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

In this paper, we consider a general Mayer optimal control problem whose dynamics is definedregionally, and, additionally, we suppose that the interface between two regions is nonsmooth in thesense that it is described by a locally Lipschitz continuous function. Our objective is to derive a hybridmaximum principle in this setting. Doing so, we consider a sequence of mollifiers which allows us toapproximate uniformly the interface between two regions by a sequence of smooth functions. This makespossible to apply the hybrid maximum principle on a sequence of approximated optimal control problemsinvolving the smooth interface in place of the nonsmooth one. By passing to the limit as the approximationparameter tends to zero, we obtain the desired necessary optimality conditions in the form of a nonsmoothhybrid maximum principle.