Continuous-Time Consensus under Persistent Connectivity and Slow Divergence of Reciprocal Interaction Weights

In this paper, we develop new results on consensus for continuous-time multi-agent systems. We introduce the assumptions of persistent connectivity of the interaction graph and of slow divergence of reciprocal interaction weights. Persistent connectivity can be considered as the counterpart of the notion of ultimate connectivity used in discrete-time consensus protocols. Slow divergence of reciprocal interaction weights generalizes the case on uniformly bounded weights on weakly symmetric graphs. We show that under these two new assumptions, the continuous-time consensus protocol succeeds: the states of all the agents converge asymptotically to a common value. We also provide two examples that make us think that both of our assumptions are tight.