A stabilizable switched linear system does not necessarily admit a smooth homogeneous Lyapunov function

The contribution of this paper is twofold. Firstly, an example of a (positive) linear switched system that can be stabilized, via a controlled switching signal, but does not admit a smooth and positively homogeneous control Lyapunov function, is provided. The spectral properties of the subsystem matrices and of the Lyapunov candidates of the convex differential inclusion associated with the switched system, are thoroughly investigated. Secondly, by taking inspiration from the example, new feedback stabilization techniques for stabilizable positive switched systems are provided.