Abstract

In this paper, the moment exponential synchronization problems of drive-response stochastic memristor neural networks are studied via a state feedback controller. The dynamics of the memristor neural network are nonidentical, consisting of both asymmetrically nondelayed and delayed coupled, state-dependent, and subject to exogenous stochastic perturbations. The pth moment exponential synchronization of these drive-response stochastic memristor neural networks is guaranteed under some testable and computable sufficient conditions utilizing differential inclusion theory and Filippov regularization. Finally, the correctness and effectiveness of our theoretical results are demonstrated through a numerical example.

1. Introduction

Neural networks, which simulate the structure of neurons and synapses in the human brain with mathematical models and combines multilevel conduction to simulate the interconnection structure of neurons, have now been widely used in artificial intelligence. On one hand, the development of the neural network is based on the understanding of biological brain to simulate its working mechanism more closely, such as the proposal and development of the third generation artificial neural network-spiking neural network; the mathematical modeling to simulate the nerve conduction system in the brain and the goal is to understand the way of brain signal transmission in order to help understand the way the brain works from the perspective of computational simulation.

Memristor is one of the best ways for hardware to realize synapses in artificial neural networks. The first nanoscale memristor device is made and set off a boom in memristor research in 2008. Memristors have been utilized as electronic synaptic devices and exhibit a migration of ions that bear striking resemblance to the diffusion procedure of neurotransmitters across neural synapses. This has led to the common tendency of utilizing memristors to character synapses in neural networks, which is broadly employed to store synaptic weights. Extensive experiments have shown that synapses in neural networks simulated by memristors will have great promising advantages. To be specific, the memristor, as a fundamental passive device, has nanoscale size and nonvolatility, achieving the consecutive changes in synaptic weights during the simulation of neural synapses, and also facilitating the integration of computation and storage. Moreover, a neural network structure that is highly integrated can be built by virtue of the memristors, which endow artificial neural networks with the ability not only to learn and memorize but also to perform more various functions.

And then, the modeling of the memristor neural network for the study of its rich dynamic behavior has become a hot research direction because of its broad application prospect. Over the past decade or so, the research studies put more energy into such a few aspects, such as signal processing [1], intelligent vehicle [2], and chaotic circuits [3].

During the past decade, there has been an increasing interest in diverse synchronization problems in complex dynamical networks considering their extensive applications in practice [4–16]. The process of synchronization in a network of dynamic nodes refers to the convergence of all nodes towards a shared behavior, which is driven by specific coupling and/or control protocols. So far, a number of results associated with the synchronization of memristor neural networks have been shown [17–24]. Furthermore, based on the consideration of synchronization efficiency, the selection and improvement of the control strategy is still the focus of our attention. What is worth celebrating is that, many peers have provided abundant case references and conclusions for synchronization problems, including state feedback control [25], impulsive control [26], and quantized control [27]. And the practical application about drive-response stochastic memristor neural networks is more widespread and useful than the memristor neural networks due to the inevitability of unknown stochastic factors, for example, as the authors expound in their studies about the stochastic memristor neural networks and their synchronization [26, 28, 29], the MNN’s dynamical behavior is exceptionally sensitive to unexpected stochastic disturbance due to frequent changes, which are caused by signal transmission anomalies or external environmental factors.

More relevant research results are as follows: Guo et al. [30] discussed the synchronization issue of multiple memristor neural networks with time delays by establishing two new integrate-differential inequalities. Zhu et al. [31] studied the synchronization problem of the master and slave memristor neural networks via an event-based impulsive scheme and considered certain state-dependent triggering conditions of nonlinear and linear continuous-time dynamic systems. We have noticed that the pth moment synchronization is more general than the mean square synchronization, where the power does not need to be greater than since it still gives a metric value for the random variables. For , this is also referred to as convergence in the mean sense and for , this is referred to as convergence in the mean square sense.

As another unexpected factor, time delay is a must to be dealt within the synchronization control process of most complex dynamic networks, including drive-response stochastic memristor neural networks, as it can negatively affect the signal transmission and dynamic behavior. The time delay neural network is an architecture of multilayer artificial neural networks that is specifically designed for classifying patterns with shift-invariance, and for modeling the context at each layer of the network. Moreover, stochastic disturbance and uncertainty occur in some random environments and influence the performance and synchronization efficiency of the system [32].

