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.

有关离散时间Cucker的大多数现有文献$-$Smale模型专注于渐进式植绒行为。当通信权重范围很长时，对于任何初始数据而言，渐近植绒都会成立。实际上，在这种情况下，每个代理的速度将指数收敛至相同的极限。但是，当通信权重在短范围内时，对于常规初始数据不存在渐近植绒。在本说明中，我们将证明在短距离通信情况下任何初始数据的速度收敛性。我们首先提出了一种关于速度收敛的新策略，然后表明了速度-位置矩的一个重要不等式，据此，我们将成功证明速度的收敛并获得两种通信权重的收敛速度。除了，对于某些特殊的初始数据，我们表明速度的极限可能彼此不同。仿真结果验证了理论结果的正确性。