Geometric and asymptotic properties associated with linear switched systems

Consider a continuous-time linear switched system on Rn 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 Poincaré-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 R3 associated with a pair of Hurwitz matrices {A, A+bcT }. After pointing out a fatal gap in Barabanov’s proof we partially recover his result by alternative arguments.