First order sweeping process with subsmooth sets

In this paper, we introduce a perturbed first order non-convex sweeping process for a class of subsmooth moving sets depending on the time and the state. In the first result we study the existence of solution and the compactness of the attainable set, the perturbation considered here is an upper semi-continuous set-valued mapping with nonempty closed convex values unnecessarily bounded. In the second result, we prove the existence of solution for the autonomous problem under assumptions that do not require the convexity of the values of the perturbation and that weaken the assumption on the upper semi-continuity. Then, we deduce a solution of the time optimality problem and we describe the attainable set.