American Institute of Mathematical Sciences

Advanced
September  2013, 3(3): 323-345. doi: 10.3934/mcrf.2013.3.323

Asymptotic stability of uniformly bounded nonlinear switched systems

Philippe Jouan 1, and  Said Naciri 1,

1. 

Université de Rouen, Laboratoire de Mathématiques Raphaël Salem, CNRS, UMR 6085, Avenue de luniversité, BP 12, 76801 Saint-Etienne du Rouvray Cedex, France, France

Received  October 2012 Revised  March 2013 Published  September 2013

We study the asymptotic stability properties of nonlinear switched systems under the assumption of the existence of a common weak Lyapunov function.
    We consider the class of nonchaotic inputs, which generalize the different notions of inputs with dwell-time, and the class of general ones. For each of them we provide some sufficient conditions for asymptotic stability in terms of the geometry of certain sets.
Keywords: weak Lyapunov functions, omega-limit sets., asymptotic stability, Switched systems, chaotic signals.
Mathematics Subject Classification: Primary: 93C10, 93D20; Secondary: 37N3.
Citation: 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
References:
[1]

A. Yu. Aleksandrov, A. A. Kosov and A. V. Platonov, On the asymptotic stability of switched homogeneous systems,, Systems & Control Letters, 61 (2012), 127.  doi: 10.1016/j.sysconle.2011.10.008.  Google Scholar

[2]

D. Angeli, B. Ingalls, E. D. Sontag and Y. Wang, Uniform global asymptotic stability of differential inclusions,, Journal of Dynamical and Control Systems, 10 (2004), 391.  doi: 10.1023/B:JODS.0000034437.54937.7f.  Google Scholar

[3]

A. Bacciotti and L. Mazzi, An invariance principle for nonlinear switched systems,, Systems & Control Letters, 54 (2005), 1109.  doi: 10.1016/j.sysconle.2005.04.003.  Google Scholar

[4]

M. Balde and P. Jouan, Geometry of the limit sets of linear switched systems,, SIAM J. Optimization and Control, 49 (2011), 1048.  doi: 10.1137/100793153.  Google Scholar

[5]

M. Balde and P. Jouan, Stability of linear switched systems with quadratic bounds and observability of bilinear systems, preprint,, , ().   Google Scholar

[6]

U. Boscain, G. Charlot and M. Sigalotti, Stability of planar nonlinear switched systems,, Discrete and Continuous Dynamical Systems, 15 (2006), 415.  doi: 10.3934/dcds.2006.15.415.  Google Scholar

[7]

J. Carr, "Applications of Centre Manifold Theory,", Applied Mathematical Sciences, (1981).   Google Scholar

[8]

J. Hespanha, Uniform stability of switched linear systems: Extensions of lasalle's invariance principle,, IEEE Trans. Automat. Control, 49 (2004), 470.  doi: 10.1109/TAC.2004.825641.  Google Scholar

[9]

J. L. Mancilla-Aguilar and R. A. García, An extension of lasalle's invariance principle for switched systems,, Systems & Control Letters, 55 (2006), 376.  doi: 10.1016/j.sysconle.2005.07.009.  Google Scholar

[10]

C. Marle, "Systèmes Dynamiques: Une Introduction,", Ellipses, (2003).   Google Scholar

[11]

U. Serres, J.-C. Vivalda and P. Riedinger, On the convergence of linear switched systems,, IEEE Trans. Automat. Control, 56 (2011), 320.  doi: 10.1109/TAC.2010.2054950.  Google Scholar

[12]

E. D. Sontag, "Mathematical Control Theory. Deterministic Finite-Dimensional Systems,", 2nd edition, (1998).   Google Scholar

show all references

References:
[1]

A. Yu. Aleksandrov, A. A. Kosov and A. V. Platonov, On the asymptotic stability of switched homogeneous systems,, Systems & Control Letters, 61 (2012), 127.  doi: 10.1016/j.sysconle.2011.10.008.  Google Scholar

[2]

D. Angeli, B. Ingalls, E. D. Sontag and Y. Wang, Uniform global asymptotic stability of differential inclusions,, Journal of Dynamical and Control Systems, 10 (2004), 391.  doi: 10.1023/B:JODS.0000034437.54937.7f.  Google Scholar

