Hybrid maximum principle for regional optimal control problems with non-smooth interfaces

Overview & Synopsis

In this paper, we consider a general Mayer optimal control problem whose dynamics is defined regionally, and, additionally, we suppose that the interface between two regions is nonsmooth in the sense that it is described by a locally Lipschitz continuous function. Our objective is to derive a hybrid maximum principle in this setting. Doing so, we consider a sequence of mollifiers which allows us to approximate uniformly the interface between two regions by a sequence of smooth functions. This makes possible to apply the hybrid maximum principle on a sequence of approximated optimal control problems involving the smooth interface in place of the nonsmooth one. By passing to the limit as the approximation parameter tends to zero, we obtain the desired necessary optimality conditions in the form of a nonsmooth hybrid maximum principle.