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 yx,z,u the
solution of the previous Cauchy problem and: v(x,z):=u∈Vinf{∫0+∞e−λsL(yx,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,∇xv(x,z))+Dzv(x,z),z˙>=0 in the sense of the
theory of viscosity solutions in infinite-dimensions of M. Crandall and P.-L.
Lions