# A note on stability conditions for planar switched systems

Abstract : This paper is concerned with the stability problem for the planar linear switched system $\dot x(t)=u(t)A_1x(t)+(1-u(t))A_2x(t)$, where the real matrices $A_1,A_2\in \R^{2\times 2}$ are Hurwitz and $u(\cdot) [0,\infty[\to\{0,1\}$ is a measurable function. We give coordinate-invariant necessary and sufficient conditions on $A_1$ and $A_2$ under which the system is asymptotically stable for arbitrary switching functions $u(\cdot)$. The new conditions unify those given in previous papers and are simpler to be verified since we are reduced to study 4 cases instead of 20. Most of the cases are analyzed in terms of the function $\Gamma(A_1,A_2)=\frac{1}{2}(\tr(A_1) \tr(A_2)- \tr(A_1A_2))$.
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https://hal.archives-ouvertes.fr/hal-00323682
Contributor : Paolo Mason <>
Submitted on : Wednesday, September 24, 2008 - 11:14:04 AM
Last modification on : Friday, January 1, 2021 - 4:02:05 PM
Long-term archiving on: : Tuesday, September 21, 2010 - 5:09:47 PM

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### Citation

Moussa Balde, Ugo Boscain, Paolo Mason. A note on stability conditions for planar switched systems. International Journal of Control, Taylor & Francis, 2009, 82 (10), pp.1882-1888. ⟨10.1080/00207170902802992Actions⟩. ⟨hal-00323682v2⟩

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