Integral Formulation and the Br\'ezis-Ekeland-Nayroles-Type Principle for Prox-Regular Sweeping Processes

TL;DR

This paper introduces a new integral formulation for sweeping processes driven by prox-regular sets, establishing equivalence with the differential formulation and providing a variational characterization for stability and approximation in nonconvex settings.

Contribution

It develops a unified bounded-variation solution concept and a Brézis-Ekeland-Nayroles-type variational principle for prox-regular sweeping processes, extending analysis to nonconvex, discontinuous constraints.

Findings

Proves equivalence between integral and differential formulations under mild regularity conditions.

Establishes a variational residual characterization for solutions, enabling stability analysis.

Provides a framework for approximation and stability in nonconvex sweeping processes.

Abstract

We study sweeping processes in a Hilbert space driven by time-dependent uniformly prox-regular sets, allowing the moving constraint to exhibit discontinuities of bounded variation. We introduce a new integral formulation for bounded-variation trajectories, given by a global variational inequality tested against continuous admissible trajectories, and we compare it with the standard differential-measure formulation, in which the differential measure of the trajectory is constrained by the proximal normal cone. In the prox-regular (generally nonconvex) framework, the variational inequality necessarily includes a quadratic correction term reflecting the hypomonotonicity of proximal normal cones. Under mild regularity assumptions on the moving set, including lower semicontinuity in time, uniform prox-regularity of the values, and a selection-extension property guaranteeing a rich class of test trajectories (satisfied, for instance, in the convex case and for bounded prox-regular sets), we prove that the new integral formulation is equivalent to the differential-measure formulation. This yields a unified bounded-variation notion of solution for prox-regular sweeping processes. We further establish a Br\'ezis-Ekeland-Nayroles-type variational characterization via a prox-regular variational residual: the residual is nonpositive along every admissible trajectory, and solutions are exactly those trajectories for which this residual attains its maximal value, namely zero. As a consequence, we prove a stability result: a uniform limit of admissible trajectories with vanishing residual is a solution of the limit sweeping process. The resulting variational framework provides a robust tool for stability and approximation analyses in the prox-regular, nonconvex setting.