Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction

TL;DR

This paper investigates the stability of periodic solutions in sweeping processes with moving boundaries and applies the results to model soft crawling locomotors, demonstrating system stabilization and defining an average velocity based on gait.

Contribution

It improves existing stability results by establishing stronger convergence and applies these findings to analyze and prove the stabilization of soft locomotor models.

Findings

Proved stronger $W^{1,2}$ convergence for periodic solutions.

Established system stabilization to periodic orbits.

Defined an average asymptotic velocity depending on gait.

Abstract

We study the asymptotic stability of periodic solutions for sweeping processes defined by a polyhedron with translationally moving faces. Previous results are improved by obtaining a stronger $W^{1,2}$ convergence. Then we present an application to a model of crawling locomotion. Our stronger convergence allows us to prove the stabilization of the system to a running-periodic (or derivo-periodic, or relative-periodic) solution and the well-posedness of an average asymptotic velocity depending only on the gait adopted by the crawler. Finally, we discuss some examples of finite-time versus asymptotic-only convergence.