Quasi-Synchronization of Nonidentical Fractional-Order Memristive Neural Networks via Impulsive Control

This paper investigates the quasi-synchronization of nonidentical fractional-order memristive neural networks (FMNNs) via impulsive control. Based on a newly provided fractional-order impulsive systems comparison lemma, the average impulsive interval deﬁnition, and the Laplace transform, some quasi-synchronization conditions are obtained with fractional order 0 < α < 1. In addition, the error convergence rates and error boundary are also obtained. Finally, one simulation example is presented to show the validity of our results.


Introduction
Memristor was predicted as the fourth circuit element describing the relationship between magnetic flux and voltage by professor Chua [1] in 1971.
Fractional calculus originated in the 17 th century and is a generalization of integer-order calculus operations to arbitrary order calculus operations [13]. Scholars introduce fractional calculus into the study of MNNs and formed fractional-order memristive neural networks (FMNNs) model [14][15][16]. FMNNs can describe the memory properties of neurons more accurately and achieve many results in synchronization and stability [17][18][19][20]. e global Mittag-Leffler stabilization of a class of FMNNs with time delays was discussed under a state feedback control in [17]. Chen and Ding [20] via a sliding mode controller and fractional-order Lyapunov direct method studied projective synchronization of nonidentical FMNNs.
On the other hand, synchronization means several systems share a common dynamical process. However, connection weights of FMNNs changed according to the state dynamics and the synchronization of derive-response systems will be destroyed. When the parameter mismatches are small enough, we can control the synchronization error in a small region, which is called quasi-synchronization [21][22][23][24]. Recently, the impulsive control method has been widely used in the quasi-synchronization of chaotic systems. He et al. used a distributed impulsive control studying the quasi-synchronization problem of drive-response heterogeneous networks in [21] and the number of controlled nodes was also considered. Tang et al. [24] derived criteria for quasi-synchronization of nonidentical coupled Lur'e networks by delayed impulsive comparison principle, where synchronization errors for different impulsive effects with different functions were evaluated. erefore, it is very meaningful to study the synchronization problem in the case of parameter mismatch [25][26][27][28].
To the best of our knowledge, the existing results about quasi-synchronization are concentrated on the integer-order systems, results about fractional-order systems are very few. Motivated by this, we use an impulsive controller to study the quasi-synchronization of FMNNs with parameter mismatches. e main innovation points of this paper are as follows: (1) a generalized fractional-order comparison lemma is provided in this paper, which plays a central role in the prove. (2) Our results connected with the fractionalorder α, which more reflects the characteristics of FMNNs.
1.1. Notations. R is the space of real number, N is the space of nonnegative integers, C is the space of complex number, R n denotes a dimensional Euclidean space, and R m×n is the set of m × n real matrices. For any matrix B, B T denotes the transpose of B and I denotes the identity matrix. For any algebraic operations, matrices are assumed to have compatible dimensions. e notation ‖ · ‖ denotes the matrix 2norm or the Euclidean vector norm. Let C([−∞, 0], R n ) denotes the family of continuous functions from [−∞, 0] to R n . λ min (A) and λ max (A) denote the minimum and maximum eigenvalue of A, respectively. sign(·) is the symbolic function; co b, c { } represents the closure of convex hull generated by real numbers b and c. Define χ(t) as a continuous function except at some finite number of points t k at which χ(t + k ) � χ(t k ) and χ(t − k ) exist; then the set of piecewise right continuous function

Caputo Fractional-Order Calculus and Mittag-Leffler
Function. Caputo fractional operator plays an important role in the fractional systems and has been more practical in physical than the Riemann-Liouville fractional operation. erefore, we use Caputo fractional operator as the main tool in this paper.
Definition 2 (see [13]). e Caputo fractional derivative of order α for a function f(t) is defined by where n − 1 < α < n; n is a positive integer.
□ Lemma 2 (see [28]). x(t) ∈ R n is a derivable and continuous function, for t ≥ t 0 ; we have where M ∈ R n×n is a positive definite matrix. e Mittag-Leffler function is defined in the following.

System
Description. Now, we consider the following FMNNs as a drive system: where i, j ∈ N; α is the fractional order; x i (t) represents the state variable of the ith neuron; d i > 0 denotes the selffeedback connection weight; x 0 � (x 01 , x 02 , . . . , x 0n ) T is the initial value of state vector; f j (·) denote the activation functions; and b ij (x j (t)) are memristive connection weights satisfy the following conditions: where the switching jumps ℸ xj > 0, b * ij , and b * * ij are all constants. e corresponding response system is described as where θ i (t) is the impulsive controller, and the memristorbased connection weights b ij (y j (t)) satisfy the following condition: where i, j ∈ N and switching jumps ℸ yj > 0, b * ij , b * * ij are known constants.
For system (14), by applying the theories of set-valued maps and differential inclusions, we have the following differential inclusion form: . e set-valued maps are defined as and, besides, there exist measurable functions Similarly, there exist measurable functions To ensure the uniqueness and existence of the solution of system (20) and (21), we make the following assumption.
ere exists a positive constant L � j such that for all x, y ∈ R, j ∈ N.
Lemma 4 (see [30]). Let Assumption 1 be satisfied; then the following inequalities hold: Remark 1. To reduce the conservativeness of our results, the constraints on activation functions as the synchronization error and the impulsive controller is designed as Discrete Dynamics in Nature and Society where a i are nonnegative constants, i ∈ N; impulsive intensity μ k ≠ 0; δ(·) denotes the Dirac impulsive function; and the time series ζ � t 1 , t 2 , . . . is a strictly increasing sequence of impulsive instants which satisfies t k−1 < t k and lim k⟶+∞ t k � +∞.
With considering the impulsive controller (24), the controlled error neural networks could be rewritten as follows: where e i (t) is right-hand continuous at t � t k and the initial conditions

