Measurable Viability Theorems and the Hamilton-Jacobi-Bellman Equation

Abstract 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 u t + H(t, x, u x ) = 0 with the Hamiltonian H measurable with respect to time, locally Lipschitz with respect to x , and convex in the last variable.