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We study the time optimal control problem with a general target $\mathcal S$for a class of differential inclusions that satisfy mild smoothness andcontrollability assumptions. In particular, we do not require Petrov'scondition at the boundary of $\mathcal S$. Consequently, the minimum timefunction $T(\cdot)$ fails to be locally Lipschitz---never mindsemiconcave---near $\mathcal S$. Instead of such a regularity, we use anexterior sphere condition for the hypograph of $T(\cdot)$ to develop theanalysis. In this way, we obtain dual arc inclusions which we apply to show theconstancy of the Hamiltonian along optimal trajectories and other optimalityconditions in Hamiltonian form. We also prove an upper bound for the Hausdorffmeasure of the set of all nonlipschitz points of $T(\cdot)$ which implies thatthe minimum time function is of special bounded variation.

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