Pierre-Yves Chevalier

Pierre-Yves Chevalier, Julien M. Hendrickx, Raphaël M. Jungers

We consider the problem of determining the existence of a sequence of matrices driving a discrete-time consensus system to consensus. We transform this problem into one of the existence of a product of the transition (stochastic) matrices that has a positive column. We then generalize some results from automata theory to sets of stochastic matrices. We obtain as a main result a polynomial-time algorithm to decide the existence of a sequence of matrices achieving consensus.

Pierre-Yves Chevalier, Julien M. Hendrickx, Raphaël M. Jungers

We address the problem of determining if a discrete time switched consensus system converges for any switching sequence and that of determining if it converges for at least one switching sequence. For these two problems, we provide necessary and sufficient conditions that can be checked in singly exponential time. As a side result, we prove the existence of a polynomial time algorithm for the first problem when the system switches between only two subsystems whose corresponding graphs are undirected, a problem that had been suggested to be NP-hard by Blondel and Olshevsky.