On a nonconvex, nonsmooth control system

Abstract

In this paper we study boundary trajectories of a control system and what happens to them after convexifying or, in other words, relaxing the system. We present an example of a very regular control system and its trajectory which goes along the boundary of reachable sets but belongs to the interior of reachable sets of the corresponding convexified system. A boundary trajectory may fall into the interior of reachable sets after relaxing the system but still remains “extremal” for the relaxed system. We give a simple proof of a version of the maximum principle which says that a boundary trajectory of a nonconvex, nonsmooth system satisfies the maximum principle not only with every ordinary but also with every relaxed control corresponding to it. This strenghens significantly the maximum principle: we present an example in which a trajectory can be eliminated as a candidate to be a boundary trajectory using relaxed controls but cannot be eliminated using only ordinary controls. Consider a control system governed by the equation