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

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.

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.

DOI : https://doi.org/10.5802/tsg.275
Classification:  32C20,  37N35,  93D20
Keywords:
@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},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {28},
     year = {2009-2010},
     pages = {1-12},
     doi = {10.5802/tsg.275},
     language = {fr},
     url = {http://www.numdam.org/item/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, Volume 28 (2009-2010) , pp. 1-12. doi : 10.5802/tsg.275. http://www.numdam.org/item/TSG_2009-2010__28__1_0/

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