On the Stability of Positive Linear Switched Systems Under Arbitrary Switching Laws

We consider n-dimensional positive linear switched systems. A necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is Hurwitz. Several researchers conjectured that for positive linear switched systems this condition is also sufficient. Recently, Gurvits, Shorten, and Mason showed that this conjecture is true for the case n = 2, but is not true in general. Their results imply that there exists some minimal integer np such that the conjecture is true for all n < n_p, but is not true for n = n_p. We show that n_p = 3.

I. Introduction

Consider the linear switched system

$$\mathdot{\mmb x}(t)=A_{\sigma(t)}{\mmb x}(t),\quad{\mmb x}(0)={\mmb x}_{0}\eqno{\hbox{(1)}}$$ where , , and is a piecewise constant function of time, referred to as the switching law. Roughly speaking, this models a system that may switch between the two linear subsystems: and .