SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 130-148, January 2021.
We present a simple proof of asymptotic consensus in the discrete Hegselmann--Krause model and flocking in the discrete Cucker--Smale model with normalization and variable delay. This proof utilizes the convexity of the normalized communication weights and a Gronwall--Halanay-type inequality. The main advantage of our method, compared to previous approaches to the delay Hegselmann--Krause model, is that it does not require any restriction on the maximal time delay, or the initial data, or decay rate of the influence function. From this point of view the result is optimal. For the Cucker--Smale model it provides an analogous result in the regime of unconditonal flocking with sufficiently slowly decaying communication rate, but still without any restriction on the length of the maximal time delay. Moreover, we demonstrate that the method can be easily extended to the mean-field limits of both the Hegselmann--Krause and Cucker--Smale systems, using appropriate stability results on the measure-valued solutions.

中文翻译：

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具有归一化和时滞的Hegselmann-Krause和Cucker-Smale模型中渐近共识的简单证明

SIAM应用动力系统杂志，第20卷，第1期，第130-148页，2021年1月。
我们提供了离散Hegselmann-Krause模型中渐近共识的简单证明，以及具有归一化和可变时滞的离散Cucker-Smale模型中的植绒。该证明利用归一化通信权重的凸性和Gronwall-Halanay型不等式。与先前的延迟Hegselmann-Krause模型方法相比，我们的方法的主要优势在于，它不需要对最大时间延迟，初始数据或影响函数的衰减率进行任何限制。从这个角度来看，结果是最佳的。对于Cucker-Smale模型，它在无条件植绒的情况下提供了类似的结果，通讯速率已足够缓慢地衰减，但对最大时间延迟的长度仍然没有任何限制。此外，