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

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.


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][2][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][11][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][24][25][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 neutraltype 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.R  and R × denote the 푛-dimensional Euclidean space and the set of all 푛 × 푛 real matrices, respectively.The superscript 푇 denotes matrix transposition, tr(⋅) denotes the trace of the corresponding matrix, and 퐼 denotes the identity matrix.휆 max and 휆 min denote the maximum and minimum eigenvalues of a real symmetric matrix.diag(⋅ ⋅ ⋅ ) stands for the block diagonal matrix.
where 퐸[⋅] stands for the correspondent expectation operator with respect to the given probability measure 푃. co{푢, V} denotes the closure of a convex hull generated by real numbers 푢 and V or real matrices 푢 and V.

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

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 , and a positive scalar 휆 such that the LMIs hold: where And the adaptive feedback controller is designed as where the feedback strength 퐾 = diag(푘 1 , 푘 2 , . . ., 푘  ) is updated by the following law: with arbitrary constant 휑  > 0 (푖 = 1, 2, . . ., 푛).
Remark 15.When we remove the stochastic perturbations, our models become the model in [37], so our models are the extension of the model in [37].Because the stochastic perturbations are unavoidable in practice, our models are more general and useful in practice.
will reduce to a general network.What is more, when we remove the neutral terms, our models become the model in [10,27], so our models are the extension of the model in [10,27].Because the neutral terms are important and complicated, our models are more general and useful in practice.

Stochastic Adaptive Synchronization for the Two Coupled Memristive Neural Networks via the Linear Feedback Control
Theorem 17.Under Assumptions 1-6, two coupled memristive neural networks ( 6) and ( 7) can be synchronized for almost every initial data, if there exist positive diagonal matrices where And the linear feedback controller is designed as where 퐾 = diag(푘 1 , 푘 2 , . . ., 푘  ), 푘  > 0 is the feedback gain.
Proof.We consider the following Lyapunov-Krasovskii functions: where Then, the proof of Theorem 17 is similar to Theorem 13, so the proof process is omitted here.
Remark 19.When 퐷 = 0, the systems are no longer neutraltype neural networks.We find that adaptive synchronization of other types of neural networks model has been researched (see [38,39]).We can also get the synchronization results from Theorem 17 when 퐷 = 0.
Remark 20.When 푊(푡) = 0, the systems no longer have distributed time-varying delays.Our models become the model in [40]; we can also get the synchronization results, so our models are the extension of the model in [40] and they are more general than that.

Numerical Simulation
In this section, a numerical example is given to illustrate the effectiveness of Theorem 13.Consider a two-dimensional synchronization error system (8) with and  So the conditions of Theorem 13 are satisfied, and we conclude that two coupled memristive neural networks ( 6) and ( 7) can be synchronized for almost every initial data.Now by taking the initial date as 푒(0) = [0.4,0.5]  , 퐾(0) = [10,15]  and 휑 1 = 0.1, 휑 2 = 0.2, we can draw the dynamic curves of the error system, the evolution of adaptive coupling strength 푘 1 , 푘 2 , and the Brownian motion 휔(푡), respectively, as Figures 1-3. Figure 1 shows that two coupled memristive neural networks (6) and (7) are synchronized.

Conclusions and Discussion
In this paper, by applying LaSalle-type invariance principle for stochastic differential delays equations, the stochastic differential inclusions theory, Lyapunov functional, and linear matrix inequalities method, 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.Even though the synchronization problem of stochastic MNNs has been studied, there are few studies on the synchronization problem of stochastic neutral-type MNNs.Neutral terms are taken into account in this paper, which make the model have wider application and make research more meaningful.So we generalized the synchronization problem of MNNs.The effectiveness of our results has been illustrated by a numerical example.Furthermore, exponential synchronization and passivity of this model can be discussed in the near future.