Canonical Construction of Barabanov Polytope Norms and Antinorms for Sets of Matrices

For a given family of matrices F , we present a methodology to construct (complex) polytope Barabanov norms (and, when F is nonnegative, polytope Barabanov monotone norms and antinorms). Invariant Barabanov norms have been introduced by Barabanov and constitute an important instrument to analyze the joint spectral radius of a set of matrices. In particular, they played a key role in the disprovement of the well-known finiteness conjecture. Moreover, they allow to determine the so-called most unstable switching laws, which are of main interest when studying switched systems. However, although the Barabanov norms have been studied extensively, even in very simple cases it is very difficult to construct them and determine the shape of their unit balls (see, e.g., Kozyakin (2010)) and there are only few examples where they have been actually computed explicitely. The reasons of this is that they are not defined in a constructive way. Another interesting issue is concerned with the existence of a polytope Barabanov norm, that is a Barabanov norm with polytope unit ball for a finite family of matrices having the finiteness property. It is well known that for such families the existence of an extremal norm is somehow generic, but no results exist about Barabanov norms. In this paper we give a canonical procedure which associates a polytope extremal norm constructed by using the methodologies described in Guglielmi, Wirth and Zennaro (2005) and Guglielmi and Protasov (2013), to a polytope Barabanov norm so that we have a general procedure which determines suck kind of norm in the majority of cases. As a consequence, the existence of a polytope Barabanov norm has the same genericity of an extremal polytope norm. Moreover, we extend the result to polytope antinorms, which have been recently introduced to compute the lower spectral radius of a finite family of matrices having an invariant cone.