Prox-regular sweeping processes with bounded retraction

TL;DR

This paper establishes the existence and uniqueness of solutions for prox-regular sweeping processes driven by moving sets with bounded retraction, and reduces the bounded retraction case to the Lipschitz case via reparametrization.

Contribution

It proves solution uniqueness under prox-regularity conditions and introduces a reparametrization technique to simplify the analysis of bounded retraction cases.

Findings

Unique solutions exist for the sweeping process under certain prox-regularity conditions.

The bounded retraction case can be reduced to the Lipschitz case using reparametrization.

The catching-up algorithm effectively solves the Lipschitz continuous case.

Abstract

The aim of this paper is twofold. On one hand we prove that the Moreau's sweeping process driven by a uniformly prox-regular moving set with local bounded retraction has a unique solution provided that the coefficient of prox-regularity is larger than the size of any jump of the driving set. On the other hand we show how the case of local bounded retraction can be easily reduced to the $1$-Lipschitz continuous case: indeed we first solve the Lipschitz continuous case by means of the so called ``catching-up algorithm", and we reduce the local bounded retraction case to the Lipschitz one by using a reparametrization technique for functions with values in the family of prox-regular sets.