Solution Regularity in Optimal Control via Subgradient Analysis of the Value Function

Lipschitz regularity for solutions to nonsmooth problems in the calculus of variations is considered, where we allow for extended-real-valued Lagrangians. Conditions for regularity are given in terms of subdifferentiability properties of the value function. In particular, it is shown that there exists a Lipschitz minimizer as long as the limiting subgradient of the value function is nonempty at the point in question. The results improve upon previous results in the literature, even in the case of finite Lagrangians.