# American Institute of Mathematical Sciences

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

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.
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

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##### 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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