# Barabanov norms, Lipschitz continuity and monotonicity for the max algebraic joint spectral radius

## TL;DR

This paper explores the properties of the max algebraic joint spectral radius (JSR), establishing its relation to matrix norms, existence of monotone Barabanov norms, and its continuity and monotonicity characteristics within certain matrix sets.

## Contribution

It extends classical results to max algebra, demonstrating the existence of Barabanov norms and establishing Lipschitz and Hölder continuity of the max algebraic JSR.

## Key findings

- Max algebraic JSR can be characterized via induced matrix norms.
- A monotone Barabanov norm exists for irreducible matrix sets.
- The max algebraic JSR is locally Lipschitz and Hölder continuous.

## Abstract

We present several results describing the interplay between the max algebraic joint spectral radius (JSR) for compact sets of matrices and suitably defined matrix norms. In particular, we extend a classical result for the conventional algebra, showing that the JSR can be described in terms of induced norms of the matrices in the set. We also show that for a set generating an irreducible semigroup (in a cone-theoretic sense), a monotone Barabanov norm always exists. This fact is then used to show that the max algebraic JSR is locally Lipschitz continuous on the space of compact irreducible sets of matrices with respect to the Hausdorff distance. We then prove that the JSR is Hoelder continuous on the space of compact sets of nonnegative matrices. Finally, we prove a strict monotonicity property for the max algebraic JSR that echoes a fact for the classical JSR.