Consensus problems for linear time-varying swarm systems with topology variances and multiple time delays

Consensus problems for linear swarm systems with time-varying interaction topologies and multiple time delays are investigated. Several necessary and sufficient conditions for consensus are given, which transform consensus problems into asymptotically stable problems. Explicit expressions of consensus functions are presented on the basis of the different impacts of initial states of agents, initial states of protocols and time delays. Especially, it is shown that if interaction topologies associated with different time delays are balanced, then consensus functions are uniquely determined by initial states. Numerical simulations are presented to demonstrate theoretical results.