۱- Introduction

۲- Preliminaries

۳- Problem formulation

۴- Bipartite leader-following synchronization of network with a differentiable node-delay

۵- Bipartite leader-following synchronization of network with a non-differentiable node-delay

۶- Numerical results

۷- Conclusions

References

Abstract

This paper considers the bipartite leader-following synchronization in a signed network composed by an array of coupled delayed neural networks by utilizing the pinning control strategy and M-matrix theory, where the communication links between neighboring nodes of the network can be either positive or negative. Under the assumption that the node-delay is bounded and differentiable, a sufficient condition in terms of a low-dimensional linear matrix inequality is derived for reaching bipartite leader-following synchronization in the signed network, based on which a simple algebraic formula is further given to estimate an upper bound of the node-delay. When the node-delay is bounded and non-differentiable, some criteria are established by using the descriptor method and the reciprocally convex approach such that the bipartite leader-following synchronization problem for the signed network can be successfully solved. Finally, numerical simulations are provided to illustrate the effectiveness of theoretical analysis.

Introduction

In the past few decades, the delayed neural networks (DNNs) have been successfully applied to solve many practical problems such as speech recognition (Waibel (1989)), image processing (W¨ohler & Anlauf (1999)), optimization (Liu, Cao, & Xia (2005)), cryptography (Yu & Cao (2006)), and secure communication (Lakshmanan et al. (2018)). Since Pecora and Carroll laid the foundation for chaotic synchronization in the 1990s (Pecora & Carroll (1990)), the synchronization problem for a class of DNNs, including Cohen-Grossberg, cellular and memristive neural networks, has been one of the most active research topics and has been intensively investigated by lots of researchers via different control approaches (see Huang et al. (2014); Yang & Ho (2016); Wan et al. (2016); Zhang, Zhao, & Huang (2016); Li et al. (2018), and the references therein). With the rapid development of communication technology and computer science, much effort has been devoted to the synchronization problem of coupled delayed neural networks (CDNNs), which is much more challenging than that of a single delayed neural network. Chen, Zhou, & Liu (2004) studied the synchronization in an array of symmetrically interconnected neural networks with a time delay. Lu, Ho, & Liu (2007) further considered the synchronization in CDNNs whose coupling matrices may not necessarily be symmetric. Yu, Cao, & L¨u (2008) investigated the synchronization problem for CDNNs with a discrete communication delay. Cao, Chen, & Li (2008) and Zhang & Gao (2017) studied the synchronization in CDNNs whose communication delays include both discrete and distributed delays. Yang, Guo, & Wang (2017) explored the synchronization in CDNNs with impulsive interactions between neighboring nodes. Note that the synchronization in the above literature is reached via the interaction of network nodes without introducing any external forces to the network. More specifically, this kind of synchronization phenomenon can be called leaderless synchronization or self-synchronization.