Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 3, pp. 744-763.

This article is devoted to the optimal control of state equations with memory of the form: x ˙(t)=F(x(t),u(t), 0 + A(s)x(t-s)ds),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 uV { 0 + e -λs L(y x,z,u (s),u(s))ds} 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: λv(x,z)+H(x,z, x v(x,z))+D z v(x,z),z ˙=0 in the sense of the theory of viscosity solutions in infinite-dimensions of Crandall and Lions.

DOI: 10.1051/cocv/2009024
Classification: 49L20, 49L25
Keywords: dynamic programming, state equations with memory, viscosity solutions, Hamilton-Jacobi-Bellman equations in infinite dimensions
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     title = {Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {744--763},
     publisher = {EDP-Sciences},
     volume = {16},
     number = {3},
     year = {2010},
     doi = {10.1051/cocv/2009024},
     mrnumber = {2674635},
     zbl = {1195.49032},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2009024/}
}
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Carlier, Guillaume; Tahraoui, Rabah. Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 3, pp. 744-763. doi : 10.1051/cocv/2009024. http://archive.numdam.org/articles/10.1051/cocv/2009024/

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