Second-order linear switching systems with arbitrary control sets: stability and invariant norms

TL;DR

This paper provides explicit methods for analyzing stability and constructing Barabanov norms for planar linear switching systems with arbitrary control sets, revealing conditions for uniqueness and smoothness of these norms.

Contribution

It introduces explicit solutions for stability and Barabanov norms in 2D linear switching systems with arbitrary control sets, including classification and examples.

Findings

Invariant norms are unique and smooth if no dominant matrix with real spectrum exists.

Multiple non-smooth invariant norms can exist when such a dominant matrix is present.

All symmetric convex bodies can serve as unit balls for Barabanov norms in these systems.

Abstract

We show that the stability problem and the problem of constructing Barabanov norms can be resolved for planar linear switching systems in an explicit form. This can be done for every compact control set of $2 \times 2$ matrices. If the control set does not contain a dominant matrix with a real spectrum, then the invariant norm is always unique (up to a multiplier) and belongs to~$C^1$. Otherwise, there may be infinitely many such norms, including non-smooth ones. All of them can be found and classified. In particular, every symmetric convex body is a unit ball of the Barabanov norm of a suitable linear switching system. Several examples of control sets such as matrix Frobenius balls and matrix polyhedra are analysed.