On the Existence of Lipschitz Continuous Optimal Feedback Control

Abstract

We consider an optimal control problem involving a nonlinear ODE with control, an integral cost functional, and a control constraint. Our main assumptions include a coercivity condition and the condition that the optimal control is an isolated solution of the variational inequality appearing in the first-order optimality condition. We show that the optimal open-loop control is Lipschitz continuous in time; moreover, we identify the dependence of the Lipschitz constant of the optimal control on the data of the problem. Then, we establish the existence of a Lipschitz continuous optimal feedback control. As an application, we study regularity properties of the optimal value function. A main tool for obtaining these results is the property of uniform strong metric regularity.