Penalization and Necessary Optimality Conditions for a Class of Nonsmooth Sweeping Processes

In this paper we derive necessary optimality conditions for a Mayer problem involving a controlled sweeping process characterized by a moving set which is merely locally prox-regular (not necessarily smooth) and satisfies a constraint qualification condition formulated exclusively in terms of its normal vectors. We employ a penalization method based on the distance from the moving constraint, which allows convergence estimates that are uniform with respect to the control and, moreover, the strong W1,2-convergence of the approximating solutions. An example of nonsmooth mechanics with finite degrees of freedom is presented.