A Necessary and Sufficient Condition for Consensus of Continuous-Time Agents Over Undirected Time-Varying Networks

The average consensus problem of continuous-time agents in undirected time-varying networks is studied. The network is allowed to be disconnected. A notion called infinite integral connectivity is proposed. Based on the notion, a necessary and sufficient condition for achieving consensus is given. That is, when the network topology is described by an undirected time-varying graph G ( t ), the agents achieve consensus if and only if the infinite integral graph of G ( t ) over [0,∞) is connected. This criterion does not hold for directed networks.