Stabilité des systèmes à commutations du plan
Séminaire de théorie spectrale et géométrie, Tome 28 (2009-2010), pp. 1-12.

Soient X et Y deux champs de vecteurs lisses sur 2 globalement asymptotiquement stables à l’origine. Nous donnons des conditions nécessaires et des conditions suffisantes sur la topologie de l’ensemble des points où X et Y sont parallèles pour pouvoir assurer la stabilité asymptotique globale du système contrôlé non linéaire non autonome

q ˙(t)=u(t)X(q(t))+(1-u(t))Y(q(t))

où le contrôle u est une fonction mesurable arbitraire de [0,+[ dans {0,1}. Les conditions données ne nécessitent aucune intégration ou construction d’une fonction de Lyapunov pour être vérifiées, et sont robustes.

Let X and Y be two smooth vector fields on R 2 , globally asymptotically stable at the origin. We give some sufficient and some necessary conditions on the topology of the set where X and Y are parallel for global asymptotic stability of the nonautonomous and nonlinear control system

q ˙(t)=u(t)X(q(t))+(1-u(t))Y(q(t)),

where u:[0,+[{0,1} is an arbitrary measurable function. Such conditions can be verified without any integration or construction of a Lyapunov function, and are robust.

DOI : 10.5802/tsg.275
Classification : 32C20, 37N35, 93D20
Mots clés : stabilité asymptotique globale, commutations, non linéaire
Boscain, Ugo 1 ; Charlot, Grégoire 2 ; Sigalotti, Mario 3

1 tabacckludge ’Ecole polytechnique CMAP Route de Saclay 91128 Palaiseau cedex (France)
2 Université Grenoble 1 Institut Fourier 100 rue des maths BP 74 38402 St Martin d’Hères cedex (France)
3 Université Nancy 1 Institut Élie Cartan de Nancy BP 70239 54506 Vandœuvre-lès-Nancy cedex (France)
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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://archive.numdam.org/articles/10.5802/tsg.275/

[1] Abraham, R.; Robbin, J. Transversal mappings and flows, An appendix by Al Kelley, W. A. Benjamin, Inc., New York-Amsterdam, 1967 | MR | Zbl

[2] Agrachev, A. A.; Boscain, U.; Charlot, G.; Ghezzi, R.; Sigalotti, M. 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] Agrachev, A. A.; Liberzon, D. Lie-algebraic stability criteria for switched systems, SIAM J. Control Optim., Volume 40 (2001) no. 1, p. 253-269 (electronic) | MR | Zbl

[4] Angeli, D.; Ingalls, B.; Sontag, E. D.; Wang, Y. Uniform global asymptotic stability of differential inclusions, J. Dynam. Control Systems, Volume 10 (2004) no. 3, pp. 391-412 | MR | Zbl

[5] Balde, M.; Boscain, U.; Mason, P. A note on stability conditions for planar switched systems, Internat. J. Control, Volume 82 (2009) no. 10, pp. 1882-1888 | MR | Zbl

[6] Blanchini, F.; Miani, S. 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] Bonnard, B.; Charlot, G.; Ghezzi, R.; Janin, G. 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] Boscain, U. 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] Boscain, U.; Chambrion, Th.; Charlot, G. 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] Boscain, U.; Charlot, G.; Ghezzi, R.; Sigalotti, M. Lipschitz classification of two-dimensional almost-Riemannian distances on compact oriented surfaces (article soumis)

[11] Boscain, U.; Charlot, G.; Rossi, F. 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] Boscain, U.; Charlot, G.; Sigalotti, M. Stability of planar nonlinear switched systems, Discrete Contin. Dyn. Syst., Volume 15 (2006) no. 2, pp. 415-432 | MR | Zbl

[13] Boscain, U.; Piccoli, B. Optimal syntheses for control systems on 2-D manifolds, Mathématiques & Applications (Berlin) [Mathematics & Applications], 43, Springer-Verlag, Berlin, 2004 | MR | Zbl

[14] Boscain, U.; Sigalotti, M. 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] Davydov, A. A. 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] Dayawansa, W. P.; Martin, C. F. 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] Grüne, L.; Sontag, E. D.; Wirth, F. R. 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] Ingalls, B.; Sontag, E. D.; Wang, Y. An infinite-time relaxation theorem for differential inclusions, Proc. Amer. Math. Soc., Volume 131 (2003) no. 2, p. 487-499 (electronic) | MR | Zbl

[19] Liberzon, D. Switching in systems and control, Systems & Control : Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 2003 | MR | Zbl

[20] Liberzon, D.; Hespanha, J. P.; Morse, A. S. Stability of switched systems : a Lie-algebraic condition, Systems Control Lett., Volume 37 (1999) no. 3, pp. 117-122 | MR | Zbl

[21] Liberzon, D.; Morse, A. S. Basic problems in stability and design of switched systems, IEEE Control Systems Magazine, Volume 19 (1999), pp. 59-70

[22] Mason, P.; Boscain, U.; Chitour, Y. Common polynomial Lyapunov functions for linear switched systems, SIAM J. Control Optim., Volume 45 (2006) no. 1, p. 226-245 (electronic) | MR | Zbl

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