Cucker-Smale Flocking Under General Interaction Topologies∗

In the literature, the study of the Cucker-Smale (C-S) model is all restricted to several special interaction topologies. In this paper, we establish the flocking behavior of the C-S model under general interaction topologies, which contain all existing topology structures in relation to the C-S model as special cases. In particular, the topology with multiple leaders is included. The flocking results are guaranteed under some conditions on the system parameters and the initial states only. The critical exponent below which unconditional convergence holds is given, depending only on the interaction topology. It is consistent with the ones obtained with some known special topologies in the literature. The convergence results are shown by employing a new combination of perturbation method and algebraic graph theory.