  This paper deals with classes of (de)stabilizing switching signals for
switched systems. Most of the available conditions for stability of switched
systems are sufficient in nature, and consequently, their violation does not
conclude instability of a switched system. The study of instability is,
however, important for obvious reasons. Our contributions are twofold: Firstly,
we propose a class of switching signals under which a continuous-time switched
linear system is unstable. Our characterization of instability depends solely
on the asymptotic behaviour of frequency of switching, frequency of transition
between subsystems, and fraction of activation of subsystems. Secondly, we show
that our class of destabilizing switching signals is a strict subset of the
class of switching signals that does not satisfy asymptotic characterization of
stability recently proposed in the literature. This observation identifies a
gap between asymptotic characterizations of stabilizing and destabilizing
switching signals for switched linear systems. The main apparatus for our
analysis is multiple Lyapunov-like functions.
