Properties of Barabanov norms and extremal trajectories associated with continuous-time linear switched systems

Consider continuous-time linear switched systems on ℝⁿ associated with compact convex sets of matrices. When the system is irreducible and the largest Lyapunov exponent is equal to zero, a Barabanov norm always exists. This paper deals with two sets 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 system. Regarding Issue (a), we provide partial answers and propose two open problems motivated by appropriate examples. As for Issue (b), we establish, when n = 3, a Poincaré-Bendixson theorem under a regularity assumption on the set of matrices defining the system.