Fixed-time synchronization of memristor-based BAM neural networks with time-varying discrete delay

doi:10.1016/j.neunet.2017.08.012

Neural Networks

Volume 96, December 2017, Pages 47-54

https://doi.org/10.1016/j.neunet.2017.08.012 Get rights and content

Abstract

This paper is devoted to studying the fixed-time synchronization of memristor-based BAM neural networks (MBAMNNs) with discrete delay. Fixed-time synchronization means that synchronization can be achieved in a fixed time for any initial values of the considered systems. In the light of the double-layer structure of MBAMNNs, we design two similar feedback controllers. Based on Lyapunov stability theories, several criteria are established to guarantee that the drive and response MBAMNNs can realize synchronization in a fixed time. In particular, by changing the parameters of controllers, this fixed time can be adjusted to some desired value in advance, irrespective of the initial values of MBAMNNs. Numerical simulations are included to validate the derived results.

Introduction

Memristor, which was first introduced by Chua in 1971 (Chua, 1971), is regarded as the fourth fundamental circuit elementbesides resistor, capacitor and inductor. However, not until a prototype of memristor was manufactured by the scientists of Hewlett–Packard (Struko, Snider, Stewart, & Williams, 2008) in 2008 did memristor attract much attention of scholars. Memristor, which is the abbreviation of memory resistor, reflects the relationship between flux and charge. The memristance of memristor varies with the quantity of charge that has passed through the memristor (Sharifiy & Banadaki, 2010). Since memristor canremember the quantity of passed charge, it is believed that memristor has the function of memory. In view of its memory characteristic, now memristor has been used to model memristor-based neural network (MNN) Jiang et al. (2014), Wu & Zeng (2013), Wu et al. (2011), Xiao et al. (2016), Xiao et al. (2015), Xiao et al. (2016a), Xiao et al. (2016b), Zhang et al. (2017), which is the suitable candidate for simulating the human brain (Jo et al., 2010). On the other hand, bidirectional associative memory (BAM) neural network (Du, Zhong, & Zhou, 2014), which was proposed in 1988 (Kosko, 1988), has been successfully applied in various fields including automatic control, pattern recognition and artificial intelligence. Unlike other neural networks, BAM neural network is consisted of neurons distributed in two layers. The neurons distributed in one layer are fully interconnected with the neurons distributed in the other layer, while there is no interconnection between the neurons distributed in the same layer. Moreover, limited transmission speed and traffic jams make discrete delay (Shi, Liu, Tang, Zhu, & Zhong, 2016) inevitably exist in transmission, so it is necessary to consider discrete delay in the studies of neural networks. In recent years, there have been some results about the dynamic behavior analysis of delayed memristor-based BAM neural networks (MBAMNNs) Anbuvithya et al. (2015), Guo et al. (2017), Li et al. (2016), Mathiyalagan et al. (2015), Sakthivel et al. (2016), Xiao, Zhong, Li, Xu et al. (2016).

Synchronization Chen et al. (2017a), Chen et al. (2017b), Shi et al. (2016), Shi et al. (2017), Zhao et al. (2015), which means that the dynamical behaviors of coupled systems tend to an identical state, has many fascinating applications in secure communications, image processing, human heartbeat regulation and chemical reactions Milanović, & Zaghloul (1996), Tan & Ali (2001), Wen et al. (2015), etc. During the past few decades, synchronization has become a hot topic and a large number of relevant results have been obtained. However, most published results are related to asymptotical synchronization and exponential synchronization Han et al. (2016), Tong et al. (2016), both of which can be referred to as infinite-time synchronization. The infinite-time synchronization implies that synchronization can only be reached in infinite time and inherently requires persistent external control, which is undesirable in lots of application fields. Therefore, it is more valuable that the synchronization can be achieved in a finite time, which is called the settling time.

However, the disadvantage of finite-time synchronization Abdurahman et al. (2015), Chen et al. (2017a), Jiang et al. (2014), Li & Cao (2015) is that the settling time depends on the initial synchronization error. Specifically, the bigger the initial synchronization error, the larger the settling time. It means that when we want to determine the settling time of finite-time synchronization, we must know the initial values of the considered systems in advance. Unfortunately, in reality, it is usually difficult to obtain the initial values of the considered systems. To solve this problem, the concept of fixed-time stability was put forward (Polyakov, 2012).

