Is stabilization of switched positive linear systems equivalent to the existence of an Hurwitz convex combination of the system matrices?

In this paper exponential stabilizability of continuous-time positive switched systems is investigated. It is proved that, when dealing with two-dimensional systems, exponential stabilizability can be achieved if and only if there exists an Hurwitz convex combination of the (Metzler) system matrices. However, for systems of higher dimension this is not true. In general, exponential stabilizability corresponds to the existence of a (positively homogeneous, concave and co-positive) control Lyapunov function, but this function is not necessarily smooth. The existence of an Hurwitz convex combination is equivalent to the stronger condition that the system is not only exponentially stable, but it also admits a smooth control Lyapunov function. These two conditions, in turn, are equivalent to the fact that the stabilizing switching law can always be based on a linear co-positive control Lyapunov function. Finally, the characterization of exponential stabilizability is exploited to provide a description of all the “switched equilibrium points” of a positive affine switched system.