New Results on Synchronization of Fractional-Order Memristor-Based Neural Networks via State Feedback Control

)is paper addresses the synchronization issue for the drive-response fractional-order memristor-based neural networks (FOMNNs) via state feedback control. To achieve the synchronization for considered drive-response FOMNNs, two feedback controllers are introduced. )en, by adopting nonsmooth analysis, fractional Lyapunov’s direct method, Young inequality, and fractional-order differential inclusions, several algebraic sufficient criteria are obtained for guaranteeing the synchronization of the drive-response FOMNNs. Lastly, for illustrating the effectiveness of the obtained theoretical results, an example is given.


Introduction
In recent years, fractional calculus has become a useful tool in the analysis of slow relaxation phenomena. As we all know, fractional derivative has two main advantages: infinite memory and more degrees of freedom [1,2]. Hence, fractional derivative plays a critical part in the depiction of memory and hereditary characteristics of multifarious processes. Compared with dynamic systems described by the classical integer-order derivative, dynamic systems described by fractional derivative can accurately reflect the actual dynamic properties of real systems due to their memory and hereditary characteristics. Recently, as an extension of the classical integer-order calculus, fractional calculus has many practical applications in many interdisciplinary areas, such as fractional-order sinusoidal oscillators [3], transient wave propagation [4], fractional relaxation-oscillation and fractional diffusion-wave phenomena [5], drug release and absorption [6], and so on. Moreover, dynamic behaviors of fractional-order systems have attracted the attention of many researchers because of their practical applications. In the past decade, the dynamic analysis of the fractional-order systems has achieved many outstanding results [7][8][9][10].
In the past few years, dynamic behaviors of neural networks (NNs) have gained many attentions [11][12][13][14][15][16][17][18][19][20][21], since NNs have lots of applications [22,23]. In addition, fractional derivative has been introduced to NNs, and dynamic analysis of fractional-order neural networks (FONNs) has become a focus of research and many results have been obtained [24,25]. Among these dynamic behaviors, as a significant dynamic characteristic, synchronization was firstly introduced [26]. Since then, research on synchronization of NNs has become a hot topic because of their wide potential applications in a large number of real systems [27,28].
On the other hand, memristor was firstly predicted by Chua [29], and a practical memristor device was successfully obtained [30]. e memristor exhibits the characteristics of pinch hysteresis, which is possessed by the human brain. It is more practical to construct artificial NNs by replacing the resistor with the memristor, that is, memristor-based neural networks (MNNs). In recent years, dynamic behaviors of MNNs have caused widespread concern around the world [31][32][33][34]. Accordingly, it is very valuable to study the synchronization problem for fractional-order memristor-based neural networks (FOMNNs), and many excellent works have been conducted [35,36].
Inspired by the discussions given above, this paper studies the synchronization issue for FOMNNs via feedback control. Firstly, to achieve the synchronization for considered drive-response FOMNNs, two feedback controllers are introduced. en, by adopting nonsmooth analysis, fractional Lyapunov's direct method and Young inequality, and fractional-order differential inclusions, several algebraic sufficient criteria are obtained for guaranteeing the synchronization for the drive-response FOMNNs.

Preliminaries and Model Description
e following preliminaries on fractional calculus are recalled.
Definition 2 (see [37]). Given an arbitrary differentiable function χ(t), its Caputo fractional derivative is defined as where the fractional order α > 0, t ≥ t 0 , α ∈ (k − 1, k), k is a positive integer. Consider an FOMNN as follows: where a l > 0 is the self-inhibition, C 0 D α t r l (t) refers to the Caputo fractional derivative, 0 < α < 1, r l (t) refers to the state, g m (r m (t)) refers to the activation function, J l is an external input or bias, and b lm (r m (t)) is the memristor connection weight. e initial value of (3) is r(0) � (r 1 (0), r 2 (0), . . . , r n (0)) T ∈ R n and l ∈ N. e memristor connection weight b lm (r m (t)) switches among different numbers, which can be simply modeled as follows: where the switching jump Υ l > 0 and b * lm and b * * lm are constant numbers. Let en, according to the theories of differential inclusion and set-valued map, for (3), we have co b lm r m (t) g m r m (t) + J l .
e drive-response synchronization is considered. e corresponding response system of the drive system (3) is where u l denotes the state feedback controller, and the initial condition w(s) � (w 1 (0), w 2 (0), . . . , w n (0)) T and l ∈ N. Similarly, Or equivalently, Assumption 1. For ∀x, y ∈ R, and l ∈ N, the function g l (·): R ⟶ R is monotone nondecreasing and satisfies g l (0) � 0, |g l (.)| ≤ M l , and |g l (x) − g l (y)| ≤ η l |x − y|, where M l and η l , are positive constants. Next, let the synchronization error e l (t) � r l (t) − w l (t). For (3) and (8), the following synchronization error dynamics system can be obtained as where ζ m (e m (t)) � g m (r m (t)) − g m (w m (t)). According to Assumption 1, we can know that ζ m (e m ) is also monotone nondecreasing, bounded, and sgn(e m (t))ζ m (e m (t)) � |ζ m (e m (t))| ≤ η m |e m (t)|.

