On the regularity of optimal controls for state constrained problems

Abstract

In this paper we summarize new results on the regularity of optimal controls 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. Recently it has been 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 non-negative weighting parameters, provided the control constraint set is non-time varying. We show that, if the control constraint set, regarded as a time dependent multifunction, is merely Lipschitz continuous with respect to the time variable, then optimal controls can fail to be Lipschitz continuous. In these circumstance, however, a weaker regularity property (Holder continuity with Holder index 1/2) can be established. On the other hand, Lipschitz continuity of optimal controls is guaranteed for time varying control sets under a positive linear independence hypothesis, when the control constraint sets are described, at each time, by a finite collection of functional inequalities.