# American Institute of Mathematical Sciences

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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

Received  November 2018 Revised  May 2019 Published  September 2019

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.

Citation: Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019212
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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 $\dot{x} = -\alpha x$
The parameters $\phi_0$ and $\phi_*.$