Asymptotic stability of uniformly bounded nonlinear switched systems

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

Asymptotic stability of uniformly bounded nonlinear switched systems

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

Abstract
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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

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