# American Institute of Mathematical Sciences

May  2006, 15(2): 415-432. doi: 10.3934/dcds.2006.15.415

## Stability of planar nonlinear switched systems

 1 SISSA, via Beirut 2-4 34014 Trieste 2 ACSIOM, I3M, CC51, Université Montpellier II, Place Eugène Bataillon, 34095 Montpellier Cedex 5 3 Institut Élie Cartan, UMR 7502 UHP/CNRS/INRIA, POB 239, 54506 Vandœuvre-lès-Nancy, France

Received  March 2005 Revised  December 2005 Published  March 2006

Let $X$ and $Y$ be two smooth vector fields on $\R^2$, globally asymptotically stable at the origin, and consider the time-dependent nonlinear 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. Analyzing the topology of the set where $X$ and $Y$ are parallel, we give some sufficient and some necessary conditions for global asymptotic stability, uniform with respect to $u(.)$. Such conditions can be verified without any integration or construction of a Lyapunov function, and they do not change under small perturbations of the vector fields.
Citation: Ugo Boscain, Grégoire Charlot, Mario Sigalotti. Stability of planar nonlinear switched systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 415-432. doi: 10.3934/dcds.2006.15.415
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