4.4 Article

FINITE-TIME CLUSTER SYNCHRONIZATION OF COUPLED DYNAMICAL SYSTEMS WITH IMPULSIVE EFFECTS

Journal

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
Volume 26, Issue 7, Pages 3595-3620

Publisher

AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/dcdsb.2020248

Keywords

Cluster synchronization; coupled dynamical systems; finite-time stability; impulsive effects; differential inequality

Funding

  1. National Natural Science Foundation of China [11902137]
  2. China Postdoctoral Science Foundation [2019M651633]
  3. Key Project of Natural Science Foundation of China [61833005]

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In this paper, the finite-time cluster synchronization problem for coupled dynamical systems in networks is investigated. Two novel Lyapunov-based finite-time stability results are proposed using impulsive differential equation theory, differential inequality method, and synchronization and desynchronization impulsive effects. By adding impulsive control input, the attractive domain of finite-time stability can be enlarged, while destabilizing impulses can reduce the attractive domain. Both impulsive control and continuous feedback control are effective in stabilizing systems.
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.

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