Optimal Control of the Sweeping Process with a Nonsmooth Moving Set

In this paper we prove a fully nonsmooth Pontryagin maximum principle for optimal control problems driven by a sweeping process with drift x is an element of f(t, x, u) -N-C(t)(x). The setting we study is an optimal control problem of Mayer type in which the optimization procedure is carried out by choosing a control function u(t) from a class of admissible controls U. The choice of u is an element of U modifies the drift f and the related solution x(t) to the perturbed sweeping process. Here, for the first time, we are able to prove a Pontryagin maximum principle in the case in which the moving set C(t) is both nonsmooth and nonconvex by using a novel exact penalization technique which is able to exploit the controllability properties of the dynamics.