Based on the discussion mentioned above, the main contributions are outlined as follows: this paper studies the pth moment exponential synchronization of the drive-response stochastic memristor neural networks with time-varying delay. In addition, our model is extended to incorporate the stochastic perturbations, thereby increasing its generality and comprehensiveness compared to previous studies. Different from the mean-square synchronization and almost sure synchronization, we consider the pth moment exponential synchronization of the drive-response memristor neural networks with stochastic perturbations. By using the differential inclusion theory and the Filippov regularization, certain more general synchronization criteria are obtained. The proposed feedback control, the second term of which is capable of eliminating the influences of both mismatched and state-dependent arguments, is simpler and more flexible than the existing results.

The organization of this paper is as follows: in Section 2, we present the model of the drive-response memristor neural network subject to stochastic perturbations, along with some definitions and assumptions. The sufficient condition for the moment synchronization is derived in Section 3. Section 4 includes numerical simulations.

2. Preliminaries and Problem Definition

2.1. Notations

Let and be the set of real numbers and the Euclidean space, respectively. Given a vector or matrix, denotes its standard 2-norm and implies the entries’ absolute values, that is, , let be the smallest eigenvalue of . The maximum element of a vector is denoted as and set is defined as .

Let be a complete probability space equipped with a filtration , that is, right continuous with and contains all the sets. shall denote the collection of continuous functions from to with the uniform norm and the family of all measurable, -valued stochastic variables such that , where means the corresponding expectation operator regarding the given probability measure .

2.2. Problem Formulation

A memristor neural network subject to stochastic perturbations and time-varying delays is formulated by the stochastic differential functional equations [33].where is system’s voltage, stands for the reset rate, is an active function, is the varying time delay, denotes the noise strength function, represents the standard one-dimensional Brownian motion satisfying and , and the connection weights and are defined as follows:where is the memductance of memristor parallel with capacitor , and are the memductances of the memristor and , is the switching jump bound, and , , , , , and are constants with , , and . We assume that for all and the identical condition of equation (1) satisfies .

Let , , , , , , and . Then, system (1) can be written in the vector form as follows:and the response stochastic memristor neural networks with controllers are as follows:where , ,and are defined similarly, and represents the controller to be developed later. is the initial conditions for system (4). Usually, for .

Our objective is to achieve the moment exponential synchronization by designing a suitable controller .

Definition 1. The drive system (3) and response system (4) are said to be pth moment exponentially synchronized ifwhere and , for any initial data .

Remark 2. Compared with the exponential synchronization [18, 19, 30], the moment exponential synchronization is used to measure the systems’ subject to the stochastic noise, which is in practice. The pth moment exponential synchronization is more general than the mean square synchronization [34], where .
The control scheme is designed as follows:where in which and will be determined later.

Remark 3. The control is basic and useful to deal with the pth moment exponential synchronization problem, which includes two terms the first term adjusting by a proportional gain constant is proportional control to produce an output value that is proportional to the current error value and the second term adjusting by a proportional gain constant is capable of eliminating the influences of mismatched and state-dependent arguments concurrently, which is simpler and more flexible than the existing results.
Utilizing the differential inclusion theory and the Filippov regularization, equation (1) has been rewritten as follows:Therefore, some measurable functions and exist from the measurable selection theorem [35] such thatThe response system (4) can be similarly written as follows:with certain measurable functions , , and .
Denote and take the following error stochastic memristor neural network into consideration.where , , , , and .Set . Then, the error system (10) is converted into the following form:where , , , , , ,, and , and .

Assumption 4. , , and are measurable, satisfying the Lipschitz condition, , there exist positive constants and such thatfor all . And there exist positive constants such thatfor all .

Lemma 5. For any and scalar , we have

Lemma 6 (see [36]). Let over the interval andwhere , , , and are continuous functions. If there exists a positive constant such thatthenwhere .

Lemma 7 (see [36]). Considering an stochastic differential equation, we have

Let denote the family of all non-negative functions on , which are twice continuously differentiable in and once differentiable in . If , define an operator from to bywhere , , and . If , then for any ,as long as the expectations of the integrals exist.