Excess-continuous prox-regular sweeping processes

TL;DR

This paper extends the theory of Moreau's sweeping processes by proving existence and uniqueness of solutions when the moving set is continuous in time with respect to the excess, a more natural topology, and without requiring boundary continuity.

Contribution

It introduces a novel existence and uniqueness result for sweeping processes driven by prox-regular sets continuous in excess, broadening the class of admissible moving constraints.

Findings

Proves unique solutions exist under excess continuity.

Removes the need for boundary continuity of the moving set.

Extends applicability to wider classes of moving constraints.

Abstract

In this paper we consider the Moreau's sweeping processes driven by a time dependent prox-regular set $C(t)$ which is continuous in time with respect to the asymmetric distance $e$ called the excess, defined by $e(A,B) := \sup_{x \in A} d(x,B)$ for every pair of sets $A$, $B$ in a Hilbert space. As observed by J.J. Moreau in his pioneering works, the excess provides the natural topological framework for sweeping process. Assuming a uniform interior cone condition for $C(t)$, we prove that the associated sweeping process has a unique solution, thereby improving the existing result on continuous prox-regular sweeping processes in two directions: indeed, in the previous literature $C(t)$ was supposed to be continuous in time with respect to the symmetric Hausdorff distance instead of the excess and also its boundary $\partial C(t)$ was required to be continuous in time, an assumption which we completely drop. Therefore our result allows to consider a much wider class of continuously moving constraints.