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

Guillaume Carlier; Rabah Tahraoui

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 14, Issue: 4, page 725-743
  • ISSN: 1292-8119

Abstract

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The aim of this paper is to study problems of the form: i n f ( u ∈ V ) J ( u ) with J ( u ) : = ∫ 0 1 L ( s , y u ( s ) , u ( s ) ) d s + g ( y u ( 1 ) ) where V is a set of admissible controls and yu is the solution of the Cauchy problem: x ˙ ( t ) = 〈 f ( . , x ( . ) ) , ν t 〉 + u ( t ) , t ∈ ( 0 , 1 ) , x ( 0 ) = x 0 and each ν 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.

How to cite

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Carlier, Guillaume, and Tahraoui, Rabah. "On some optimal control problems governed by a state equation with memory." ESAIM: Control, Optimisation and Calculus of Variations 14.4 (2008): 725-743. <http://eudml.org/doc/250320>.

@article{Carlier2008,
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 yu 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. },
author = {Carlier, Guillaume, Tahraoui, Rabah},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Optimal control; memory; optimal control},
language = {eng},
month = {1},
number = {4},
pages = {725-743},
publisher = {EDP Sciences},
title = {On some optimal control problems governed by a state equation with memory},
url = {http://eudml.org/doc/250320},
volume = {14},
year = {2008},
}

TY - JOUR
AU - Carlier, Guillaume
AU - Tahraoui, Rabah
TI - On some optimal control problems governed by a state equation with memory
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/1//
PB - EDP Sciences
VL - 14
IS - 4
SP - 725
EP - 743
AB - 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 yu 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.
LA - eng
KW - Optimal control; memory; optimal control
UR - http://eudml.org/doc/250320
ER -

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