The p th Moment Exponential Synchronization of Drive-Response Memristor Neural Networks Subject to Stochastic Perturbations

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


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 artifcial 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 artifcial 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 artifcial neural networks. Te frst nanoscale memristor device is made and set of 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 difusion procedure of neurotransmitters across neural synapses. Tis 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 specifc, 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 artifcial 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][5][6][7][8][9][10][11][12][13][14][15][16]. Te process of synchronization in a network of dynamic nodes refers to the convergence of all nodes towards a shared behavior, which is driven by specifc coupling and/or control protocols. So far, a number of results associated with the synchronization of memristor neural networks have been shown [17][18][19][20][21][22][23][24]. Furthermore, based on the consideration of synchronization efciency, 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 driveresponse 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-diferential 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 p does not need to be greater than 1 since it still gives a metric value for the random variables. For p � 1, this is also referred to as convergence in the mean sense and for p � 2, 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 afect the signal transmission and dynamic behavior. Te time delay neural network is an architecture of multilayer artifcial neural networks that is specifcally 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 infuence the performance and synchronization efciency 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 driveresponse stochastic memristor neural networks with timevarying delay. In addition, our model is extended to incorporate the stochastic perturbations, thereby increasing its generality and comprehensiveness compared to previous studies. Diferent 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 diferential inclusion theory and the Filippov regularization, certain more general synchronization criteria are obtained. Te proposed feedback control, the second term of which is capable of eliminating the infuences of both mismatched and state-dependent arguments, is simpler and more fexible than the existing results.
Te 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 defnitions and assumptions. Te sufcient condition for the moment synchronization is derived in Section 3. Section 4 includes numerical simulations.

Notations.
Let R and R n be the set of real numbers and the n − di mensional Euclidean space, respectively. Given a vector or matrix, ‖·‖ denotes its standard 2-norm and |·| implies the entries' absolute values, that is, |A| � (|a ij |) m×n , let λ min (A) be the smallest eigenvalue of A. Te maximum element of a vector x is denoted as max x { } and set 1, 2, . . . , n { } is defned as N. Let (Ω, F, F t t≥0 , P) be a complete probability space equipped with a fltration F t t≥0 , that is, right continuous with F 0 and contains all the P − null sets. C([− τ, 0]; R n ) shall denote the collection of continuous functions ϕ from [− τ, 0] to R n with the uniform norm ‖ϕ‖ 2 � sup − τ≤s≤0 ϕ(s) T ϕ(s) and where Ε means the corresponding expectation operator regarding the given probability measure Ρ.

Problem Formulation.
A memristor neural network subject to stochastic perturbations and time-varying delays is formulated by the stochastic diferential functional equations [33].
Complexity where x i (t) ∈ R is ith system's voltage, di(xi(t)) > 0 stands for the reset rate, f j (·) is an active function, τj(t) is the varying time delay, σi(xi(t)) denotes the noise strength function, ω(t) represents the standard one-dimensional Brownian motion satisfying Εdω � 0 and Ε(dω) 2 � dt, and the connection weights aij(xj(t)) and b ij (xj(t)) are defned as follows: where W i (x i (t)) is the memductance of memristor parallel with capacitor C i , W a ij and W b ij are the memductances of the memristor R f ij and R g ij , T i > 0 is the switching jump bound, , ..., σ(x n (t))] T . Ten, system (1) can be written in the vector form as follows: and the response stochastic memristor neural networks with controllers are as follows: where , a ij (y i (t)),and b ij (y j (t)) are defned similarly, and u i (t) represents the controller to be developed later.
Our objective is to achieve the pth moment exponential synchronization by designing a suitable controller u i (t). Defnition 1. Te drive system (3) and response system (4) are said to be pth moment exponentially synchronized if where K > 0 and κ > 0, for any initial data Remark 2. Compared with the exponential synchronization [18,19,30], the pth moment exponential synchronization is used to measure the systems' subject to the stochastic noise, which is in practice. Te pth moment exponential synchronization is more general than the mean square synchronization [34], where p � 2.
Te control scheme is designed as follows: where K � diag n k i in which k i and η will be determined later.

