Discrete Dynamics in Nature and Society

Volume 2018, Article ID 8126127, 13 pages

https://doi.org/10.1155/2018/8126127

## Analysis of Adaptive Synchronization for Stochastic Neutral-Type Memristive Neural Networks with Mixed Time-Varying Delays

Department of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, China

Correspondence should be addressed to Zuoliang Xiong; moc.361@1061gnoix

Received 21 November 2017; Revised 21 January 2018; Accepted 30 January 2018; Published 19 April 2018

Academic Editor: Zhengqiu Zhang

Copyright © 2018 Desheng Hong 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.

#### Abstract

Linear feedback control and adaptive feedback control are proposed to achieve the synchronization of stochastic neutral-type memristive neural networks with mixed time-varying delays. By applying the stochastic differential inclusions theory, Lyapunov functional, and linear matrix inequalities method, we obtain some new adaptive synchronization criteria. A numerical example is given to illustrate the effectiveness of our results.

#### 1. Introduction

During the last few years, as we know neural networks have been widely researched in control, image processing, associative memory design, pattern recognition, information science, and so on (see [1–3]). Chua firstly predicted the memristor as the fourth fundamental electrical circuit element in 1971 [4]. In 2008, Hewlett-Packard research team [5] obtained a practical memristor device and exhibited its characteristic, such as nanoscale and the memory ability. It has been shown that memristors can be used to work as biological synapses in artificial neural network and replace resistor to simulate the human brain in memristor-based neural networks (MNNs) model, which would benefit many practical applications (see [6, 7]).

It is well know that time delays present complex and unpredictable behaviors in practice often caused by finite switching speeds of the amplifiers, which may affect the stability of the system and even results in oscillation, divergence, and instability phenomena. Therefore, much effort has been devoted to analyze dynamic behaviors of MNNs with various types of times delays (see [8, 9]); constant time delays and the time-varying delays have been studied in [10–12]. The investigations of MNNs discussed consider the discrete delays in [13]. However, since the neural signal propagation is often distributed during a certain time period in the presence of an amount of parallel pathways with a variety of axon sizes and lengths, hence, the authors in [14, 15] have concentrated on the mixed delays.

On the other hand, in reality, the fluctuations from the release of neurotransmitters or other probabilistic causes may affect the stability property in the nervous system and synaptic transmission. So the stability analysis with stochastic perturbation has aroused great interest of many researchers (see [16, 17]). It is natural and important that systems containing some information are not only related to the derivative of the current state, but also have a great relationship with the previous derivative, which is called neural-type neural networks (see [9, 18, 19]).

Recently synchronization and antisynchronization of memristor-based neural networks have received great attention due to their potential, such as secure communication information science and biological technology [20]. But the networks are not always able to synchronize by themselves. Then, various effective control approaches and techniques have been proposed for synchronization, such as impulsive control, feedback control, adaptive control, and intermittent control (see [21, 22]). And a lot of achievements have been made in the stability and synchronization problem of MNNs, including exponential synchronization, lag synchronization, and finite time synchronization (see [23–26]).

Motivated by the above discussion, even though the synchronization problem of stochastic MNNs has been studied, there are few studies on the synchronization problem of stochastic neutral-type MNNs. So in this paper we focus our minds on the adaptive synchronization for neutral-type MNNs with mixed time-varying delays to bridge the gap. By applying the stochastic differential inclusions theory, Lyapunov functional, and linear matrix inequalities method, we obtain some new adaptive synchronization criteria.

This paper is organized as follows. In Section 2, we introduce the model and some preliminaries. The main theoretical results are derived in Section 3. In Section 4, a numerical simulation is presented to verify our obtained results. Finally, conclusion is given in Section 5.

Throughout this paper, solutions of all the systems considered are intended in the Filippov’s sense. and denote the -dimensional Euclidean space and the set of all real matrices, respectively. The superscript denotes matrix transposition, denotes the trace of the corresponding matrix, and denotes the identity matrix. and denote the maximum and minimum eigenvalues of a real symmetric matrix. stands for the block diagonal matrix. denote the family of all measurable, -valued stochastic variables , such that , where stands for the correspondent expectation operator with respect to the given probability measure denotes the closure of a convex hull generated by real numbers and or real matrices and .

#### 2. Preliminaries

In this paper, the following stochastic neutral-type memristive neural network with mixed time-varying delays is described by () with initial conditions , where is the voltage of the capacitor , , , and are neuron activation functions, and is the external constant input. and are self-feedback connection matrices and , (), , , and represent memristor-based weights: Here denote the memductances of memristors , According to the pinched hysteretic loops of property of memristors, we setwhere the switching jumps , , , , , , are constants. , , and are memristive connection weights, which represent the neuron interconnection matrix, respectively. If , , and are constants, system (1) will reduce to a general network. Let , , , , , , , , , , , , , , , , , for . represent the time-varying transmission delays. Since , , are discontinuous, in this paper, the solutions of all the following systems are illustrated in Filippov’s sense. By applying theory of differential inclusions and set-valued maps in system (1), this can be written as follows: where the set-valued maps are defined as follows:or equivalently, there exist , , , and , , , such that We consider system (6) as the drive system. Similarly, the response system is where is the controller, is an -dimensional Brownian motion defined on the complete probability space with a natural filtration , and is the noise intensity matrix, where satisfies Let be the synchronization error, where , , From (6) and (7), we can get the following synchronization error system: where , , . To prove our main results, the following assumptions and lemmas are needed.

*Assumption 1 (see [27]). *There exist diagonal matrices and , satisfying , , for all .

*Assumption 2. *There exist positive constants , and , such that

*Remark 3. *The assumption strong condition can be weaken; please refer to [28, 29] Assumptions 2 and 1.

*Assumption 4. *, there exist positive constants , such thatwhere and

*Assumption 5 (see [30]). *There exist positive matrices , and , such thatfor all and

*Assumption 6. *The matrix satisfies , where is the spectral radius of .

*Definition 7 (see [31]). *The two coupled memristive neural networks (6) and (7) are said to be stochastic synchronization for almost every initial data if for every

Lemma 8 (see [32]). *For any vectors , the inequality holds, in which is any matrix with .*

Lemma 9 (see [33]). *For any positive definite matrix , scalar , and vector function such that the integration concerned is well defined, then*

Lemma 10 (see [34]). *If , then *

*Remark 11. *There are some other convenient and useful inequality techniques; refer to [1] Lemmas , , , and .

Lemma 12 (see [35]). *Given matrices , where and , then if and only if*

#### 3. Main Results

In this section, the stochastic synchronization for the two coupled memristive neural networks (6) and (7) is investigated under Assumptions 1–6.

##### 3.1. Stochastic Adaptive Synchronization for the Two Coupled Memristive Neural Networks via the Adaptive Feedback Control

Theorem 13. *Under Assumptions 1–6, the two coupled memristive neural networks (6) and (7) can be synchronized for almost every initial data, if there exist positive diagonal matrices , positive definite matrices , and a positive scalar such that the LMIs hold: whereAnd the adaptive feedback controller is designed as where the feedback strength is updated by the following law: with arbitrary constant .*

*Proof. *We consider the following Lyapunov-Krasovskii functions: where By Itô formula, it follows thatwhere , , , and From Lemma 8, we get Utilizing Lemma 9 yields It follows from Assumption 5 and (14) that By Itô formula, we have From Assumption 1, it follows that where are positive diagonal matrices and for Condition (15) yields Substituting inequalities (23)–(30) into (22), we obtain where with