[3]

A. Bacciotti and L. Mazzi, An invariance principle for nonlinear switched systems,, Systems & Control Letters, 54 (2005), 1109.  doi: 10.1016/j.sysconle.2005.04.003.  Google Scholar

[4]

M. Balde and P. Jouan, Geometry of the limit sets of linear switched systems,, SIAM J. Optimization and Control, 49 (2011), 1048.  doi: 10.1137/100793153.  Google Scholar

[5]

M. Balde and P. Jouan, Stability of linear switched systems with quadratic bounds and observability of bilinear systems, preprint,, , ().   Google Scholar

[6]

U. Boscain, G. Charlot and M. Sigalotti, Stability of planar nonlinear switched systems,, Discrete and Continuous Dynamical Systems, 15 (2006), 415.  doi: 10.3934/dcds.2006.15.415.  Google Scholar

[7]

J. Carr, "Applications of Centre Manifold Theory,", Applied Mathematical Sciences, (1981).   Google Scholar

[8]

J. Hespanha, Uniform stability of switched linear systems: Extensions of lasalle's invariance principle,, IEEE Trans. Automat. Control, 49 (2004), 470.  doi: 10.1109/TAC.2004.825641.  Google Scholar

[9]

J. L. Mancilla-Aguilar and R. A. García, An extension of lasalle's invariance principle for switched systems,, Systems & Control Letters, 55 (2006), 376.  doi: 10.1016/j.sysconle.2005.07.009.  Google Scholar

[10]

C. Marle, "Systèmes Dynamiques: Une Introduction,", Ellipses, (2003).   Google Scholar

[11]

U. Serres, J.-C. Vivalda and P. Riedinger, On the convergence of linear switched systems,, IEEE Trans. Automat. Control, 56 (2011), 320.  doi: 10.1109/TAC.2010.2054950.  Google Scholar

[12]

E. D. Sontag, "Mathematical Control Theory. Deterministic Finite-Dimensional Systems,", 2nd edition, (1998).   Google Scholar

[1]

Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671
[2]

Bruce Kitchens, Michał Misiurewicz. Omega-limit sets for spiral maps. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 787-798. doi: 10.3934/dcds.2010.27.787
[3]

Andrew D. Barwell, Chris Good, Piotr Oprocha, Brian E. Raines. Characterizations of $\omega$-limit sets in topologically hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1819-1833. doi: 10.3934/dcds.2013.33.1819
[4]

Hongyong Cui, Peter E. Kloeden, Meihua Yang. Forward omega limit sets of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1103-1114. doi: 10.3934/dcdss.2020065
[5]

Michael Schönlein. Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4053-4069. doi: 10.3934/dcds.2017172
[6]

Jianfeng Lv, Yan Gao, Na Zhao. The viability of switched nonlinear systems with piecewise smooth Lyapunov functions. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2020048
[7]

Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231
[8]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195
[9]

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
[10]

José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781
[11]

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
[12]

Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631
[13]

Noah H. Rhee, PaweŁ Góra, Majid Bani-Yaghoub. Predicting and estimating probability density functions of chaotic systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 297-319. doi: 10.3934/dcdsb.2017144
[14]

Huijuan Li, Junxia Wang. Input-to-state stability of continuous-time systems via finite-time Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 841-857. doi: 10.3934/dcdsb.2019192
[15]

Volodymyr Pichkur. On practical stability of differential inclusions using Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1977-1986. doi: 10.3934/dcdsb.2017116
[16]

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
[17]

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
[18]

Marc Chamberland, Anna Cima, Armengol Gasull, Francesc Mañosas. Characterizing asymptotic stability with Dulac functions. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 59-76. doi: 10.3934/dcds.2007.17.59
[19]

Gunther Dirr, Hiroshi Ito, Anders Rantzer, Björn S. Rüffer. Separable Lyapunov functions for monotone systems: Constructions and limitations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2497-2526. doi: 10.3934/dcdsb.2015.20.2497
[20]

Jóhann Björnsson, Peter Giesl, Sigurdur F. Hafstein, Christopher M. Kellett. Computation of Lyapunov functions for systems with multiple local attractors. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4019-4039. doi: 10.3934/dcds.2015.35.4019

2018 Impact Factor: 1.292

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences