Stability properties of a class of positive switched systems with rank one difference

Given a single-input continuous-time positive system, described by a pair (A,b), with A a diagonal matrix, we investigate under what conditions there exists a state-feedback law u(t)=c'x(t) that makes the resulting controlled system positive and asymptotically stable, by this meaning that A+bc⊤ is Metzler and Hurwitz. In the second part of this note we assume that the state-space model switches among different state-feedback laws (c_i',i=1,2,…,p) each of them ensuring the positivity, and show that the asymptotic stability of this type of switched system is equivalent to the asymptotic stability of all its subsystems, while its stabilizability is equivalent to the existence of an asymptotically stable subsystem.