Fixed-time synchronization/stability means that synchronization/stability can be achieved in a fixed time for any initial values. Because its settling time is a fixed constant regardless of the initial values of the considered systems, fixed-time synchronization/stability is essential for some special application scenarios, such as power system (Ni, Liu, Liu, & Hu, 2016) and rigid spacecraft (Jiang, Hu, & Friswell, 2016). So it is important to study the fixed-time synchronization/stability of nonlinear systems. Sufficient conditions that could guarantee the fixed-time stability of nonlinear systems were first given in Polyakov (2012). Ref. Wan, Cao, Wen, and Yu (2015) studied the fixed-time synchronization of Cohen–Grossberg neural networks. Ref. Hu, Yu, Chen, Jiang, and Huang (2017) investigated the fixed-time stability of dynamical systems. Moreover, fixed-time synchronization of coupled neural networks with discontinuous activations was also studied in Hu et al. (2017). The problem of fixed-time synchronization for complex-valued neural networks with discontinuous activations was solved in Ding, Cao, Alsaedi, Alsaadi, and Hayat (2017). Fixed-time synchronization of MNNs with discrete delay was first investigated in Cao and Li (2017). In summary, up to now, fixed-time synchronization/stability is still a new research area and there are only very few related results.

In the existing literatures on MBAMNNs, most of them focused on the asymptotical synchronization/stability and exponential synchronization/stability. The exponential synchronization of MBAMNNs was investigated in Mathiyalagan et al. (2015) via impulsive control. Ref. Sakthivel et al. (2016) studied the asymptotical anti-synchronization of MBAMNNs. The asymptotical synchronization of MBAMNNs was considered in Anbuvithya et al. (2015). The exponential stability of MBAMNNs was studied in Li et al. (2016). Ref. Guo et al. (2017) investigated the exponential stability of complex-valued MBAMNNs. Only Ref. Xiao, Zhong, et al. (2016) was related to the finite-time synchronization of fractional-order MBAMNNs. However, as far as we know, there is still no research on the fixed-time synchronization of MBAMNNs, which is the objective of this paper.

Inspired by the aforementioned analysis, this paper aims to investigate the fixed-time synchronization of MBAMNNs with discrete delay. The main contributions of this paper are embodied in the following aspects: (1) In the light of the double-layer structure of MBAMNNs, two similar controllers are designed for the fixed-time synchronization of MBAMNNs. In particular, once the parameters of controllers have been determined, the settling time of fixed-time synchronization can be estimated, irrespective of the initial values of MBAMNNs. (2) The finite-time synchronization of MBAMNNs with discrete delay is also studied in this paper. To the best of our knowledge, this problem has not been considered yet. (3) For activation functions involved in this paper, we only assume they are bounded, which is a relatively weak assumption.

The remainder of this paper is organized as follows. In Section 2, some necessary preliminaries are presented. The main theoretical results are derived in Section 3. In Section 4, numerical simulations are included to verify the effectiveness of the derived theoretical results. Section 5 draws the conclusion.

Section snippets

Preliminaries

Consider the following MBAMNN model with time-varying delay Mathiyalagan et al. (2015), Sakthivel et al. (2016) : $\begin{matrix} \{\begin{matrix} {\dot{x}}_{i} (t) = - σ_{i} (x_{i} (t)) x_{i} (t) + \sum_{j = 1}^{m} a_{j i} (x_{i} (t)) f_{j} (y_{j} (t)) \\ + \sum_{j = 1}^{m} b_{j i} (x_{i} (t)) f_{j} (y_{j} (t - τ (t))), \\ {\dot{y}}_{j} (t) = - ρ_{j} (y_{j} (t)) y_{j} (t) + \sum_{i = 1}^{n} c_{i j} (y_{j} (t)) g_{i} (x_{i} (t)) \\ + \sum_{i = 1}^{n} d_{i j} (y_{j} (t)) g_{i} (x_{i} (t - τ (t))), \end{matrix} \end{matrix}$ $i = 1, 2, \dots, n$ , $j = 1, 2, \dots, m$ , where $x_{i} (t)$ and $y_{j} (t)$ represent the voltages of capacitors $C_{i}$ and ${\hat{C}}_{j}$ , respectively; $σ_{i} (\cdot) > 0$ and $ρ_{j} (\cdot) > 0$ denote the rates of neuron self-inhibition; $f_{j} (\cdot)$ and $g_{i} (\cdot)$ are the activation

