Local regularity of the value function in optimal control
Résumé
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.