Construction of invariant Lyapunov norms for planar switching systems

Abstract

We consider the problem of the stability of linear dynamical switching systems. It is known that an irreducible $d$-dimensional system always has an invariant Lyapunov norm (a Barabanov norm), which determines the stability of the system and the rate of growth of its trajectories. We prove that in the case of $d=2$ the invariant norm is a piecewise analytic function and can be constructed explicitly for every finite system of matrices. The method of construction, an algorithm for computing the Lyapunov exponent and a way for deciding the stability of the system are presented. A complete classification of invariant norms for planar systems is derived. A criterion of the uniqueness of an invariant norm of a given system is proved, and some special norms (norms generated by polygons and so on) are investigated. Bibliography: 30 titles.