Flocking of non-identical Cucker-Smale models on general coupling network

Abstract

The purpose of the paper is to investigate the flocking behavior of the discrete-time Cucker-Smale(C-S) model under general interaction network topologies with agents having their free-will accelerations. We prove theoretically that if the free-will accelerations of agents are summable, then, for any given initial conditions, the solution achieves flocking with a finite moving speed by suitably choosing the time step as well as the communication rate of the system or the strength of the interaction between agents. In particular, if the communication rate \begin{document}$ \beta $\end{document} of the system is subcritical, i.e., \begin{document}$ \beta $\end{document} is less than a critical value \begin{document}$ \beta_c $\end{document} , then flocking holds for any initial conditions regardless of the strength of the interaction between agents. While, if the communication rate is critical ( \begin{document}$ \beta = \beta_c $\end{document} ) or supercritical ( \begin{document}$ \beta > \beta_c $\end{document} ), then flocking can only be achieved by making the strength of the interaction large enough. We also present some numerical simulations to support our obtained theoretical results.