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

This article is devoted to the optimal control of state equations with memory of the form: ?[x(t) = F(x(t),u(t), \int_0^{+\infty} 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: $$v(x,z):=\inf_{u\in V} \{\int_0^{+\infty} e^{-\lambda s} L(y_{x,z,u}(s), u(s))ds \}$$ 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: $$\lambda v(x,z)+H(x,z,\nabla_x v(x,z))+D_z v(x,z), \dot{z} >=0$$ in the sense of the theory of viscosity solutions in infinite-dimensions of M. Crandall and P.-L. Lions.