Continuous-time consensus under persistent connectivity and slow   divergence of reciprocal interaction weights
  In this paper, we present 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 assumption of cut-balanced interactions. We show that under these two assumptions, the continuous-time consensus protocol succeeds: the states of all the agents converge asymptotically to a common value. Moreover, our proof allows us to give an estimate of the rate of convergence towards the consensus. We also provide two examples that make us think that both of our assumptions are tight. 
Samuel Martin, Antoine Girard
math.OC