#### Abstract

This paper investigates projective synchronization of nonidentical fractional-order memristive neural networks (NFMNN) via sliding mode controller. Firstly, based on the sliding mode control theory, a new fractional-order integral sliding mode controller is designed to ensure the occurrence of sliding motion. Furthermore, according to fractional-order differential inequalities and fractional-order Lyapunov direct method, the trajectories of the system converge to the sliding mode surface to carry out sliding mode motion, and some sufficient criteria are obtained to achieve global projective synchronization of NFMNN. In addition, the conclusions extend and improve some previous works on the synchronization of fractional-order memristive neural networks (FMNN). Finally, a simulation example is given to verify the effectiveness and correctness of the obtained results.

#### 1. Introduction

In 1971, Professor Chua theoretically predicted the existence of memristive element [1]. In the following years, his team presented the basic characteristics, synthesis principles, and applications of memristor [2, 3]. Until 2008, researchers from Hewlett-Packard Co. firstly made nanomemristor devices, which triggered an upsurge in memristor research [4]. As the fourth basic circuit element, memristor has the characteristics that the other three basic components (resistance, capacitance, and inductance) cannot replicate. It can not only remember the number of charges flowing through it, but change its resistance by controlling the change of current. What is more important, these capabilities can be maintained even when power is cut off. Therefore, memristor has broadened application prospects in computer science [5], bioengineering [6], neural networks [7], electronic engineering [8], communication engineering [9], and so on.

Neural networks have the abilities of self-learning and self-adapting through training. The design of weights is crucial for neural networks. The hardware implementation of weight requires a long-term memory. With the development of memristor, it is feasible to introduce memristor into neural networks to solve the above problem. At present, the memristive neural networks (MNN) have been widely studied in information processing [10], image processing [11], artificial intelligence [12], and other fields [13–15]. As we all know, fractional-order calculus is a generalization of traditional integer-order calculus. Fractional-order calculus is not local and can describe memory property of neuron and dependence of the historical data. Meanwhile, fractional-order models can describe and model a real system more accurately than the classical integer-order models [16, 17]. By introducing fractional-order differential operators into integer-order MNN model, a new fractional-order memristive neural networks (FMNN) model can be established. Such systems can truly reflect the essential features of the system and extend the capability of neural networks. So far, many researchers have devoted themselves to investigating FMNN and a series of results about FMNN have appeared, such as [18–23].

It was Huygens, the inventor of pendulum, who discovered synchronization for the first time. Until 1990, Pecora and Carroll of the U.S. Naval Laboratory put forward the master-slave chaotic synchronization scheme, which realized the synchronization of two chaotic systems in the circuit and promoted the theoretical study of chaotic synchronization and chaotic control [24]. So far, there are many types of synchronization, such as complete synchronization [25–27], antisynchronization [28], lag synchronization [29, 30], generalized synchronization [31], projective synchronization [32–35], and phase and antiphase synchronization [36]. The scale factor of projective synchronization can be set flexibly, which enhances the uncertainty of slave system and improves communication security. Therefore, it is necessary to investigate projective synchronization of FMNN.

To the best of our knowledge, the existing results of projective synchronization about FMNN are identical system. In fact, functions and parameters of master-slave system are often mismatched, which is inevitable. Although various forms of control strategies have been designed to solve the problems of system’s synchronization [25, 27, 30, 37–40], they are difficult to deal with projective synchronization of NFMNN. In order to solve such problems, the fractional-order sliding mode controller is introduced, which can overcome the uncertainty of systems and has strong robustness to disturbance, especially for the control of nonlinear systems. The sliding mode controller needs the following three elements: firstly, any trajectories reach the sliding mode surface in free time; secondly, there is a sliding mode region on the sliding mode surface; thirdly, the sliding mode motion is asymptotic stable.

Motivated by the above discussions, we focus on the projective synchronization of nonidentical fractional-order memristive neural networks in this paper. The main contributions of this paper are as follows: In this paper, projective synchronization of NFMNN is studied by designing a new type of fractional-order integral sliding mode controller. By using the fractional-order differential inequality and fractional-order Lyapunov direct method, the projective synchronization criteria of NFMNN are obtained. At the same time, the obtained criteria can realize the complete synchronization and antisynchronization of NFMNN, stability, and projective synchronization of FMNN. Comparing with the conclusions in [25, 33], our results on projective synchronization of FMNN achieve a valuable improvement and are less conservative.

