Local regularity of the value function in optimal control

It is well-known that the value function V of a Bolza optimal control problem fails to be everywhere differentiable. In this paper, however, we show that, if V is proximally subdifferentiable at ( t , x ) , then it is smooth on a neighborhood of ( t , x ) . Our result yields that V stays smooth on a neighborhood of any optimal trajectory starting at a point where the proximal subdifferential is nonempty. This leads to sufficient conditions for the regularity of optimal trajectories and optimal controls.