Exponential convergence of multiagent systems with lack of connection

Finding conditions ensuring consensus, i.e. convergence to a common value, for a networked system is of crucial interest, both for theoretical reasons and applications. This goal is harder to achieve when connections between agents are temporarily lost. Here, we prove that known conditions (introduced by Moreau) ensure an exponential convergence to consensus, with explicit rate of convergence. The key result is related to the length of the graph (i.e. the number of connections to reach a common agent): if this is large, then convergence is slow. This general result also provides conditions for convergence of second-order cooperative systems with lack of connections.