Measurable Viability Theorems and the Hamilton-Jacobi-Bellman Equation

We prove viability and invariance theorems for systems with dynamics depending on time in a measurable way and having time dependent state constraints: x′(t) F(t, x(t)), x(t) P(t). In the above t P(t) is an absolutely continuous set-valued map and (t, x) F(t, x) is a set-valued map which is measurable with respect to t and upper semicontinuous (or continuous, or locally Lipschitz) with respect to x. For this aim we investigate infinitesimal generators of reachable maps and the Lebesgue points of set-valued maps. The results are applied to define and to study lower semicontinuous solutions of the Hamilton-Jacobi-Bellman equation ut + H(t, x, ux) = 0 with the Hamiltonian H measurable with respect to time, locally Lipschitz with respect to x, and convex in the last variable