  We investigate the stabilizability of discrete-time linear switched systems,
when the sole control action of the controller is the switching signal, and
when the controller has access to the state of the system in real time. Despite
their apparent simplicity, determining if such systems are stabilizable appears
to be a very challenging problem, and basic examples have been known for long,
for which the stabilizability question is open. We provide new results allowing
us to bound the so-called stabilizability radius, which characterizes the
stabilizability property of discrete-time linear switched systems. These
results allow us to compute significantly improved explicit lower bounds on the
stabilizability radius for the above-mentioned examples. As a by-product, we
exhibit a discontinuity property for this problem, which brings theoretical
understanding of its complexity.
