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

Giovanni Colombo; Paolo Gidoni; Emilio Vilches

doi:10.3934/dcds.2021135

Article Contents

2022, Volume 42, Issue 2: 737-757. Doi: 10.3934/dcds.2021135

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

1.
Department of Mathematics "Tullio Levi-Civita", University of Padua, via Trieste 63, 35121 Padova (Italy)
2.
Institute of Information Theory and Automation, Czech Academy of Sciences, Pod vodárenskou věží 4, CZ-182 08 Praha 8 (Czech Republic)
3.
Instituto de Ciencias de la Ingeniería, Universidad de O'Higgins, Av. Libertador Bernardo O'Higgins 611, 2841959 Rancagua (Chile)

^*Corresponding author
^*Corresponding author

Received: March 2021

Revised: July 2021

Early access: September 2021

Published: February 2022

G.C. was partially supported by Padua University grant SID 2018 "Controllability, stabilizability and infimum gaps for control systems", prot. BIRD 187147, and by GNAMPA of INdAM. P.G. was partially supported by by the GAČR–FWF grant 19-29646L. E.V. was partially supported by ANID-Chile under grant Fondecyt de Iniciación N$ ^{\circ} $ 11180098.

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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.

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Mathematics Subject Classification: Primary: 70K42, 34A60; Secondary: 34D45, 37C25.

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Figure 1. The model of soft crawler

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Figure 2. The orbit (red) of a solution $ z(t) $ of counterexample 5.2, converging only asymptotically to a constant solution $ \bar z $. The set $ C(t) $ is periodically sliding vertically. Notice that the solution will stop for progressively longer intervals at the edges of the orbit. The set $ \mathcal{Z} $ of periodic solutions is the set of constant functions with value in the light-gray triangle

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Figure 3. The orbit (red) of a solution $ z(t) $ of counterexample 5.3, converging only asymptotically to a periodic solution

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