On nonconvex differential inclusions whose state is constrained in the closure of an open set. Applications to dynamic programming (Q1854085)

The authors present two main results which extend the Filippov theorem to nonconvex differential inclusions with constrained state variable. One involves a mapping \(F\) being Lipschitz in \((t,x)\) and the boundary of the set of state variable constraints \(\Theta \) being locally Lipschitz; the other one requires merely a measurability of \(F\) in \(t\) but the boundary of \(\Theta \) has to be sufficiently smooth. The theorems are applied to prove some continuity properties of the value function for the Boltz problem with state constraints and with the control set depending on both time and state.