Lipschitzness of Minimal-Time Functions in Constrained Continuous-Time Systems with Applications to Reachability Analysis

The minimal-time function with respect to a closed set for a constrained continuous-time system provides the first time that a solution starting from a given initial condition reaches that set. In this paper, we propose infinitesimal necessary and sufficient conditions for the minimal-time function to be locally Lipschitz. As an application of our results, we show that, in constrained continuous-time systems, the Lipschitz continuity of the minimal-time function with respect to the boundary of the set where the solutions are defined plays a crucial role on the Lipschitz continuity of the reachable set.