Volume 29, 2023
|Number of page(s)||42|
|Published online||17 November 2023|
Formation of a nontrivial finite-time stable attractor in a class of polyhedral sweeping processes with periodic input
Institute of Mathematics of the Czech Academy of Sciences, Žitná 609/25, 115 67, Praha 1, Czech Republic
2 Department of Mathematical Sciences, the University of Texas at Dallas, 800 West Campbell Road, Richardson, TX 75080, USA
* Corresponding author: email@example.com
Accepted: 12 October 2023
We consider a differential inclusion known as a polyhedral sweeping process. The general sweeping process was introduced by J.-J. Moreau as a modeling framework for quasistatic deformations of elastoplastic bodies, and a polyhedral sweeping process is typically used to model stresses in a network of elastoplastic springs. Krejčí’s theorem states that a sweeping process with periodic input has a global attractor which consists of periodic solutions, and all such periodic solutions follow the same trajectory up to a parallel translation. We show that in the case of polyhedral sweeping process with periodic input the attractor has to be a convex polyhedron χ of a fixed shape. We provide examples of elastoplastic spring models leading to structurally stable situations where χ is a one- or two- dimensional polyhedron. In general, an attractor of a polyhedral sweeping process may be either exponentially stable or finite-time stable and the main result of the paper consists of sufficient conditions for finite-time stability of the attractor, with upper estimates for the settling time. The results have implications for the shakedown theory.
Mathematics Subject Classification: 34A60 / 34D45 / 93D40 / 47J22 / 74C05
Key words: sweeping process / finite-time stability / convex polyhedra / elastoplasticity / differential inclusions / shakedown theory
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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