Sweeping process by prox-regular sets in Riemannian Hilbert manifolds

TL;DR

This paper extends the theory of sweeping processes to Riemannian Hilbert manifolds, introducing local prox-regularity and proving well-posedness of perturbed processes in this geometric setting.

Contribution

It generalizes proximal normal cone and prox-regularity concepts to infinite-dimensional Riemannian manifolds and establishes well-posedness results for sweeping processes in this context.

Findings

Prox-regularity implies hypomonotonicity of the proximal normal cone.

Metric projection onto locally prox-regular sets is single-valued nearby.

Well-posedness of perturbed sweeping processes is proven under new geometric conditions.

Abstract

In this paper, we deal with sweeping processes on (possibly infinite-dimensional) Riemannian Hilbert manifolds. We extend the useful notions (proximal normal cone, prox-regularity) already defined in the setting of a Hilbert space to the framework of such manifolds. Especially we introduce the concept of local prox-regularity of a closed subset in accordance with the geometrical features of the ambient manifold and we check that this regularity implies a property of hypomonotonicity for the proximal normal cone. Moreover we show that the metric projection onto a locally prox-regular set is single-valued in its neighborhood. Then under some assumptions, we prove the well-posedness of perturbed sweeping processes by locally prox-regular sets.