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

Guillaume Carlier; Rabah Tahraoui

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 16, Issue: 3, page 744-763
- ISSN: 1292-8119

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topCarlier, Guillaume, and Tahraoui, Rabah. "Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory." ESAIM: Control, Optimisation and Calculus of Variations 16.3 (2010): 744-763. <http://eudml.org/doc/250723>.

@article{Carlier2010,

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.
},

author = {Carlier, Guillaume, Tahraoui, Rabah},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Dynamic programming; state equations with memory; viscosity solutions; Hamilton-Jacobi-Bellman equations in infinite dimensions; dynamic programming},

language = {eng},

month = {7},

number = {3},

pages = {744-763},

publisher = {EDP Sciences},

title = {Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory},

url = {http://eudml.org/doc/250723},

volume = {16},

year = {2010},

}

TY - JOUR

AU - Carlier, Guillaume

AU - Tahraoui, Rabah

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/7//

PB - EDP Sciences

VL - 16

IS - 3

SP - 744

EP - 763

AB -
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.

LA - eng

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

UR - http://eudml.org/doc/250723

ER -

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