Geometric and asymptotic properties associated with linear switched systems

TL;DR

This paper investigates the geometric and asymptotic properties of linear switched systems, focusing on Barabanov norms, their uniqueness, convexity, and the long-term behavior of extremal solutions, especially in three-dimensional cases.

Contribution

It provides new insights into Barabanov norms, addresses open problems on their properties, and establishes a Poincaré-Bendixson theorem for three-dimensional systems under certain conditions.

Findings

Existence of Barabanov norms when the Lyapunov exponent is zero.

Partial answers to the uniqueness and convexity of Barabanov norms.

A Poincaré-Bendixson theorem for 3D linear switched systems.

Abstract

Consider continuous-time linear switched systems on R^n associated with compact convex sets of matrices. When the system is irreducible and the largest Lyapunov 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 system together with extremal trajectories starting at every point, that is trajectories of the linear switched system with constant norm). 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 linear switched system. Regarding Issue (a), we provide partial answers and propose four open problems motivated by appropriate examples. As for Issue (b), we establish, when n = 3, a Poincar\'e-Bendixson theorem under a regularity assumption on the set of matrices defining the system. Moreover, we revisit the noteworthy result of N.E. Barabanov [5] dealing with the linear switched system on R^3 associated with a pair of Hurwitz matrices {A, A + bcT }. We first point out a fatal gap in Barabanov's argument in connection with geometric features associated with a Barabanov norm. We then provide partial answers relative to the asymptotic behavior of this linear switched system.