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

Abstract 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 $W^{1,2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> -convergence of the approximating solutions. An example of nonsmooth mechanics with finite degrees of freedom is presented.