A continuation principle for periodic BV-continuous state-dependent sweeping processes

TL;DR

This paper establishes the existence of periodic solutions in state-dependent BV-continuous sweeping processes using a catching-up algorithm, extending understanding of their solvability and periodic behavior.

Contribution

It introduces a new approach to prove solvability and periodic solutions for BV-continuous state-dependent sweeping processes, including near boundary equilibria.

Findings

Existence of solutions for initial value problems.

Existence of at least one T-periodic solution.

Periodic solutions near boundary equilibria.

Abstract

We consider a Caratheodory differential equation with a state-dependent convex constraint that changes BV-continuously in time (a perturbed BV-continuous state-dependent sweeping processes). By setting up an appropriate catching-up algorithm we prove solvability of the initial value problem. Then, for sweeping processes with $T$-periodic right-hand-sides, we prove the existence of at least one $T$-periodic solution. Finally, we further consider a $T$-periodic sweeping process which is close to an autonomous sweeping process with a constant constraint and prove the existence of a $T$-periodic solution specifically located near the boundary switched equilibrium of the autonomous sweeping process.