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: $\text{x′(t) ∈ F(t, x(t)), x(t) ∈ P(t).}$ 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 $\text{u}{}_{\mathrm{t}}\text{+ H(t, x, u}{}_{\mathrm{x}}\text{) = 0}$ with the Hamiltonian H measurable with respect to time, locally Lipschitz with respect to x, and convex in the last variable.