  Every irreducible discrete-time linear switching system possesses an
invariant convex Lyapunov function (Barabanov norm), which provides a very
refined analysis of trajectories. Until recently that notion remained rather
theoretical apart from special cases. In 2015 N.Guglielmi and M.Zennaro showed
that many systems possess at least one simple Barabanov norm, which moreover,
can be efficiently computed. In this paper we classify all possible Barabanov
norms for discrete-time systems. We prove that, under mild assumptions, such
norms are unique and are either piecewise-linear or piecewise quadratic. Those
assumptions can be verified algorithmically and the numerical experiments show
that a vast majority of systems satisfy them. For some narrow classes of
systems, there are more complicated Barabanov norms but they can still be
classified and constructed. Using those results we find all trajectories of the
fastest growth. They turn out to be eventually periodic with special periods.
Examples and numerical results are presented.