Main results

To achieve the fixed-time synchronization of MBAMNNs (1) and (4), the following controllers should be added on MBAMNN (4) : $\begin{matrix} u_{i} (t) & = - k_{1 i} e_{i}^{x} (t) - k_{2 i} s i g n (e_{i}^{x} (t)) - k_{3 i} s i g n (e_{i}^{x} (t)) {| e_{i}^{x} (t) |}^{p} \\ - k_{4 i} s i g n (e_{i}^{x} (t)) {| e_{i}^{x} (t) |}^{q}, \\ v_{j} (t) & = - l_{1 j} e_{j}^{y} (t) - l_{2 j} s i g n (e_{j}^{y} (t)) - l_{3 j} s i g n (e_{j}^{y} (t)) {| e_{j}^{y} (t) |}^{p} \\ - l_{4 j} s i g n (e_{j}^{y} (t)) {| e_{j}^{y} (t) |}^{q}, \end{matrix}$ $i = 1, 2, \dots, n$ , $j = 1, 2, \dots, m$ , where $k_{1 i}$ , $k_{2 i}$ , $l_{1 j}$ and $l_{2 j}$ are constants to be determined later, $k_{3 i}$ , $k_{4 i}$ , $l_{3 j}$ and $l_{4 j}$ are any positive constants, $0 < p < 1$ and $q > 1$ .

Theorem 1

Suppose Assumptions A1 and

Numerical simulations

In this section, we give two numerical examples to verify the effectiveness of the theoretical results.

Example 1

Consider the following MBAMNN model with discrete delay: $\begin{matrix} \{\begin{matrix} {\dot{x}}_{i} (t) = - σ_{i} (x_{i} (t)) x_{i} (t) + \sum_{j = 1}^{2} a_{j i} (x_{i} (t)) f_{j} (y_{j} (t)) \\ + \sum_{j = 1}^{2} b_{j i} (x_{i} (t)) f_{j} (y_{j} (t - τ (t))), \\ {\dot{y}}_{j} (t) = - ρ_{j} (y_{j} (t)) y_{j} (t) + \sum_{i = 1}^{2} c_{i j} (y_{j} (t)) g_{i} (x_{i} (t)) \\ + \sum_{i = 1}^{2} d_{i j} (y_{j} (t)) g_{i} (x_{i} (t - τ (t))), \end{matrix} \end{matrix}$ $i = 1, 2$ , $j = 1, 2$ , where $T_{1} = T_{2} = 1$ , ${\hat{T}}_{1} = {\hat{T}}_{2} = 2$ , $σ_{1}^{'} = 1$ , $σ_{1}^{''} = 1.5$ , $σ_{2}^{'} = 0.8$ , $σ_{2}^{''} = 1$ , $a_{11} (\cdot) = 1.4$ , $a_{21} (\cdot) = 0.8$ , $b_{22} (\cdot) = 0.7$ , $c_{22} (\cdot) = 0.1$ , $d_{22} (\cdot) = - 0.3$ , $a_{12}^{'} = - 0.4$ ,

Conclusions

This paper discusses the fixed-time synchronization of MBAMNNs with discrete delay. For fixed-time synchronization, the drive and response systems can achieve synchronization in a fixed time for any initial values. Considering the double-layer structure of MBAMNNs, two similar controllers are designed to add on the response MBAMNN. Based on Lyapunov stability theories, several fixed-time synchronization criteria are established successfully. In particular, the settling time of fixed-time

Acknowledgments

The work is supported by the National Key Research and Development Program (Grant Nos. 2016YFB0800602 and 2016YFB0800604), the National Natural Science Foundation of China (Grant Nos. 61573067 and 61472045), and the Beijing City Board of Education Science and Technology Key Project (Grant No. KZ201510015015).

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