Lipschitz regularity of the minimum time function of differential inclusions with state constraints

For control systems, the local regularity of the minimum time function τ_min in the absence of state constraints has been extensively studied and related both to inward-pointing conditions and to small-time controllability in the neighborhood of a closed target C. In the presence of state constraints, assessing this regularity is crucial to ensure the existence of solutions when perturbing the initial condition. In this paper, we prove, without imposing the inclusion C⊂IntK, that, for differential inclusions with closed state constraints K and under general assumptions, τ_min is locally Lipschitz continuous on its domain which is open in K. We discuss as well extensions to nonautonomous systems and to point targets.