Remark 3.
Te control is basic and useful to deal with the pth moment exponential synchronization problem, which includes two terms the frst term adjusting by a proportional gain constant K 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 μ is capable of eliminating the infuences of mismatched and state-dependent arguments concurrently, which is simpler and more fexible than the existing results.
Utilizing the diferential inclusion theory and the Filippov regularization, equation (1) has been rewritten as follows: Terefore, some measurable functions Te response system (4) can be similarly written as follows: with certain measurable functions θ 4 . Denote e i (t) � y i (t) − x i (t) and take the following error stochastic memristor neural network into consideration. 4 Complexity Set e(t) � [e 1 (t), e 2 (t), ..., e n (t)] T . Ten, the error system (10) is converted into the following form: where − τ(t))) and σ(e(t)) � diag n×1 σ i (e i (t)) , and Assumption 4. f i (·), σ i (·), and i ∈ 1, 2, ..., n { } are measurable, satisfying the Lipschitz condition, i.e., there exist positive constants l i and h i such that for all x, y ∈ R. And there exist positive constants m i , i ∈ Ν such that for all x ∈ R.
If V ∈ C 2,1 (R + × R n ; R + ), defne an operator LV from R + × R n to R by where LV(s, x(s))ds, (21) as long as the expectations of the integrals exist.

Main Result
Tis section investigates the pth moment exponential synchronization of the proposed systems.
Te selection of the gain K and η to guarantee the pth moment exponential synchronization of system (1) with system (4) will be given in the following theorem: Theorem . We assumed that Assumption 4 is satisfed. If the gain K satisfes where u A , u B , ρ A , and ρ B are positive constants satisfying and M � diag n m i , then system (1) and system (4) can reach the pth moment exponential synchronization with control scheme (6).

))dω(t). (23)
Integrating from t 0 to t and taking the expectations, it holds in which Because η � max D T + (μ A + μ B )M in (6), one has 2y T (t) ∆A(t)f(x(t)) + ∆B(t)f(x(t − τ(t))) + I n sgn(e(t)) D T − ηsgn(e(t)) + σ T (e(t))σ(e(t)) 6 Complexity By means of Lemma 5 and (25), inequality (26) turns to By virtue of (27) and (28), we can get Using the assumptions of identical conditions for the drive and response drive-response stochastic memristor neural networks, there exists a positive constant N 0 such that ‖e(t)‖ ≤ N 0 for t ∈ [t 0 − τ 2 , t 0 ]. Te stochastic diferential function is given as under the identical initial condition as V(t). Applying the results with respect to the existent and unique solutions for stochastic diferential equations, it follows that equation (30) involves an only solution where t ∈ [t 0 , Nτ 1 ] for any natural numbers N following the mathematical induction. By using H € older ′ s inequality, (see [36] p.5) one can obtain where t ∈ [t 0 , T] for any constants T > t 0 .
□ Remark 9. We present a rigorous proof of the main result in this paper, where the moment inequalities and martingale inequalities are crucial in demonstrating the abovementioned theorem. Te pth moment exponential synchronization is more general than the mean square case.
By some calculations, one can obtain that u A � 12.7, ρ A � 6.6, u B � 20.7, and ρ B � 7.4. We set K � 10I 5 and η � 0.1, which means the conditions in Teorem 8 holds. Moreover, Figure 1 shows that the trajectories reach synchronization.

Conclusion
In this paper, we studied the synchronization problem for the drive-response memristor neural networks subject to stochastic perturbations and time-varying delays. Te phenomenon of the pth moment exponential synchronization is obtained. Some testable and computable sufcient conditions are derived to ensure the pth moment exponential synchronization of these drive-response stochastic memristor neural networks utilizing the Filippov theory and the Lyapunov stable theory, which is more general than the mean-square synchronization.
In future, we will improve on the control strategy, such as data-sampled control, quantization, intermittent control, and impulsive control. And the results will be extended to the pth moment synchronization of the general networked dynamical systems.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.