Optimality conditions and regularity results for time optimal control problems with differential inclusions

We study the time optimal control problem with a general target S for a class of differential inclusions that satisfy mild smoothness and controllability assumptions. In particular, we do not require Petrov’s condition at the boundary of S. Consequently, the minimum time function T(·) fails to be locally Lipschitz— never mind semiconcave—near S. Instead of such a regularity, we use an exterior sphere condition for the hypograph of T(·) to develop the analysis. In this way, we obtain dual arc inclusions which we apply to show the constancy of the Hamiltonian along optimal trajectories and other optimality conditions in Hamiltonian form. We also prove an upper bound for the Hausdorff measure of the set of all non-Lipschitz points of T(·) which implies that the minimum time function is of special bounded variation (SBV).