Convergence of velocities for the short range communicated discrete-time Cucker–Smale model

Most existing literature about the discrete-time Cucker−Smale model focus on the asymptotic flocking behavior. When the communication weight has a long range, asymptotic flocking holds for any initial data. Actually, the velocity of every agent will exponentially converge to the same limit in this case. However, when the communication weight has a short range, asymptotic flocking does not exist for general initial data. In this note, we will prove the convergence of velocities for any initial data in the short range communication case. We first propose a new strategy about the convergence of velocities, and then show an important inequality about the velocity–position moment, according to which we will successfully prove the convergence of velocities and obtain the convergence rates for two kinds of communication weights. Besides, for some special initial data we show that the limits of velocities can be different from each other. Simulation results are given to validate the theoretical results.