Semiconcavity for Optimal Control Problems with Exit Time

In this paper a semiconcavity result is obtained for the value function of an optimal exit time problem. The related state equation is of general form ẏ(t) = f(y(t), u(t)), y(t) ∈ Rn, u(t) ∈ U ⊂ Rm. However, suitable assumptions are needed relating f with the running and exit costs. The semiconcavity property is then applied to obtain necessary optimality conditions, through the formulation of a suitable version of the Maximum Principle, and to study the singular set of the value function.