Semiconcavity of the value function for exit time problems with nonsmooth target

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Abstract
We prove semiconcavity of the value function of a nonlinear optimal control problem where the cost functional depends on the arrival time of the trajectory on a given target set. We make suitable smoothness assumptions on the dynamics of the system, while the target set can be completely general. As a corollary, we prove differentiability of the value function for a class of linear systems.