Global exponential synchronization of delayed memristive neural networks with reaction–diffusion terms

Abstract

This paper investigates the global exponential synchronization problem of delayed memristive neural networks (MNNs) with reaction–diffusion terms. First, by utilizing the pinning control technique, two novel kinds of control methods are introduced to achieve synchronization of delayed MNNs with reaction–diffusion terms. Then, with the help of inequality techniques, pinning control technique, the drive–response concept and Lyapunov functional method, two sufficient conditions are obtained in the form of algebraic inequalities, which can be used for ensuring the exponential synchronization of the proposed delayed MNNs with reaction–diffusion terms. Moreover, the obtained results based on algebraic inequality complement and improve the previously known results. Finally, two illustrative examples are given to support the effectiveness and validity of the obtained theoretical results.

Introduction

The definition of memristor was first given by Professor Chua (Chua, 1971) in 1971. It is known to us that capacitor, inductor and resistor are the three basic circuit elements, Chua believed that the circuit should exist a fourth fundamental circuit element and called it as memristor. A research team from the Hewlett–Packard Lab achieved a great breakthrough in memristor after more than thirty years, they realized the practical memristor device and published their achievements in Strukov, Snider, Stewart, and Williams (2008). Since then, memristor has attracted more and more scientists’ research interests. Memristor is a passive and nonlinear circuit element, which demonstrates the relationship between flux and charge. Recently more and more memristor devices have been proposed and deeply investigated (Jo et al., 2010, Wen, Xie et al., 2018, Xiao et al., 2018, Xie, Wen et al., 2018, Yakopcic et al., 2011). When the circuit is turned off, memristor can memorize the latest passed value of electric charge. Due to the attractive property, it can be worked as synaptic weights in artificial neural networks and an ideal candidate to simulate the synapse (Iu et al., 2011, Wen, Wei, Yan et al., 2019, Wen, Wei, Yang et al., 2019, Xie et al., 2019, Zeng et al., 2018). Then we can apply the model of MNNs to simulate the human brain (Bao et al., 2018, Cao et al., 2019, Dong et al., 2019, Li et al., 2019, Wei and Cao, 2018, Wen et al., 2017, Wen, Hu et al., 2019, Wen et al., 2016, Wen, Xiao et al., 2018, Yan et al., 2019, Yang et al., 2018).

Recently, the dynamic behaviors of MNNs have became a hot research topic, more and more researchers focus their attention on it and made significant progress in it. In Cao, Cao, Wen, Huang and Zeng (2019) and Li, Cao, and Tu (2016), the authors investigated the passivity problem of delayed MNNs. Ding, Wang, and Zhang (2018a) focused on the dissipativity problem of stochastic MNNs. In this publications (Cai and Huang, 2018, Liu, Wang et al., 2018), the authors showed their interest in analyzing the stability of MNNs. Synchronization as one of the important dynamic behavior of neural networks has been widely applied in many areas, such as secure communication (Lakshmanan et al., 2018), image processing (Prakash et al., 2016, Wen et al., 2015), information science (Yang, Feng, Feng and Cao, 2017) and so on. And for the synchronization problem of neural networks, there are many strategies to synchronize neural networks (Cao et al., 2016, Li and Cao, 2018, Senan et al., 2017, Wei et al., 2018, Wen, Liu et al., 2019). So it is vital and valuable to consider the synchronization problems of MNNs, many researchers have done lots of work on it (Ding et al., 2018b, Li and Cao, 2016a, Sun et al., 2019, Tang et al., 2012, Wang, Cao, Guo et al., 2020, Wang, Cao, Huang et al., 2020, Wang et al., 2019, Wang et al., 2015, Yang, Cao et al., 2017, Yang et al., 2015, Zhang and Yang, 2017). Zhang and Yang (2017) adopted the intermittent control to achieve lag synchronization of MNNs. In Li and Cao (2016a), Li et al. paid their attention to the problem of lag synchronization of MNNs via $\omega$-Measure. Yang et al. (2015) proposed a novel multiple MNNs by nonlinear coupling and investigated the robust synchronization of it.

