Some remarks on the stabilisation problem and the stabilisability radius for linear switched systems

We discuss stabilisability issues for discrete-time linear switched systems, assuming that one can use the switching law as a control function either in feedback form or as an open loop control. Stabilisability properties may be characterised through suitable indices such as the joint spectral subradius and the stabilisability radius. The paper focuses on the numerical computation of these quantities, especially for what concerns lower bounds, and investigates conditions under which they are different. Our analysis shed light on some counterintuitive phenomena related to stabilisation of discrete-time linear switched systems. In particular, one can construct examples of systems which are (exponentially) stabilisable for every initial condition, yet no periodic stabilising switching law exists, except for a set of initial conditions of measure zero.