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

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    *Corresponding author 

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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  • 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.

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


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  • Figure 1.  The model of soft crawler

    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

    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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