Soient et deux champs de vecteurs lisses sur globalement asymptotiquement stables à l’origine. Nous donnons des conditions nécessaires et des conditions suffisantes sur la topologie de l’ensemble des points où et sont parallèles pour pouvoir assurer la stabilité asymptotique globale du système contrôlé non linéaire non autonome
où le contrôle est une fonction mesurable arbitraire de dans . Les conditions données ne nécessitent aucune intégration ou construction d’une fonction de Lyapunov pour être vérifiées, et sont robustes.
Let and be two smooth vector fields on , globally asymptotically stable at the origin. We give some sufficient and some necessary conditions on the topology of the set where and are parallel for global asymptotic stability of the nonautonomous and nonlinear control system
where is an arbitrary measurable function. Such conditions can be verified without any integration or construction of a Lyapunov function, and are robust.
Mots clés : stabilité asymptotique globale, commutations, non linéaire
@article{TSG_2009-2010__28__1_0, author = {Boscain, Ugo and Charlot, Gr\'egoire and Sigalotti, Mario}, title = {Stabilit\'e des syst\`emes \`a commutations du plan}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {1--12}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {28}, year = {2009-2010}, doi = {10.5802/tsg.275}, language = {fr}, url = {http://www.numdam.org/articles/10.5802/tsg.275/} }
TY - JOUR AU - Boscain, Ugo AU - Charlot, Grégoire AU - Sigalotti, Mario TI - Stabilité des systèmes à commutations du plan JO - Séminaire de théorie spectrale et géométrie PY - 2009-2010 SP - 1 EP - 12 VL - 28 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/tsg.275/ DO - 10.5802/tsg.275 LA - fr ID - TSG_2009-2010__28__1_0 ER -
%0 Journal Article %A Boscain, Ugo %A Charlot, Grégoire %A Sigalotti, Mario %T Stabilité des systèmes à commutations du plan %J Séminaire de théorie spectrale et géométrie %D 2009-2010 %P 1-12 %V 28 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/tsg.275/ %R 10.5802/tsg.275 %G fr %F TSG_2009-2010__28__1_0
Boscain, Ugo; Charlot, Grégoire; Sigalotti, Mario. Stabilité des systèmes à commutations du plan. Séminaire de théorie spectrale et géométrie, Tome 28 (2009-2010), pp. 1-12. doi : 10.5802/tsg.275. http://www.numdam.org/articles/10.5802/tsg.275/
[1] Transversal mappings and flows, An appendix by Al Kelley, W. A. Benjamin, Inc., New York-Amsterdam, 1967 | MR | Zbl
[2] Two-dimensional almost-Riemannian structures with tangency points, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 27 (2010) no. 3, pp. 793-807 | Numdam | MR | Zbl
[3] Lie-algebraic stability criteria for switched systems, SIAM J. Control Optim., Volume 40 (2001) no. 1, p. 253-269 (electronic) | MR | Zbl
[4] Uniform global asymptotic stability of differential inclusions, J. Dynam. Control Systems, Volume 10 (2004) no. 3, pp. 391-412 | MR | Zbl
[5] A note on stability conditions for planar switched systems, Internat. J. Control, Volume 82 (2009) no. 10, pp. 1882-1888 | MR | Zbl
[6] A new class of universal Lyapunov functions for the control of uncertain linear systems, IEEE Trans. Automat. Control, Volume 44 (1999) no. 3, pp. 641-647 | MR | Zbl
[7] The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry, Journal of Dynamical and Control Systems, Volume 17 (2011), pp. 141-161
[8] Stability of planar switched systems : the linear single input case, SIAM J. Control Optim., Volume 41 (2002) no. 1, p. 89-112 (electronic) | MR | Zbl
[9] Nonisotropic 3-level quantum systems : complete solutions for minimum time and minimum energy, Discrete Contin. Dyn. Syst. Ser. B, Volume 5 (2005) no. 4, p. 957-990 (electronic) | MR | Zbl
[10] Lipschitz classification of two-dimensional almost-Riemannian distances on compact oriented surfaces (article soumis)
[11] Existence of planar curves minimizing length and curvature, Proceedings of the Steklov Institute of Mathematics, Volume 270 (2010) no. 1, pp. 43-56 | MR
[12] Stability of planar nonlinear switched systems, Discrete Contin. Dyn. Syst., Volume 15 (2006) no. 2, pp. 415-432 | MR | Zbl
[13] Optimal syntheses for control systems on 2-D manifolds, Mathématiques & Applications (Berlin) [Mathematics & Applications], 43, Springer-Verlag, Berlin, 2004 | MR | Zbl
[14] High-order angles in almost-Riemannian geometry, Actes du Séminaire de Théorie Spectrale et Géométrie. Vol. 25. Année 2006–2007 (Sémin. Théor. Spectr. Géom.), Volume 25, Univ. Grenoble I, Saint, 2008, pp. 41-54 | Numdam | MR | Zbl
[15] Qualitative theory of control systems, Translations of Mathematical Monographs, 141, American Mathematical Society, Providence, RI, 1994 (Translated from the Russian manuscript by V. M. Volosov) | MR | Zbl
[16] A converse Lyapunov theorem for a class of dynamical systems which undergo switching, IEEE Trans. Automat. Control, Volume 44 (1999) no. 4, pp. 751-760 | MR | Zbl
[17] Asymptotic stability equals exponential stability, and ISS equals finite energy gain—if you twist your eyes, Systems Control Lett., Volume 38 (1999) no. 2, pp. 127-134 | MR | Zbl
[18] An infinite-time relaxation theorem for differential inclusions, Proc. Amer. Math. Soc., Volume 131 (2003) no. 2, p. 487-499 (electronic) | MR | Zbl
[19] Switching in systems and control, Systems & Control : Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 2003 | MR | Zbl
[20] Stability of switched systems : a Lie-algebraic condition, Systems Control Lett., Volume 37 (1999) no. 3, pp. 117-122 | MR | Zbl
[21] Basic problems in stability and design of switched systems, IEEE Control Systems Magazine, Volume 19 (1999), pp. 59-70
[22] Common polynomial Lyapunov functions for linear switched systems, SIAM J. Control Optim., Volume 45 (2006) no. 1, p. 226-245 (electronic) | MR | Zbl
Cité par Sources :