The Minimal Time Problem of a Sweeping Process with a Discontinuous Perturbation

We investigate the minimal time problem where the dynamic data involves an autonomous sweeping process perturbed by a discontinuous dissipative Lipschitz multifunction. We show that under mild assumptions the minimal time function can be characterized in terms of Hamilton-Jacobi inequalities, one of which incorporates a limiting component that captures the discontinuous nature of the perturbation. Continuity of the minimal time function is proven under a Petrov-type condition.