Main Results
In this section, two state feedback controllers are provided for achieving synchronization of FOMNNs. e following two state feedback controllers are provided: where λ l , σ l and c l denote the control parameters, l ∈ N.

Theorem 1. By holding Assumption 1, the drive FOMNN (3)
is globally synchronized with the response FOMNN (8) via the controller (15), if there are two positive constants ς 1 and ς 2 so that Proof. Consider the Lyapunov function, Also, by calculating the Caputo fractional derivative along the trajectory of (11) with 0 < α < 1, we get By utilizing Lemma 2, we get Similarly, Substituting (20), (21) into (19), According to (17), we choose a constant 9 > 0 so that From (22) and (23), we get By utilizing Lemma 1, we can know namely, According to (26) and Definition 3, we can obtain that the drive FOMNN (3) is globally synchronized with the response FOMNN (8) via controller (15). e proof of eorem 1 is completed. □ Theorem 2. By holding Assumption 1, the drive FOMNN (3) is globally synchronized with the response FOMNN (8) via controller (16), if there are positive constants ρ l so that Proof. Firstly, we can know that, on t ∈ [0, +∞), e l (t) is a differentiable and continuous function. erefore, (d|e l (t)|)/dt is piecewise continuous, and lim s⟶t + ((d|e l (s)|)/ds) exits for ∀t ∈ R + . en, we choose the Lyapunov function: Now, by calculating the Caputo fractional derivative along the trajectory of (11) with 0 < α < 1, we get From (27), it follows that According to (28), we choose a constant 9 > 0 so that From (31) and (32), By utilizing Lemma 1, we can know that is, and then, According to (36) and Definition 3, we can obtain that the drive FOMNN (3) is globally synchronized with the response FOMNN (8) under the state feedback controller (16). e proof of eorem 2 is completed.
It is well known that eigenvalue of the system matrix has a tight relation with dynamics. Next, we will obtain synchronization conditions according to it. □ Theorem 3. By holding Assumption 1, the drive FOMNN (3) is globally synchronized with the response FOMNN (8) via controller (15), if where Proof. Firstly, according to the matrix theory, we can easily know that −θ is also the eigenvalue of Ξ if θ is its eigenvalue. erefore, the maximum eigenvalues of matrix Ξ is greater than zero, that is, λ max (Ξ) > 0.

(42)
By utilizing Lemma 1, we can know and then, n l�1 e 2 (t) 2 According to (45) and Definition 3, we can obtain that the drive FOMNN (3) is globally synchronized with the response FOMNN (8) under the state feedback controller (15). e proof of eorem 3 is completed. □ Remark 1. Free-weighting parameters ς 1 and ς 2 are introduced in synchronization criterion (17) of eorem 1. Also, free-weighting parameters ς 1 and ς 2 can be used to reduce the conservativeness of the synchronization criterion.

Remark 2.
e synchronization criteria obtained in this paper only depend on their system parameters, which are simpler in form than linear matrix inequalities [40]. In addition, these algebraic synchronization criteria are easy to check, quick to calculate, and help greatly reduce the computational burden.

Numerical Simulation
Now, we provide a numerical simulation to illustrate the effectiveness of results.

Conclusions
In this paper, the synchronization issue for FOMNNs has been investigated via state feedback control. To achieve the synchronization for the considered drive-response FOMNNs, feedback controllers are first introduced. en, by adopting nonsmooth analysis, fractional Lyapunov's direct method and Young inequality, and fractional-order differential inclusions, several algebraic sufficient criteria are obtained for guaranteeing the synchronization for the driveresponse FOMNNs. Finally, an example is given to illustrate the effectiveness of the theoretical results. In future research, theoretical results here will be used to address the state estimation of FOMNNs.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  (8) with the state feedback controller u 1 (t) � 3.8e 1 (t), u 2 (t) � 4e 2 (t).