differentialand algebraic dynamic constraints

Published online by Cambridge University Press:  15 March 2005

Boris S. Mordukhovich  and
Lianwen Wang
Boris S. Mordukhovich
Affiliation:
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. boris@math.wayne.edu
Lianwen Wang
Affiliation:
Department of Mathematics and Computer Science, Central Missouri State University, Warrensburg, MO 64093, USA; lwang@cmsu1.cmsu.edu
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Abstract

This paper concerns constrained dynamic optimization problemsgoverned by delay control systems whose dynamic constraints are described by bothdelay-differential inclusions and linear algebraic equations. This is a new class ofoptimal control systems that, on one hand, may be treated as a specific type ofvariational problems for neutral functional-differential inclusions while, on the otherhand, is related to a special class of differential-algebraic systems with a generaldelay-differential inclusion and a linear constraint link between “slow” and “fast”variables. We pursue a twofold goal: to study variational stability for this class ofcontrol systems with respect to discrete approximations and to derive necessaryoptimality conditions for both delayed differential-algebraic systems under considerationand their finite-difference counterparts using modern tools of variational analysis andgeneralized differentiation. The authors are not familiar with any results in thesedirections for such systems even in the delay-free case. In the first part of the paperwe establish the value convergence of discrete approximations as well as the strongconvergence of optimal arcs in the classical Sobolev space W -1,2. Then using discreteapproximations as a vehicle, we derive necessary optimality conditions for the initialcontinuous-time systems in both Euler-Lagrange and Hamiltonian forms via basicgeneralized differential constructions of variational analysis.

Keywords

Optimal controlvariational analysisfunctional-differential inclusionsof neutral typedifferential and algebraic dynamic constraintsdiscrete approximationsgeneralized differentiationnecessary optimality conditions.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 2 , April 2005 , pp. 285 - 309
Copyright
© EDP Sciences, SMAI, 2005

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