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

Carlier, Guillaume; Tahraoui, Rabah

ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 3, pp. 744-763.

This article is devoted to the optimal control of state equations with memory of the form: $\dot{x}\left(t\right)=F(x\left(t\right),u\left(t\right),{\int }_0^{+\infty }A\left(s\right)x(t-s)\mathrm{d}s),t>0,$ with initial conditions $x\left(0\right)=x,x(-s)=z\left(s\right),s>0$. Denoting by $y_{x,z,u}$ the solution of the previous Cauchy problem and: $v(x,z):={inf}_{u\in V}\left\{{\int }_0^{+\infty }{\mathrm{e}}^{-\lambda s}L(y_{x,z,u}\left(s\right),u\left(s\right))\mathrm{d}s\right\}$ 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: $\lambda v(x,z)+H(x,z,{\nabla }_{x}v(x,z))+\langle D_{z}v(x,z),\dot{z}\rangle =0$ in the sense of the theory of viscosity solutions in infinite-dimensions of Crandall and Lions.

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