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Mathematical Problems in Engineering
Volume 2018, Article ID 9126183, 24 pages
Research Article

Synchronization of a Class of Memristive Stochastic Bidirectional Associative Memory Neural Networks with Mixed Time-Varying Delays via Sampled-Data Control

1School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing 100083, China
2Beijing Key Laboratory of Knowledge Engineering for Materials Science, Beijing 100083, China
3North China University of Science and Technology, Tangshan 063009, China
4Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China
5Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
6Department of Electrical Engineering and Computer Science, Cleveland State University, Cleveland, OH 44115, USA

Correspondence should be addressed to Weiping Wang; moc.621@888666ayihs and Xiong Luo; nc.ude.btsu@oulx

Received 27 December 2017; Accepted 18 March 2018; Published 30 April 2018

Academic Editor: Jean Jacques Loiseau

Copyright © 2018 Manman Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The paper addresses the issue of synchronization of memristive bidirectional associative memory neural networks (MBAMNNs) with mixed time-varying delays and stochastic perturbation via a sampled-data controller. First, we propose a new model of MBAMNNs with mixed time-varying delays. In the proposed approach, the mixed delays include time-varying distributed delays and discrete delays. Second, we design a new method of sampled-data control for the stochastic MBAMNNs. Traditional control methods lack the capability of reflecting variable synaptic weights. In this paper, the methods are carefully designed to confirm the synchronization processes are suitable for the feather of the memristor. Third, sufficient criteria guaranteeing the synchronization of the systems are derived based on the derive-response concept. Finally, the effectiveness of the proposed mechanism is validated with numerical experiments.

1. Introduction

Associate memory is one of the most significant activities of human brain, which can be applied in study of brain-like systems [1], intelligent thinking for intelligent robots [2], and so on. Since Kosko discussed the concept of bidirectional associative memory neural networks (BAMNNs) [3] in 1988, BAMNNs occupied the great researchers’ time and have been studied for several years. Nowadays, due to the wide applications in signal processing, associative memory, pattern recognition, and so on [4], chaos control and synchronization of BAMNNs have been intensively investigated. Owing to the special characters of the memristor [5], researchers have replaced resistor with memristor in large scale integration circuits to construct the MBAMNNs [6]. MBAMNNs are more suitable for mimicking the associative memory process of human brain contrast with the BAMNNs. Thus, more and more researchers build the MBAMNNs models for investigating a variety of applications [710].

In practical MBAMNNs systems, an ever-present phenomenon is the threshold of the sensitive memristor with a nonlinear drift effect [11], such as the voltages, current, and magnetic flux. It is well known that the presence of the threshold input may drastically deteriorate the desired performance, even inducing the inaccuracy of closed-loop systems under investigation. Hence, it is important to take the nonlinear characteristic of the MBAMNNs into consideration in the dynamical systems. In general, two common methods are utilized to cope with the nonlinear characteristic of the MBAMNNs. One is treating the parameters as constants [12, 13], and the other is making a study of antisynchronization [14]. Recently, the problem of nonlinear characteristic has also been considered to the field of memristive neural networks (MNNs) [1522]. Although the importance of nonlinear characteristic has been rather well recognized, but few related results have been reported on the synchronization of MBAMNNs. This is the first motivation of the present paper.

Synchronization is an elementary collective phenomenon that enables coherent behavior in neural networks (NNs), where neurons coact with each other and achieve a common dynamic behavior. Synchronization is of great significance for its potential applications in many areas, including harmonic oscillation generation, biology systems, and secure communication. Looking through the literatures on the synchronization of MBAMNNs, one can find that most of the results are based on the two kinds of continuous-time control strategies: the state feedback control [2328] and adaptive control [29]. A prerequisite of these approaches is that the controllers must obtain signals from sensors in a continuous way [30]. This will increase the control cost heavily and cause a waste of communication bandwidth [31].

