On some optimal control problems governed by a state equation with memory
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: $in{f}_{\left(u\in V\right)}J\left(u\right)$ with $J\left(u\right):={\int }_{0}^{1}L\left(s,{y}_{u}\left(s\right),u\left(s\right)\right)\mathrm{d}s+g\left({y}_{u}\left(1\right)\right)$ where $V$ is a set of admissible controls and ${y}_{u}$ is the solution of the Cauchy problem: $\stackrel{˙}{x}\left(t\right)=〈f\left(.,x\left(.\right)\right),{\nu }_{t}〉+u\left(t\right),t\in \left(0,1\right)$, $x\left(0\right)={x}_{0}$ and each ${\nu }_{t}$ is a nonnegative measure with support in $\left[0,t\right]$. 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.

DOI : https://doi.org/10.1051/cocv:2008005
Classification : 34K35,  49K25
Mots clés : optimal control, memory
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title = {On some optimal control problems governed by a state equation with memory},
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Carlier, Guillaume; Tahraoui, Rabah. On some optimal control problems governed by a state equation with memory. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 725-743. doi : 10.1051/cocv:2008005. http://www.numdam.org/articles/10.1051/cocv:2008005/

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