Uniqueness of consensus functions for time-delayed swarm systems with time-varying topologies

• The consensus function is uniquely determined if topologies are balanced.
• An explicit expression of the consensus function is given.
• The influences of initial states of agents and protocols are determined.

This paper shows that consensus functions for high-order linear swarm systems with time-varying interaction topologies and time delays are uniquely determined by initial states if interaction topologies are balanced. In addition, an explicit expression of the consensus function is presented, and the different influences of initial states of agents and protocols on the consensus function are determined respectively.