May  2006, 15(2): 415-432. doi: 10.3934/dcds.2006.15.415

Stability of planar nonlinear switched systems

1. 

SISSA, via Beirut 2-4 34014 Trieste

2. 

ACSIOM, I3M, CC51, Université Montpellier II, Place Eugène Bataillon, 34095 Montpellier Cedex 5

3. 

Institut Élie Cartan, UMR 7502 UHP/CNRS/INRIA, POB 239, 54506 Vandœuvre-lès-Nancy, France

Received  March 2005 Revised  December 2005 Published  March 2006

Let $X$ and $Y$ be two smooth vector fields on $\R^2$, globally asymptotically stable at the origin, and consider the time-dependent nonlinear system $\dot q(t)=u(t)X(q(t))+(1-u(t))Y(q(t))$, where $u:[0,\infty)\to\{0,1\}$ is an arbitrary measurable function. Analyzing the topology of the set where $X$ and $Y$ are parallel, we give some sufficient and some necessary conditions for global asymptotic stability, uniform with respect to $u(.)$. Such conditions can be verified without any integration or construction of a Lyapunov function, and they do not change under small perturbations of the vector fields.
Citation: Ugo Boscain, Grégoire Charlot, Mario Sigalotti. Stability of planar nonlinear switched systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 415-432. doi: 10.3934/dcds.2006.15.415
[1]

Philippe Jouan, Said Naciri. Asymptotic stability of uniformly bounded nonlinear switched systems. Mathematical Control & Related Fields, 2013, 3 (3) : 323-345. doi: 10.3934/mcrf.2013.3.323

[2]

Moussa Balde, Ugo Boscain. Stability of planar switched systems: The nondiagonalizable case. Communications on Pure & Applied Analysis, 2008, 7 (1) : 1-21. doi: 10.3934/cpaa.2008.7.1

[3]

Balázs Boros, Josef Hofbauer, Stefan Müller, Georg Regensburger. Planar S-systems: Global stability and the center problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 707-727. doi: 10.3934/dcds.2019029

[4]

Gregory Berkolaiko, Cónall Kelly, Alexandra Rodkina. Sharp pathwise asymptotic stability criteria for planar systems of linear stochastic difference equations. Conference Publications, 2011, 2011 (Special) : 163-173. doi: 10.3934/proc.2011.2011.163

[5]

Honglei Xu, Kok Lay Teo, Weihua Gui. Necessary and sufficient conditions for stability of impulsive switched linear systems. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1185-1195. doi: 10.3934/dcdsb.2011.16.1185

[6]

Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 199-208. doi: 10.3934/dcdsb.2017010

[7]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727

[8]

E. Cabral Balreira, Saber Elaydi, Rafael Luís. Local stability implies global stability for the planar Ricker competition model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 323-351. doi: 10.3934/dcdsb.2014.19.323

[9]

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

[10]

Lin Du, Yun Zhang. $\mathcal{H}_∞$ filtering for switched nonlinear systems: A state projection method. Journal of Industrial & Management Optimization, 2018, 14 (1) : 19-33. doi: 10.3934/jimo.2017035

[11]

Teng Wang, Yi Wang. Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation. Kinetic & Related Models, 2019, 12 (3) : 637-679. doi: 10.3934/krm.2019025

[12]

Christian Lax, Sebastian Walcher. A note on global asymptotic stability of nonautonomous master equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2143-2149. doi: 10.3934/dcdsb.2013.18.2143

[13]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019212

[14]

J.E. Muñoz Rivera, Reinhard Racke. Global stability for damped Timoshenko systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1625-1639. doi: 10.3934/dcds.2003.9.1625

[15]

Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991

[16]

Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063

[17]

Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations & Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493

[18]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325

[19]

Shiwang Ma, Xiao-Qiang Zhao. Global asymptotic stability of minimal fronts in monostable lattice equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 259-275. doi: 10.3934/dcds.2008.21.259

[20]

Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 203-209. doi: 10.3934/dcdss.2020011

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]