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

0578413 - MÚ 2024 RIV FR eng J - Journal Article
Gudoshnikov, Ivan - Makarenkov, O. - Rachinskii, D.
Formation of a nontrivial finite-time stable attractor in a class of polyhedral sweeping processes with periodic input.
ESAIM-Control Optimisation and Calculus of Variations. Roč. 29, November (2023), č. článku 84. ISSN 1292-8119. E-ISSN 1262-3377
R&D Projects: GA ČR(CZ) GA20-14736S
Grant - others:AV ČR(CZ) L100192151
Institutional support: RVO:67985840
Keywords : sweeping process * finite-time stability * convex polyhedra * elastoplasticity
OECD category: Pure mathematics
Impact factor: 1.4, year: 2022
Method of publishing: Open access
https://doi.org/10.1051/cocv/2023074

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.
Permanent Link: https://hdl.handle.net/11104/0347408