# American Institute of Mathematical Sciences

July  2021, 26(7): 3595-3620. doi: 10.3934/dcdsb.2020248

## Finite-time cluster synchronization of coupled dynamical systems with impulsive effects

 1 School of Mathematics, Southeast University, Nanjing 210096, China 2 Department of Mathematics, Luoyang Normal University, Luoyang 471934, China 3 Department of Applied Mathematics, Changsha University of Science and Technology, Changsha 410114, China

* Corresponding author: Jinde Cao

Received  December 2019 Revised  June 2020 Published  July 2021 Early access  August 2020

In our paper, the finite-time cluster synchronization problem is investigated for the coupled dynamical systems in networks. Based on impulsive differential equation theory and differential inequality method, two novel Lyapunov-based finite-time stability results are proposed and be used to obtain the finite-time cluster synchronization criteria for the coupled dynamical systems with synchronization and desynchronization impulsive effects, respectively. The settling time with respect to the average impulsive interval is estimated according to the sufficient synchronization conditions. It is illustrated that the introduced settling time is not only dependent on the initial conditions, but also dependent on the impulsive effects. Compared with the results without stabilizing impulses, the attractive domain of the finite-time stability can be enlarged by adding impulsive control input. Conversely, the smaller attractive domain can be obtained when the original system is subject to the destabilizing impulses. By using our criteria, the continuous feedback control can always be designed to finite-time stabilize the unstable impulsive system. Several existed results are extended and improved in the literature. Finally, typical numerical examples involving the large-scale complex network are outlined to exemplify the availability of the impulsive control and continuous feedback control, respectively.

Citation: Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248
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##### References:
Phase plots of (a). the system (61) and (b). the system (62) in Example 1
Time histories of (a). the coupled system without control input, (b-d). the variables $x_{i1}$, $x_{i2}$ and $x_{i3}$ of the coupled system with synchronization impulsive effect in Example 1
Under control input, time histories of (a). the error function $E(t)$ in Eq. (63), (b-d). the variables $e_{i1}$, $e_{i2}$ and $e_{i3}$ of the synchronization error system in Example 1
With desynchronization impulses, time histories of (a-c). the variables $x_{i1}$, $x_{i2}$ and $x_{i3}$ of the coupled system with nonidentical nodes (61) and (62), (d-f). the variables $e_{i1}$, $e_{i2}$ and $e_{i3}$ of the synchronization error system in Example 1
Time histories of (a-c). the variables $x_{i1}$, $x_{i2}$ and $x_{i3}$ of the complex networks, (d-f). the variables $e_{i1}$, $e_{i2}$ and $e_{i3}$ of the synchronization error system in Example 2
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