Nonconvex integro-differential sweeping process with applications

TL;DR

This paper studies a new type of integro-differential sweeping process involving non-convex sets and integral perturbations, proving well-posedness and exploring applications in nonlinear systems and electrical circuits.

Contribution

It introduces a novel integro-differential sweeping process with non-convex sets, establishing existence, uniqueness, and continuity of solutions, and applies these results to complex systems.

Findings

Proved unique existence of solutions for the process.

Established solution continuity with respect to initial conditions.

Applied results to nonlinear integro-differential systems and electrical circuits.

Abstract

In this paper, we analyze and discuss the well-posedness of a new variant of the so-called sweeping process, introduced by J.J. Moreau in the early 70's \cite{More71} with motivation in plasticity theory. In this variant, the normal cone to the (mildly non-convex) prox-regular moving set $C(t)$, supposed to have an absolutely continuous variation, is perturbed by a sum of a Carath\'{e}odory mapping and an integral forcing term. The integrand of the forcing term depends on two time-variables, that is, we study a general integro-differential sweeping process of Volterra type. By setting up an appropriate semi-discretization method combined with a new Gronwall-like inequality (differential inequality), we show that the integro-differential sweeping process has one and only one absolutely continuous solution. We also establish the continuity of the solution with respect to the initial value. The results of the paper are applied to the study of nonlinear integro-differential complementarity systems which are combination of Volterra integro-differential equations with nonlinear complementarity constraints. Another application is concerned with non-regular electrical circuits containing time-varying capacitors and nonsmooth electronic device like diodes. Both applications represent an additional novelty of our paper.