Local regularity of the value function in optimal control

It is well-known that the value function VV of a Bolza optimal control problem fails to be everywhere differentiable. In this paper, however, we show that, if VV is proximally subdifferentiable at (t,x)(t,x), then it is smooth on a neighborhood of (t,x)(t,x). Our result yields that VV 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.