Geometric and asymptotic properties associated with linear switched systems

Yacine Chitour 1 Moussa Gaye 2, 3 Paolo Mason 4
1 Division Systèmes - L2S
L2S - Laboratoire des signaux et systèmes : 1289
2 GECO - Geometric Control Design
CNRS - Centre National de la Recherche Scientifique : UMR7641, X - École polytechnique, Inria Saclay - Ile de France
Abstract : Consider a continuous-time linear switched system on $\mathbb{R}^n$ associated with a compact convex set of matrices. When it is irreducible and its largest Lyapunov exponent is zero there always exists a Barabanov norm associated with the system. This paper deals with two types of issues: $(a)$ properties of Barabanov norms such as uniqueness up to homogeneity and strict convexity; $(b)$ asymptotic behaviour of the extremal solutions of the linear switched system.Regarding Issue $(a)$, we provide partial answers and propose four related open problems. As for Issue $(b)$, we establish, when $n=3$, a Poincar\'e-Bendixson theorem under a regularity assumption on the set of matrices. We then revisit a noteworthy result of N.E. Barabanov describing the asymptotic behaviour of linear switched system on $\mathbb{R}^3$ associated with a pair of Hurwitz matrices $\{A,A+bc^T\}$. After pointing out a fatal gap in Barabanov's proof we partially recover his result by alternative arguments.
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Yacine Chitour, Moussa Gaye, Paolo Mason. Geometric and asymptotic properties associated with linear switched systems. Journal of Differential Equations, Elsevier, 2015, 259 (11), pp.5582-5616. ⟨10.1016/j.jde.2015.07.001 ⟩. ⟨hal-01064241v2⟩

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