Regularity of Solution Maps of Differential Inclusions Under State Constraints

Abstract

Consider a differential inclusion under state constraints

$$x'(t) \in F(t,x(t)), \; x(t) \in K,$$

where $F$ is an unbounded set-valued map with closed and convex images, which is measurable in $t$ and $k(t)$-Lipschitz in $x$ (with $k(\cdot)\in L^1$) and $K\subset\mathbb{R}^{n}$ is a closed set with smooth boundary. We provide sufficient conditions for the set-valued map $\xi_0 \leadsto {\cal S}^K_{[t_0,T\,]}(\xi_0)$ associating to each initial point $\xi_0 \in K$ the set of all solutions to the above constrained differential inclusion starting at $\xi_0$ to be pseudo-Lipschitz on $K$. This result is applied to investigate local Lipschitz continuity of the value function for the constrained Bolza problem of optimal control theory.