Flocking of a Cucker–Smale Type Model with Compactly Supported Interaction Functions

How to analyze flocking behaviors of a multi-agent system with local interaction functions is a challenging problem in theory. Motsch and Tadmor in 2011 also stressed the significance to assume that the interaction function is rapidly decaying or cut-off at a finite distance (cf. Motsch and Tadmor in J. Stat. Phys. 2011). In this paper, we study the flocking behavior of a Cucker–Smale type model with compactly supported interaction functions. Using properties of a connected stochastic matrix, together with an elaborate analysis on perturbations of a linearized system, we obtain a sufficient condition imposed only on model parameters and initial data to guarantee flocking. Moreover, it is shown that the system achieves flocking at an exponential rate.