Besides the above control strategies, pinning control is an important and effective tool to achieve synchronization of neural networks (Gong et al., 2016, He et al., 2017, Hu et al., 2014, Liu, Song et al., 2018, Lü et al., 2019, Shi et al., 2016, Wang, Huang et al., 2018, Yu et al., 2009). Instead of controlling all nodes, pinning control aims at controlling partial nodes to achieve synchronization of systems, which is different from the normal control strategy. In Wang, Huang et al. (2018), the authors designed generalized pinning controllers to realize the pinning synchronization of proposed systems. In recent years, researchers made significant progress in investigating synchronization of MNNs via pinning control (Feng et al., 2018, Guo et al., 2016, Li et al., 2018, Yang, Luo et al., 2017). Guo et al. (2016) investigated the global exponential synchronization of MNNs with external noise via two kinds of distributed controls and introduced a novel pinning impulsive control law to achieve synchronization of proposed systems. In Li et al. (2018), robust synchronization of MNNs is obtained by means of pinning control. Yang, Luo et al. (2017) paid their attention on pinning synchronization of delayed MNNs and put forward a novel pinning method to synchronize the drive–response systems.

However, these works investigated the synchronization problem of MNNs via pinning control without taking the diffusion effects into account. It should be noted that the diffusion phenomena cannot be avoided once electrons transport in asymmetric electromagnetic fields. Thus, it is necessary and valuable to introduce the concept of reaction–diffusion into neural networks. In Cao et al., 2018, Gan, 2012, Rakkiyappan and Dharani, 2017, Shanmugam et al., 2018, Wang, Zhang et al., 2018 and Zhang and Zeng (2019), various types of neural networks with reaction–diffusion terms have been investigated, for example the stochastic reaction–diffusion neural networks (Gan, 2012). Moreover, the authors (Shanmugam et al., 2018) designed novel adaptive controllers to synchronize reaction–diffusion neural networks and applied it to secure communication. In Li and Cao, 2016b, Li and Wei, 2016, Liu et al., 2019, Tu et al., 2017 and Wu, Zhang, Li, and Yao (2015), researchers focused their attention on synchronization or stability of delayed MNNs with reaction–diffusion terms. Meanwhile, in real systems, the time delays have great impact on the dynamic behavior of it (Wu et al., 2016, Xie, Yue et al., 2018, Xu et al., 2018). Therefore, it is valuable to investigate the synchronization problem of MNNs with time delays and reaction–diffusion terms. Unlike other control strategies, pinning control scheme only needs to control a small fraction of neurons of neural networks. Besides, in large-scale neural networks, it is too costly and impractical to add controller to all neurons. Furthermore, it is difficult to design effective pinning controllers to achieve synchronization of MNNs because of the strong mismatch characteristic, especially considering diffusion effects. How to synchronize MNNs with reaction–diffusion terms, how to improve the control practicability and reduce the control cost, these are challenge and valuable problems.

Inspired by the above discussion, to narrow the gap, this paper intends to investigate the global exponential synchronization of delayed MNNs with reaction–diffusion terms. The main contributions of this present paper are concluded as: (1) In order to synchronize the delayed MNNs with reaction–diffusion terms and parameters mismatch, we introduce two control schemes, and two sufficient conditions are obtained in the form of algebraic inequality, which can ensure the global exponential synchronization of the proposed system. (2) To effectively reduce the control cost, we design two novel kinds of controllers. For the adaptive controllers, it is more flexible and helps to determine which neurons should first be pinned. For the state-feedback controllers, there are less parameters of controllers which need to be determined. (3) These two control methods can be improved to normal control schemes, which can improve the control practicability.

The rest of this paper is organized as follows. Preliminaries and the mathematical model of delayed MNNs with reaction–diffusion terms are given in Section 2. Next, two novel kinds of control schemes are introduced and two sufficient conditions for globally exponential synchronization are discussed in Section 3. In Section 4, two numerical simulations are given to illustrate the obtained theoretical results. At last, conclusions and future work are drawn in Section 5.

Notations: Let $\mathbb{R}=(-\infty ,+\infty )$ and ${\mathbb{R}}^n$ be the $n$-dimensional Euclidean space. $\mathbb{X}=\{x={(x_1,x_2,\dots ,x_s)}^T\mid \left|x_q\right|<l_q,q=1,2,\dots ,s\}$ is an open bounded domain with smooth boundary $\partial \mathbb{X}$ and $\bar{\mathbb{X}}=\mathbb{X}\cup \partial \mathbb{X}$, and the mes$\mathbb{X}$ is used to represent the measure of $\mathbb{X}$. For any $w(x,t)={(w_1(x,t),w_2(x,t),\dots ,w_n(x,t))}^T\in {\mathbb{R}}^n$, the 2-norm of $w(x,t)$ is defined as the follows: ${\Vert w(x,t)\Vert }_2={\left({\int }_{\mathbb{X}}{\sum }_{i=1}^n{w}_i^2(x,t)dx\right)}^{\frac{1}{2}}$. $C^1\left(\mathbb{X}\right)$ denotes the continuous differential function space which defines on $\mathbb{X}$.