Tight Bounds for Consensus Systems Convergence

We analyze the asymptotic convergence of all infinite products of matrices taken in a given finite set by looking only at finite or periodic products. It is known that when the matrices of the set have a common nonincreasing polyhedral norm, all infinite products converge to zero if and only if all infinite periodic products with periods smaller than a certain value converge to zero. Moreover, bounds on that value are available [1]. We provide a stronger bound that holds for both polyhedral norms and polyhedral seminorms. In the latter case, the matrix products do not necessarily converge to 0, but all trajectories of the associated system converge to a common invariant subspace. We prove that our bound is tight for all seminorms. Our work is motivated by problems in consensus systems, where the matrices are stochastic (nonnegative with rows summing to one), and hence always share a same common nonincreasing polyhedral seminorm. In that case, we also improve existing results.