In contrast to the continuous-time control, the sampled-data control merely makes use of the sampling signals at discrete time instants. Consequently, the sampled-data control can eliminate the continuous monitoring of system states as well as the continuous information transmission. Therefore, the sampled-data control is a more managing choice in applications. Till now, numerous results have been reported in this aspect [3033]. However, to the best of our knowledge, there are few relevant achievements that consider the sampled-data synchronization of MBAMNNs. This is the second motivation of the present paper.

Owing to the limited speed of signal transmission between the neurons, the finite switching speed between different circuit elements in hardware implementations of NNs, and the viscosity of synapses triggered by biological NNs, time delay is an inevitable phenomenon in NNs. There are many types of delays, like discrete delay, leakage time delay, distributed delay, neutral-type delay, and so on. These delays are the main factors that contribute to the oscillation, instability, and the performance degradation to the dynamical systems [34]. Therefore, the dynamic systems with time delay becomes a hot topic in the theoretical and application realms.

Besides, the actual communication between real systems is usually disturbed by a stochastic perturbation from various uncertainties. In secure communication systems, the digital signal is transmitted by switching forth and back continuously between synchronization. The stochastic perturbation will probably lead to package losses or influences the signal transmission. Hence, it is important to discuss the effect of probabilistic delays, stochastic perturbations, and so on [3538]. Therefore, it is valuable and practical to discuss the effect of the stochastic perturbations and time-varying delays on MBAMNNs.

Motivated by the foregoing discussions, this paper aims at investigating the synchronization of MBAMNNs with mixed time-varying delays and stochastic perturbations by designing a suitable sampled-data controller. The main contributions of this paper are summarized as follows.(1)We first investigate the globally asymptotic synchronization of MBAMNNs with mixed time-varying delays and stochastic perturbations.(2)According to the characters of memristor, we consider the parameters mismatch between the drive-response systems and design a suitable sampled-data controller to fit the features of the memristor.(3)We also analyze the feasible region of the sampling period according to simulations, which is significant to some potential future research. Owing to the sampled-data synchronization analysis, Lyapunov functional method, and stochastic analysis theory, the synchronization criteria of the parameters mismatched MBAMNNs are derived.

The rest of this paper is organized as follows. The systems and problems formulation are presented in Section 2. In Section 3, based on Lyapunov functional method, stochastic analysis theory, and inequality techniques, sufficient criteria that depend on such system for synchronization are obtained. Numerical simulations are demonstrated to verify the effectiveness of the obtained results in Section 4. Finally, conclusions are given in Section 5.

2. Model Description and Preliminaries

2.1. Model Description

In order to better understand the MBAMNNs, firstly we describe the circuit of a general class of BAMNNs with the architecture as shown in Figure 1. Take the th subsystem and the th subsystem as the unit of analysis so as to simplify illustration [39]; one can clearly see that the Kirchhoff’s current law (KCL) of the subsystems of BAMNNs [3] is described as the following differential equation:where and are the voltages of capacitors and , respectively. And presents the resistor between the feedback function and ; depicts the resistor between the feedback function and . Then the transmission time-varying delay is illustrated by , is the bias function or external input on the th subsystem at time , denotes the bias function or external input on the th subsystem at time , and

Figure 1: The circuits implementing of BAMNNs with transmission time-varying delay.

Remark 1. Enlightened by [2329], especially for [39], we proposed the following system which contains not only discrete time-varying delays and , but also distributed time-varying delays and . And self-inhibition weights and are also time-varying. Therefore, the obtained results are more general and practical than some existing results.

Based on the physical properties of a memristor, the proposed MBAMNNs with mixed time-varying delays are described by the following differential equations:where and denote the voltages of capacitors and at time , for , , and . and represent the self-feedback connection weight. Then , , , , , and represent the memristor-based weights. In addition, and are feedback functions, and are discrete time-varying delays, and are finite distributed time-varying delays. In addition, and denote the continuous external inputs, respectively.

According to the current-voltage characteristic and the property of a memristor, the memristive connection weights of system (3) can be modeled asin which switching jumps , , and , , , , , , for are constants, so do , , , , , , , , and .

