Inexact Catching-Up Algorithm for Moreau's Sweeping Processes

TL;DR

This paper introduces an inexact catching-up algorithm for Moreau's sweeping processes, allowing approximate projections compatible with various numerical methods, and applies it to electrical circuits with ideal diodes.

Contribution

It develops a novel inexact algorithm with a new approximate projection concept, extending convergence analysis to broader set classes and practical electrical circuit applications.

Findings

Convergence proven for prox-regular, subsmooth, and closed sets.

Implemented in electrical circuits with ideal diodes.

Recovers classical existence results and offers new numerical insights.

Abstract

In this paper, we develop an inexact version of the catching-up algorithm for sweeping processes. We define a new notion of approximate projection, which is compatible with any numerical method for approximating exact projections, as this new notion is not restricted to remain strictly within the set. We provide several properties of the new approximate projections, which enable us to prove the convergence of the inexact catching-up algorithm in three general frameworks: prox-regular moving sets, subsmooth moving sets, and merely closed sets. Additionally, we apply our numerical results to address complementarity dynamical systems, particularly electrical circuits with ideal diodes. In this context, we implement the inexact catching-up algorithm using a primal-dual optimization method, which typically does not necessarily guarantee a feasible point. Our results are illustrated through an electrical circuit with ideal diodes. Our results recover classical existence results in the literature and provide new insights into the numerical simulation of sweeping processes.