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

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

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 and 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), 127-133. doi: 10.1016/j.sysconle.2011.10.008.
[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), 391-412. doi: 10.1023/B:JODS.0000034437.54937.7f.
[3]	A. Bacciotti and L. Mazzi, An invariance principle for nonlinear switched systems, Systems & Control Letters, 54 (2005), 1109-1119. doi: 10.1016/j.sysconle.2005.04.003.
[4]	M. Balde and P. Jouan, Geometry of the limit sets of linear switched systems, SIAM J. Optimization and Control, 49 (2011), 1048-1063. doi: 10.1137/100793153.
[5]	M. Balde and P. Jouan, Stability of linear switched systems with quadratic bounds and observability of bilinear systems, preprint, arXiv:1201.4078.
[6]	U. Boscain, G. Charlot and M. Sigalotti, Stability of planar nonlinear switched systems, Discrete and Continuous Dynamical Systems, 15 (2006), 415-432. doi: 10.3934/dcds.2006.15.415.
[7]	J. Carr, "Applications of Centre Manifold Theory," Applied Mathematical Sciences, 35. Springer-Verlag, New York-Berlin, 1981.
[8]	J. Hespanha, Uniform stability of switched linear systems: Extensions of lasalle's invariance principle, IEEE Trans. Automat. Control, 49 (2004), 470-482. doi: 10.1109/TAC.2004.825641.
[9]	J. L. Mancilla-Aguilar and R. A. García, An extension of lasalle's invariance principle for switched systems, Systems & Control Letters, 55 (2006), 376-384. doi: 10.1016/j.sysconle.2005.07.009.
[10]	C. Marle, "Systèmes Dynamiques: Une Introduction," Ellipses, Paris, 2003.
[11]	U. Serres, J.-C. Vivalda and P. Riedinger, On the convergence of linear switched systems, IEEE Trans. Automat. Control, 56 (2011), 320-332. doi: 10.1109/TAC.2010.2054950.
[12]	E. D. Sontag, "Mathematical Control Theory. Deterministic Finite-Dimensional Systems," 2nd edition, Texts in Applied Mathematics, 6. Springer-Verlag, New York, 1998.

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), 127-133. doi: 10.1016/j.sysconle.2011.10.008.
[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), 391-412. doi: 10.1023/B:JODS.0000034437.54937.7f.
[3]	A. Bacciotti and L. Mazzi, An invariance principle for nonlinear switched systems, Systems & Control Letters, 54 (2005), 1109-1119. doi: 10.1016/j.sysconle.2005.04.003.
[4]	M. Balde and P. Jouan, Geometry of the limit sets of linear switched systems, SIAM J. Optimization and Control, 49 (2011), 1048-1063. doi: 10.1137/100793153.
[5]	M. Balde and P. Jouan, Stability of linear switched systems with quadratic bounds and observability of bilinear systems, preprint, arXiv:1201.4078.
[6]	U. Boscain, G. Charlot and M. Sigalotti, Stability of planar nonlinear switched systems, Discrete and Continuous Dynamical Systems, 15 (2006), 415-432. doi: 10.3934/dcds.2006.15.415.
[7]	J. Carr, "Applications of Centre Manifold Theory," Applied Mathematical Sciences, 35. Springer-Verlag, New York-Berlin, 1981.
[8]	J. Hespanha, Uniform stability of switched linear systems: Extensions of lasalle's invariance principle, IEEE Trans. Automat. Control, 49 (2004), 470-482. doi: 10.1109/TAC.2004.825641.
[9]	J. L. Mancilla-Aguilar and R. A. García, An extension of lasalle's invariance principle for switched systems, Systems & Control Letters, 55 (2006), 376-384. doi: 10.1016/j.sysconle.2005.07.009.
[10]	C. Marle, "Systèmes Dynamiques: Une Introduction," Ellipses, Paris, 2003.
[11]	U. Serres, J.-C. Vivalda and P. Riedinger, On the convergence of linear switched systems, IEEE Trans. Automat. Control, 56 (2011), 320-332. doi: 10.1109/TAC.2010.2054950.
[12]	E. D. Sontag, "Mathematical Control Theory. Deterministic Finite-Dimensional Systems," 2nd edition, Texts in Applied Mathematics, 6. Springer-Verlag, New York, 1998.

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