Nous nous intéressons à une équation d'Hamilton–Jacobi–Bellman ergodique provenant d'un problème de contrôle stochastique comprenant un nombre fini k de points en lesquels la dynamique s'annule et le lagrangien est minimal. Sous une condition de stabilisabilité, on établit que les solutions de l'équation ergodique sont uniquement déterminées par leurs valeurs en ces points et que l'ensemble des solutions est isométrique au sens de la norme sup à un ensemble convexe fermé non vide dont la dimension est majorée par k.
We consider an ergodic Hamilton–Jacobi–Bellman equation coming from a stochastic control problem in which there are exactly k points where the dynamics vanishes and the Lagrangian is minimal. Under a stabilizability assumption, we state that the solutions of the ergodic equation are uniquely determined by their value on these k points, and that the set of solutions is sup-norm isometric to a non-empty closed convex set whose dimension is less or equal to k.
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@article{CRMATH_2008__346_21-22_1149_0,
author = {Akian, Marianne and David, Beno{\^\i}t and Gaubert, St\'ephane},
title = {Un th\'eor\`eme de repr\'esentation des solutions de viscosit\'e d'une \'equation {d'Hamilton{\textendash}Jacobi{\textendash}Bellman} ergodique d\'eg\'en\'er\'ee sur le tore},
journal = {Comptes Rendus. Math\'ematique},
pages = {1149--1154},
publisher = {Elsevier},
volume = {346},
number = {21-22},
year = {2008},
doi = {10.1016/j.crma.2008.09.020},
language = {fr},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.09.020/}
}
TY - JOUR AU - Akian, Marianne AU - David, Benoît AU - Gaubert, Stéphane TI - Un théorème de représentation des solutions de viscosité d'une équation d'Hamilton–Jacobi–Bellman ergodique dégénérée sur le tore JO - Comptes Rendus. Mathématique PY - 2008 SP - 1149 EP - 1154 VL - 346 IS - 21-22 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2008.09.020/ DO - 10.1016/j.crma.2008.09.020 LA - fr ID - CRMATH_2008__346_21-22_1149_0 ER -
%0 Journal Article %A Akian, Marianne %A David, Benoît %A Gaubert, Stéphane %T Un théorème de représentation des solutions de viscosité d'une équation d'Hamilton–Jacobi–Bellman ergodique dégénérée sur le tore %J Comptes Rendus. Mathématique %D 2008 %P 1149-1154 %V 346 %N 21-22 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2008.09.020/ %R 10.1016/j.crma.2008.09.020 %G fr %F CRMATH_2008__346_21-22_1149_0
Akian, Marianne; David, Benoît; Gaubert, Stéphane. Un théorème de représentation des solutions de viscosité d'une équation d'Hamilton–Jacobi–Bellman ergodique dégénérée sur le tore. Comptes Rendus. Mathématique, Tome 346 (2008) no. 21-22, pp. 1149-1154. doi : 10.1016/j.crma.2008.09.020. http://www.numdam.org/articles/10.1016/j.crma.2008.09.020/
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⁎ Ce travail a été partiellement soutenu par le projet RFBR-CNRS numéro 05-01-02807.
