This article is devoted to studying the problem of globally and exponentially adaptive cluster synchronization for a class of Lur'e dynamical networks with multiple time-varying delays and hybrid couplings. A novel impulsive adaptive pinning feedback control protocol is proposed with fully considering the cluster-tree topology structures of the networks and only imposed on the Lur'e systems in current cluster which exist directed paths with those Lur'e systems in the other clusters. In view of the concept of average impulsive interval, the comparison principle, the extended parameter variation methods, and the reductio ad absurdum, sufficient conditions are acquired for achieving the cluster synchronization of the derivative coupled Lur'e networks with considering different functions of the impulsive effects, respectively. Furthermore, according to the designed adaptive updating law, suitable feedback control strengths are obtained, which largely save the control costs. Finally, the effectiveness of the control schemes and the theoretical results have been proved by executing two numerical simulation examples.

中文翻译：

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基于参数变化协议的具有簇树拓扑结构的 Lur'e 网络的脉冲自适应钉扎同步

本文致力于研究一类具有多个时变时滞和混合耦合的Lur'e动态网络的全局和指数自适应集群同步问题。提出了一种新的脉冲自适应钉扎反馈控制协议，充分考虑了网络的簇树拓扑结构，并且只强加于当前簇中与其他簇中的那些 Lur'e 系统存在有向路径的 Lur'e 系统。鉴于平均脉冲间隔的概念、比较原理、扩展参数变化方法和归约性，为考虑脉冲的不同函数实现导数耦合Lur'e网络的簇同步获得了充分条件。效果，分别。此外，根据设计的自适应更新律，获得合适的反馈控制强度，大大节省了控制成本。最后，通过两个数值模拟实例验证了控制方案的有效性和理论结果。