# American Institute of Mathematical Sciences

• Previous Article
Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion
• DCDS-B Home
• This Issue
• Next Article
The fast signal diffusion limit in nonlinear chemotaxis systems
March  2020, 25(3): 1129-1139. doi: 10.3934/dcdsb.2019212

## 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  March 2020 Early access  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, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212
##### References:

show all references

##### References:
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 $\phi_*.$
 [1] José A. Carrillo, Dejan Slepčev, Lijiang Wu. Nonlocal-interaction equations on uniformly prox-regular sets. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1209-1247. doi: 10.3934/dcds.2016.36.1209 [2] Alexander Tolstonogov. BV solutions of a convex sweeping process with a composed perturbation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021012 [3] Ayça Çeşmelioğlu, Wilfried Meidl. Bent and vectorial bent functions, partial difference sets, and strongly regular graphs. Advances in Mathematics of Communications, 2018, 12 (4) : 691-705. doi: 10.3934/amc.2018041 [4] Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations & Control Theory, 2020, 9 (3) : 753-772. doi: 10.3934/eect.2020032 [5] Marc Chamberland, Anna Cima, Armengol Gasull, Francesc Mañosas. Characterizing asymptotic stability with Dulac functions. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 59-76. doi: 10.3934/dcds.2007.17.59 [6] Jerzy Gawinecki, Wojciech M. Zajączkowski. Global regular solutions to two-dimensional thermoviscoelasticity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1009-1028. doi: 10.3934/cpaa.2016.15.1009 [7] Dalila Azzam-Laouir, Fatiha Selamnia. On state-dependent sweeping process in Banach spaces. Evolution Equations & Control Theory, 2018, 7 (2) : 183-196. doi: 10.3934/eect.2018009 [8] Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021026 [9] Ahmad Al-Salman, Ziyad AlSharawi, Sadok Kallel. Extension, embedding and global stability in two dimensional monotone maps. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4257-4276. doi: 10.3934/dcdsb.2020096 [10] Je-Chiang Tsai. Global exponential stability of traveling waves in monotone bistable systems. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 601-623. doi: 10.3934/dcds.2008.21.601 [11] Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234 [12] Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891 [13] Dmitrii Rachinskii. On geometric conditions for reduction of the Moreau sweeping process to the Prandtl-Ishlinskii operator. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3361-3386. doi: 10.3934/dcdsb.2018246 [14] Tan H. Cao, Boris S. Mordukhovich. Optimal control of a perturbed sweeping process via discrete approximations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3331-3358. doi: 10.3934/dcdsb.2016100 [15] Tan H. Cao, Boris S. Mordukhovich. Optimality conditions for a controlled sweeping process with applications to the crowd motion model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 267-306. doi: 10.3934/dcdsb.2017014 [16] Doria Affane, Meriem Aissous, Mustapha Fateh Yarou. Almost mixed semi-continuous perturbation of Moreau's sweeping process. Evolution Equations & Control Theory, 2020, 9 (1) : 27-38. doi: 10.3934/eect.2020015 [17] Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709 [18] Guangliang Zhao, Fuke Wu, George Yin. Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems. Mathematical Control & Related Fields, 2015, 5 (2) : 359-376. doi: 10.3934/mcrf.2015.5.359 [19] Sergiu Aizicovici, Simeon Reich. Anti-periodic solutions to a class of non-monotone evolution equations. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 35-42. doi: 10.3934/dcds.1999.5.35 [20] Sabri Bensid, Jesús Ildefonso Díaz. On the exact number of monotone solutions of a simplified Budyko climate model and their different stability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1033-1047. doi: 10.3934/dcdsb.2019005

2020 Impact Factor: 1.327