On the Regularity of Semipermeable Surfaces in Control Theory with Application to the Optimal Exit-time Problem (part Ii)

In control theory, a semipermeable surface is an (in general nonsmooth) oriented surface that, on one hand, contains solutions (the so-called barrier solutions) of the controlled system and, on the other hand, may be crossed by the solutions of this system in only one direction. Without making any assumption on the regularity of the boundary of the semipermeable surface, we show that the barrier solutions contained in this semipermeable surface satisfy the Pontryagin principle, that this surface is a Lipschitz manifold, and that it is, locally, the graph of a semiconcave function. Applying these results to the optimal exit-time function from a given open set yields, without any controllability assumption at the boundary of the open set, that this function is semiconcave on an open dense subset of its domain.