ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - July 2010 - Volume 16, Issue 03  

Research Article

Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory

Carlier, Guillaumea1 and Tahraoui, Rabaha1

a1 Université Paris Dauphine, CEREMADE, Pl. de Lattre de Tassigny, 75775 Paris Cedex 16, France. carlier@ceremade.dauphine.fr ; tahraoui@ceremade.dauphine.fr

Abstract

This article is devoted to the optimal control of state equations with memory of the form:

$\dot{x}(t)=F(x(t),u(t), \int_0^{+\infty} A(s) x(t-s) {\rm d}s), \; t>0,$

with initial conditions $x(0)=x, \; x(-s)=z(s), s>0$ .

Denoting by $y_{x, z, u}$ the solution of the previous Cauchy problem and:

$ v(x,z):=\inf_{u\in V} \lbrace \int_0^{+\infty} {\rm e}^{-\lambda s } L(y_{x,z,u}(s), u(s)){\rm d}s\rbrace $

where V is a class of admissible controls, we prove that v is the only viscosity solution of an Hamilton-Jacobi-Bellman equation of the form:

$ \lambda v(x,z)+H(x,z,\nabla_x v(x,z))+\langle D_z v(x,z), \dot{z} \rangle=0 $

in the sense of the theory of viscosity solutions in infinite-dimensions of Crandall and Lions.

(Received March 21 2008)

(Revised March 26 2009)

(Online publication July 31 2009)

Key Words:

  • Dynamic programming;
  • state equations with memory;
  • viscosity solutions;
  • Hamilton-Jacobi-Bellman equations in infinite dimensions

Mathematics Subject Classification:

  • 49L20;
  • 49L25