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 set-up, 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.