Formation of a nontrivial finite-time stable attractor in a class of polyhedral sweeping processes with periodic input

¹ 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: ivangudoshnikov@gmail.com

Received: 13 July 2023
Accepted: 12 October 2023

Abstract

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.