Cluster synchronization of community network with distributed time delays via impulsive control

Abstract

Cluster synchronization is an important dynamical behavior in community networks and deserves further investigations. A community network with distributed time delays is investigated in this paper. For achieving cluster synchronization, an impulsive control scheme is introduced to design proper controllers and an adaptive strategy is adopted to make the impulsive controllers unified for different networks. Through taking advantage of the linear matrix inequality technique and constructing Lyapunov functions, some synchronization criteria with respect to the impulsive gains, instants, and system parameters without adaptive strategy are obtained and generalized to the adaptive case. Finally, numerical examples are presented to demonstrate the effectiveness of the theoretical results.

Synchronization of delayed neural networks with hybrid coupling via impulsive control

“…As is well known, the delay-dependent criteria are less conservative than the delay-independent criteria for the systems with small delays. Therefore, for the coupled neural networks with small transmission delay τ ( t ) and distributed delay h 0 , our results are less conservative than the results obtained in He et al (2017); Leng and Wu (2016).…”

“…Remark 3: According to Theorem 3, the synchronization criteria obtained here are dependent on both the distributed delay h 0 and the upper bound of the transmission delay τ ( t ) in equation (11). In recent years, the synchronization of delayed neural networks with distributed-delay coupling has been investigated in He et al (2017) and Leng and Wu (2016). However, the synchronization criteria in Leng and Wu (2016) were independent on the distributed-delay, and the results in He et al (2017) were dependent on the distributed-delay only.…”

“…In recent years, the synchronization of delayed neural networks with distributed-delay coupling has been investigated in He et al (2017) and Leng and Wu (2016). However, the synchronization criteria in Leng and Wu (2016) were independent on the distributed-delay, and the results in He et al (2017) were dependent on the distributed-delay only. As is well known, the delay-dependent criteria are less conservative than the delay-independent criteria for the systems with small delays.…”

“…During the 1970s, some authors pointed out that the use of distributed-delays in biology can lead the models to be more tractable and more realistic than those with discrete-delays (MacDonald, 1980). It is thus necessary and important to study the synchronization problem of the coupled neural networks with distributed-delay coupling (Cao et al, 2008; He et al, 2017; Hu et al, 2014; Leng and Wu, 2016; Li et al, 2015; Yuan, 2009). Several dynamical properties of the neural networks with distributed-delay were investigated, such as passivity analysis (Wen et al, 2015; Yu and Cao, 2018), stability (Liu et al, 2006) and almost periodic solution (Jiang et al, 2015).…”

Adaptive exponential lag synchronization for neural networks with mixed delays via intermittent control

“…For such case, adaptive approach can be applied to deal with the synchronization problem [14,21,23]. Moreover, in biological neural networks, it often has a spatial extent because of the presence of parallel pathways with a variety of axon sizes and lengths [54][55][56]. Therefore, there exists a distribution of transmission delay, which is not suitable to be modeled with discrete delay.…”