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