Regularity of the state constrained minimal time function

The regularity of the state constrained minimal time function is studied. We generalize [P. Wolenski, Y. Zhuang, Proximal analysis and the minimal time function, SIAM J. Control Optim. 36 (1998) 1048–1072, Theorem 6.1] in which Wolenski and Zhuang give necessary and sufficient conditions for Lipschitz continuity of the unconstrained minimal time function, and discuss certain ramifications.