Tight bound for deciding convergence of consensus systems

摘要

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 period smaller than a certain value converge to zero. Moreover, bounds on that value are available (Lagarias and Wang, 1995).