Co-Positive Lyapunov Functions for the Stabilization of Positive Switched Systems

In this paper, exponential stabilizability of continuous-time positive switched systems is investigated. For two-dimensional systems, exponential stabilizability by means of a switching control law can be achieved if and only if there exists a Hurwitz convex combination of the (Metzler) system matrices. In the higher dimensional case, it is shown by means of an example that the existence of a Hurwitz convex combination is only sufficient for exponential stabilizability, and that such a combination can be found if and only if there exists a smooth, positively homogeneous and co-positive control Lyapunov function for the system. In the general case, exponential stabilizability ensures the existence of a concave, positively homogeneous and co-positive control Lyapunov function, but this is not always smooth. The results obtained in the first part of the paper are exploited to characterize exponential stabilizability of positive switched systems with delays, and to provide a description of all the “switched equilibrium points” of an affine positive switched system.