Exterior Sphere Condition and Time Optimal Control for Differential Inclusions

The minimum time function T(·) of smooth control systems is known to be locally semiconcave provided Petrov’s controllability condition is satisfied. Moreover, such a regularity holds up to the boundary of the target under an inner ball assumption. We generalize this analysis to differential inclusions, replacing the above hypotheses with the continuity of T(·) near the target, and an inner ball property for the multifunction associated with the dynamics. In such a weakened setup, we prove that the hypograph of T(·) satisfies, locally, an exterior sphere condition. As is well known, this geometric property ensures most of the regularity results that hold for semiconcave functions, without assuming T(·) to be Lipschitz.