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We consider the control problem with \textit{exit time}. Unlike the Bolza andMayer problems, in this problem the terminal time of the trajectories is notfixed, but it is the first time at which they reach a given closed subset -\textit{the target}. The most studied example is the \textit{optimal timeproblem}, where we want to steer a point to the target in minimal time. In thissection, we first introduce the exit time problem, then we recall the existenceof optimal controls, and some regularity results for the value function. Wethen use a suitable form of the \textit{Pontryagin maximum principle} to studysome optimality conditions and sensitivity relations for the exit time problem.The strongest regularity property for the value function that one can expect,in fairly general cases, is \textit{semiconcavity}. In this case, the valuefunction is twice differentiable almost everywhere. Furthermore, in general, itfails to be differentiable at points where there are multiple optimaltrajectories and its differentiability at a point does not guarantee continuousdifferentiability around this point. In the subsection \ref{S3} we shown that,under suitable assumptions, the nonemptiness of proximal subdifferential of thevalue function at a point implies its continuous differentiability on aneighborhood of this point.

Optimality conditions and local regularity of the value function for the optimal exit time problem