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

Tahraoui, Rabah; Carlier, Guillaume (2010), Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory, ESAIM. COCV, 16, 3, p. 744-763. http://dx.doi.org/10.1051/cocv/2009024

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00363273/en/
Date
2010
Journal name
ESAIM. COCV
Volume
16
Number
3
Publisher
EDP Sciences
Pages
744-763
Publication identifier
http://dx.doi.org/10.1051/cocv/2009024
Metadata
Show full item record
Author(s)
Tahraoui, Rabah
Carlier, Guillaume
Abstract (EN)
This article is devoted to the optimal control of state equations with memory of the form: ?[x(t) = F\left(x(t),u(t), \int_0^{+\infty} A(s) x(t-s) ds\right), \; 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: \[v(x,z):=\inf_{u\in V} \left\{ \int_0^{+\infty} e^{-\lambda s } L(y_{x,z,u}(s), u(s))ds \right\}\] 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: \[\lambda v(x,z)+H(x,z,\nabla_x v(x,z))+\=0\] in the sense of the theory of viscosity solutions in infinite-dimensions of M. Crandall and P.-L. Lions.
Subjects / Keywords
Hamilton-Jacobi-Bellman equations in infinite dimensions; Dynamic programming; state equations with memory; viscosity solutions

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