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

Carlier, Guillaume; Tahraoui, Rabah

ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 725-743

The aim of this paper is to study problems of the form: $in{f}_{(u\in V)}J\left(u\right)$ with $J\left(u\right):={\int }_0^{1}L(s,y_u\left(s\right),u\left(s\right))\mathrm{d}s+g\left(y_u\left(1\right)\right)$ where $V$ is a set of admissible controls and $y_u$ is the solution of the Cauchy problem: $\dot{x}\left(t\right)=\langle f(.,x(.)),{\nu }_t\rangle +u\left(t\right),t\in (0,1)$, $x\left(0\right)=x_0$ 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.

Classification : 34K35, 49K25
Keywords: optimal control, memory

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