Résumé en anglais

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.