Global asymptotic stability of nonconvex sweeping processes

Lakmi Niwanthi Wadippuli; Ivan Gudoshnikov; Oleg Makarenkov

doi:10.3934/dcdsb.2019212

Article Contents

2020, Volume 25, Issue 3: 1129-1139. Doi: 10.3934/dcdsb.2019212

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Global asymptotic stability of nonconvex sweeping processes

Department of Mathematical Sciences, University of Texas at Dallas, 75080 Richardson, USA

^* Corresponding author: Oleg Makarenkov
^* Corresponding author: Oleg Makarenkov

Received: October 31, 2018

Revised: April 30, 2019

Early access: September 2019

Published: March 2020

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Abstract

Building upon the technique that we developed earlier for perturbed sweeping processes with convex moving constraints and monotone vector fields (Kamenskii et al, Nonlinear Anal. Hybrid Syst. 30, 2018), the present paper establishes the conditions for global asymptotic stability of global and periodic solutions to perturbed sweeping processes with prox-regular moving constraints. Our conclusion can be formulated as follows: closer the constraint to a convex one, weaker monotonicity is required to keep the sweeping process globally asymptotically stable. We explain why the proposed technique is not capable to prove global asymptotic stability of a periodic regime in a crowd motion model (Cao-Mordukhovich, DCDS-B 22, 2017). We introduce and analyze a toy model which clarifies the extent of applicability of our result.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Illustrations of the notations of the example. The closed ball centered at $ (-1.5, 0) $ is $ \bar B_1 $ and the white ellipses are the graphs of $ S(t) $ for different values of the argument. The arrows is the vector field of $ \dot{x} = -\alpha x $

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Figure 2. The parameters $ \phi_0 $ and $ \phi_*. $

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