e Definition of Quasi-Synchronization and Average
Impulsive Interval. Now, the definitions of quasi-synchronization and average impulsive interval are given as follows.
Definition 5 (see [26]). Consider the impulsive sequence ζ � t 1 , t 2 , . . . in the time interval (t, T). ω(t, T) is represented as the number of impulsive times, if there exist positive numbers N 0 and T a , such that and then, T a is greater than the average impulsive interval of the impulsive sequence ζ.
are satisfied, then error systems (25)  Proof. Construct the following Lyapunov function: For t ∈ [t k−1 , t k )(k � 1, 2, . . .), according to Lemmas 2 and 4, we get 4 Discrete Dynamics in Nature and Society Based on (4) and (5) of eorem 1, we get For t � t k , k ∈ N, according to (25), it yields Let Λ(t) be the unique solution of the following delayed impulsive comparison system for any c > 0: When t > 0, we have v(t) ≤ Λ(t) according to Lemma 1. For t ∈ [0, t 1 ), using the Laplace transform on (32), one gets

Proof .
e same as the proof in eorem 1, we obtain Substituting (48) into (40) yields Discrete Dynamics in Nature and Society Similar to (41), we can prove the following inequality is true for all t > 0: Letting c ⟶ 0, we have en, according to (51), the error converges into with the convergence rate (h ′ /2). So far, all proofs are over.
□ Remark 2. In impulsive dynamical networks or impulsive control mechanism, the impulsive effect μ k is essential to the final results of the whole networks. Some previous works have discussed the impulsive effects in the synchronization of complex dynamical networks. Specifically, the impulsive effect μ k is always assumed to satisfy (−1, 0) or (−2, 0). However, when a fractional-order system exchanges information with another system, the impulsive effect could either play positive roles or negative roles. In this paper, the impulsive gain μ k is talked over under the value ranges: (1) −α < μ k ≤ α and μ k ≠ 0, (2) μ k ≤ − α or μ k > α, where μ k only have a relation with variable α. e range of gain μ k values was extended.
Remark 3. In this paper, the controller (24) contains the impulsive control part and the feedback control part. For the case −α < μ k ≤ α and μ k ≠ 0, the impulsive effect μ k plays a positive role in the quasi-synchronization. In this case, the feedback control gain a i could be set as zero in order to save control costs and manufacture. However, for the case μ k ≤ − α or μ k > α, the impulsive effect μ k could hinder the synchronization of neural networks or even cause the instability of the coupled neural networks. In this case, the feedback control part could control effectively reduce the adverse effects brought by the disadvantageous impulse. In addition, if μ k � −1, it implies e i (t + k ) � −e i (t − k ), which is impossible for impulsive control. If μ k � 0, e i (t + k ) � e i (t k ) � 0, which means that there is no impulsive control effect.

Illustrative Examples
In this part, one example is given to show the effectiveness of our methods. Example 1. Consider 2-dimension drive system (14) with e switching jumps and memristive connection weights of the response system (16) are the same as system (14). e initial conditions of the drive-response system are x 0i � (1.5, 1) T , y 0i � (0.5, −0.5) T . e average impulsive interval is less than T a � 0.02 and the positive constant N 0 � 1.
Case 1: when −0.9 < μ k ≤ 0.9 and μ k ≠ 0, for eorem 1, considering disadvantageous impulsive effects here, set a i � 8(i ∈ N), μ k � −0.7, so we get ι � μ 2 k � 0.49. Denoting β � 7, then we have Υ � (ln(ι/α) /T a ) + β (1/α) � −21.7099 < 0. By solving conditions (4) and (5) In Figure 1, we plot the chaotic attractors of the drive-response system with initial conditions. In Figure 2, one can find that the error is under the bound 0.1752, which is less than the error bound ‖e(t)‖ derived from the theorem. It implies the systems achieve quasi-synchronization with a given error bound. Case 2: when μ k > 0.9, for eorem 2, impulsive effect μ k � 1.3, so ι � 1.1 2 � 1.69; we set the control gain a i � 0, i ∈ N. As a similar process in Example 1, we have Υ � (ln(ι/α)/T a ) + β (1/α) − β (1/α)+1 � −28.6372 8 Discrete Dynamics in Nature and Society e chaotic attractors of the drive-response system with initial conditions are shown in Figure 3. In Figure 4, one can find that the error is under the bound 0.1647, which is less than the error bound ‖e(t)‖ derived from the theorem. It can be obviously seen that the drive-response systems achieved quasi-synchronization.

Conclusion
In this paper, the problem of quasi-synchronization of delayed FMNNs is investigated. By designing an effective impulsive controller and a new fractional comparison lemma, some effective criteria for quasi-synchronization (0 < α < 1) are established. In the future, we will consider the quasi-synchronization problem of delayed FMNNs.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.   Discrete Dynamics in Nature and Society 9