On the stability and stabilizability of a class of continuous-time 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 exist state-feedback laws u(t) = c^Tx(t) that make the resulting controlled system positive and asymptotically stable, namely A + bc^T Metzler and Hurwitz. In the second part of the paper we assume that the state-space model switches among different state-feedback laws c_i^T, i = 1,2, ... , p, each of them ensuring the positivity, and show that the asymptotic stability of the switched system is equivalent to the asymptotic stability of all the subsystems, while its stabilizability is equivalent to the existence of an asymptotically stable subsystem.