Lipschitz Continuity of the Value Function for the Infinite Horizon Optimal Control Problem Under State Constraints

Abstract

This paper investigates sufficient conditions for Lipschitz regularity of the value function for an infinite horizon optimal control problem subject to state constraints. We focus on problems with a cost functional that includes a discount rate factor and allow time dependent dynamics and Lagrangian. Furthermore, our state constraints may be unbounded and with nonsmooth boundary. The key technical result used in our proof is an estimate on the distance of a given trajectory from the set of all its viable (feasible) trajectories (provided the discount rate is sufficiently large). These distance estimates are derived under a uniform inward pointing condition on the state constraint and imply, in particular, that feasible trajectories depend on initial states in a Lipschitz way with an exponentially increasing in time Lipschitz constant. As a corollary, we show that the value function of the original problem coincides with the value function of the relaxed infinite horizon problem.