Lipschitz Continuity of Optimal Controls for State Constrained Problems

Abstract

This paper provides new conditions under which optimal controls are Lipschitz continuous for dynamic optimization problems with functional inequality constraints, a control constraint expressed in terms of a general closed convex set and a coercive cost function. It is shown that the linear independence condition on active state constraints, present in the earlier literature, can be replaced by a less restrictive, positive linear independence condition that requires linear independence merely with respect to nonnegative weighting parameters. Smoothness conditions on the data, imposed in earlier work, are also relaxed. The new conditions for Lipschitz continuity of optimal controls are obtained by a detailed analysis of the implications of first order optimality conditions in the form of a nonsmooth maximum principle.