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.