In this paper, we treat system (3) as the drive system; then the corresponding response system with stochastic perturbations is described aswhere

and are the appropriate controllers that will be proposed in order to gain the certain control objectives. Consider the following state feedback sampled-data controllers:where and are the sampled-data controllers gain matrices to be designed. and are positive scalars. Then and are discrete measurements of and at the sampling instant and , respectively. And the sampling instants satisfy the following conditions:

Remark 2. According to the discussions above, the inner connection matrices , , , , , , , , , , , , , , , and of systems (3) and (5) are varying with the state of memristance. Therefore, the MBAMNNs are considered as the state-dependent systems. When the parameters are all constants, systems (3) and (5) become a general class of BAMNNs.

We define the following error system as follows:

Remark 3. Some of the published papers studied the synchronization of MNNs [24, 25] through the following assumption to design the error system:Let denote the closure of convex hull generated by real numbers and or real matrices and . However this assumption has not always been proved to be correct. In [26], authors tried to deal with the synchronization issue of MABMNNs, but the results are unreasonable without taking the switching jumps into account.
The antisynchronization can avoid the problems mentioned above by constructing the error system as the following assumption:But it still has limitations in practical applications, such as associative memory and associative learning. It is worth mentioning that we fully consider the complex property of the inner connection and take the switching jumps into consideration. Therefore, the obtained results are more practical and less conservative than some published literatures.

2.2. Definitions and Assumptions

In order to get our primary conclusions in the next section, we make the following assumptions. For the sampling interval, one has the following.

Assumption 4 (see [40]). It is supposed that the interval between any two sampling instants is bounded by ():The constants , denote the maximum time span between and , and . The state is sampled, and , are the next update reaching the destination.
Due to the discrete terms and , it is difficult to analyze the synchronization of MBAMNNs directly. Therefore, according to the input delay approach, and are defined aswhere , , and the controllers can be designed as

Remark 5. It should be noted that [41] has made use of the sampled-data control to the MNNs, but to the best of our knowledge little attention has been paid to the MBAMNNs with mixed time-varying delays and stochastic perturbations based on sampled-data control theory, which motivates our present study.

Definition 6. Systems (3) and (5) are said to be asymptotically synchronized if and only if the error systems are globally asymptotically stable for the equilibrium points and . That is, , as , for any initial conditionswhere which denotes the Banach space of all continuous functions mapping into with 2-norm defined by , , and . They denote the Banach space of all continuous functions mapping into with 2-norm defined by , .

Assumption 7 (see [42]). The activation functions and are bounded and globally Lipschitz continuous in ; namely, there exist constants , , and for all , such thatwhere the constants , , , and can be positive numbers, negative numbers, or zero.

Assumption 8 (see [37]). There exist constants , , such that For the stochastic system [42],where is the Brownian motion and it is clearly . is the operator defined as follows:where

Assumption 9. The time-varying delays , in this paper are differential functions, wherefor all .

Lemma 10 (see [43]). Given any real matrix of appropriate dimensions, a scalar , and , the following inequality holds:

In particular, if and are vectors,

3. Main Results

In this section, we get some new sufficient conditions to ensure the synchronization of MBAMNNs by the designed sampled-data controller.

Theorem 11. Assume that Assumption 7 is satisfied; then error system (9) achieves global stable situation under the designed sampled-data feedback controller (7) with the control law as follows:where and are all positive constants.

Proof. Consider the following Lyapunov function for synchronization error system (9):whereCase 1. If , , , at time , according to the jumping rules, systems (3) and (5) are reduced to systems (28) and (29), respectively:Then we define the corresponding response systemAnd error system (9) can be rewritten:ThenWe conclude thatwhereFrom all the above discussion, we getand thenAccording to (31), one has Due to Assumptions 7 and 8, we obtain the following inequality:and thenBased on Lemma 10, we get the inequalities as follows:Thus, we infer that