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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 |
  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.
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. |
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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. |
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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. |
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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. |
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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. |
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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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