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Consider continuous-time linear switched systems on R^n associated withcompact convex sets of matrices. When the system is irreducible and the largestLyapunov exponent is equal to zero, there always exists a Barabanov norm (i.e.a norm which is non increasing along trajectories of the linear switched systemtogether with extremal trajectories starting at every point, that istrajectories of the linear switched system with constant norm). This paperdeals with two sets of issues: (a) properties of Barabanov norms such asuniqueness up to homogeneity and strict convexity; (b) asymptotic behaviour ofthe extremal solutions of the linear switched system. Regarding Issue (a), weprovide partial answers and propose four open problems motivated by appropriateexamples. As for Issue (b), we establish, when n = 3, a Poincar\'e-Bendixsontheorem under a regularity assumption on the set of matrices defining thesystem. Moreover, we revisit the noteworthy result of N.E. Barabanov [5]dealing with the linear switched system on R^3 associated with a pair ofHurwitz matrices {A, A + bcT }. We first point out a fatal gap in Barabanov'sargument in connection with geometric features associated with a Barabanovnorm. We then provide partial answers relative to the asymptotic behavior ofthis linear switched system.

Geometric and asymptotic properties associated with linear switched systems