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On some optimal control problems governed by a state equation with memory

Published online by Cambridge University Press:  18 January 2008

Guillaume Carlier  and
Rabah Tahraoui
Guillaume Carlier
Affiliation:
Université Paris Dauphine, CEREMADE, Pl. de Lattre de Tassigny, 75775 Paris Cedex 16, France; carlier@ceremade.dauphine.fr; tahraoui@ceremade.dauphine.fr
Rabah Tahraoui
Affiliation:
Université Paris Dauphine, CEREMADE, Pl. de Lattre de Tassigny, 75775 Paris Cedex 16, France; carlier@ceremade.dauphine.fr; tahraoui@ceremade.dauphine.fr
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Abstract

The aim of this paper is to study problems of the form: $inf_{(u\in V)}J(u)$ with $J(u):=\int_0^1 L(s,y_u(s),u(s)){\rm d}s+g(y_u(1))$ where V is a set of admissible controls and y u is the solution of the Cauchy problem: $\dot{x}(t) =\langle f(.,x(.)), \nu_t \rangle + u(t), t \in (0,1)$ , $x(0) = x_{\rm 0}$ and each $\nu_t$ is a nonnegative measure with support in [0,t]. After studying the Cauchy problem, we establish existence of minimizers, optimality conditions (in particular in the form of a nonlocal version of the Pontryagin principle) and prove some regularity results. We also consider the more general case where the control also enters the dynamics in a nonlocal way.

Keywords

Optimal controlmemory
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 4 , October 2008 , pp. 725 - 743
Copyright
© EDP Sciences, SMAI, 2008

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