Stabilité des systèmes à commutations du plan

Ugo Boscain; Grégoire Charlot; Mario Sigalotti

doi:10.5802/tsg.275

Ugo Boscain ¹; Grégoire Charlot ²; Mario Sigalotti 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)

Séminaire de théorie spectrale et géométrie, Volume 28 (2009-2010), pp. 1-12.

Abstract
Résumé

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

\dot{q} (t) = u (t) X (q (t)) + (1 - u (t)) Y (q (t)),

where $u : [0, + \infty [\to {0, 1}$ 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 $ℝ^{2}$ 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

\dot{q} (t) = u (t) X (q (t)) + (1 - u (t)) Y (q (t))

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

DOI: 10.5802/tsg.275

Classification: 32C20, 37N35, 93D20
Keywords: stabilité asymptotique globale, commutations, non linéaire

Author's affiliations:

Ugo Boscain ¹; Grégoire Charlot ²; Mario Sigalotti ³

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     author = {Ugo Boscain and Gr\'egoire Charlot and Mario Sigalotti},
     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},
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     doi = {10.5802/tsg.275},
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     url = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.275/}
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Ugo Boscain; Grégoire Charlot; Mario Sigalotti. 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. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.275/

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