Geometric and asymptotic properties associated with linear switched systems

Yacine Chitour (1) , Moussa Gaye (2, 3) , Paolo Mason (1)

1 L2S - Laboratoire des signaux et systèmes
2 GECO - Geometric Control Design
3 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

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.

Mots clés: switched systems; stability; Barabanov norm

Domaines: Optimisation et contrôle [math.OC]

Yacine Chitour, Moussa Gaye, Paolo Mason. Geometric and asymptotic properties associated with linear switched systems. Journal of Differential Equations, 2015, 259 (11), pp.5582-5616. ⟨10.1016/j.jde.2015.07.001⟩. ⟨hal-01064241v2⟩