The structure of this paper is summarized as follows. Some preliminaries and the system models are given in Section 2. In Section 3, some sufficient criteria are obtained to achieve global projective synchronization of NFMNN. In Section 4, the validity and correctness of the obtained results are illustrated by numerical simulation. Finally, conclusions are drawn in Section 5.

#### 2. Preliminaries and System Description

##### 2.1. Caputo Fractional-Order Calculus

*Definition 1 (see [41]). *The fractional-order integral of order for an integrable function is defined as where , , and is the Gamma function which is defined as

*Definition 2 (see [41]). *The Caputo fractional-order derivative of order for a function is defined by where and . Particularly, when ,

A series of significant properties about fractional-order calculus are listed as follows [41].

*Property 3. *For any constants and , the linearity of Caputo fraction-order calculus is expressed by

*Property 4. *If , , then where , , and .

Lemma 5 (see [32]). *If , then the following inequality holds almost everywhere: *

Lemma 6 (see [42]). *Let be a continuous and derivable function. Then, for any , one have *

By applying Lemma 6, the following formula can be obtained: where , , and .

##### 2.2. System Description

In this section, we consider a class of FMNN as the master system, which is described by the following equation: where , , denotes the state variable associated with the th neuron, , ; is self-regulation parameters of neurons; is a constant of the external input; denotes nonlinear activation function; express connection memristive weights, defined by where , denotes switching of memristor, and and are any constants. And the matrix form of the master system is given by where , , and .

Similarly, the corresponding slave system is described as where is the control input; , ; is the external input vector; , are expressed as

Assuming that the measurement output of system (12) depends on the instantaneous state, the form is as follows: where and is known constant matrix.

Compared with continuous fractional-order neural networks, the FMNN is discontinued on the right-hand side because of the introduction of memristor. To deal with this problem, we consider the concept of Filippov solution in this paper.

*Definition 7 (see [43]). *In the Filippov sense, a function is a solution of system (12) on with initial condition , if is absolutely continuous on any compact interval of , andfor , , where , or there exist such thatwhere

And in the same way, we can obtain where , , and , or there exist such thatwhere

In order to guarantee the existence and uniqueness of the solution of system (12) and (13) or fractional-order differential inclusions (17) and (20), the following assumptions are provided.

*Assumption 8. *For a.a. , there exist Lipschitz constants , ; for any , the following conditions are always established: and

*Assumption 9. * and are bounded on . Meanwhile, .

*Definition 10 (see [32]). *If there exists a nonzero constant for any solution and of master system (12) and slave system (13), respectively, one obtainsthen the globally asymptotically projective synchronization of the master system (12) and slave system (13) can be realized, where denotes the Euclidean norm and denotes the projective coefficient.

*Remark 11. *If , the global antisynchronization of system (12) and (13) can be attainable. If , system (12) and (13) can achieve globally asymptotically complete synchronization. If , there is the following form: , which shows that the slave system (13) is globally asymptotically stabilized to the origin.

##### 2.3. Mittag-Leffler Stability

In this part, the Mittag-Leffler function and Mittag-Leffler stability are given by the following content.

*Definition 12 (see [41]). *The Mittag-Leffler function with two parameters is defined as For , we have defined as

*Definition 13 (see [25]). *If is an equilibrium point of the system (12), and there are two constants and for any solution of FMNN (12) with initial value , one has In particular, , and one has

*Definition 14 (see [25]). *System (12) is expected to be globally Mittag-Leffler stable, if its equilibrium point is Mittag-Leffler stable.

Lemma 15 (see [32]). *For is an equilibrium point of the FMNN (12), if there is a Lyapunov function and class- function satisfying where and the origin is included in a domain of , so the equilibrium point of the FMNN (12) is asymptotically stable.*

#### 3. Main Results

The synchronization error is defined as . From the master system (12) and slave system (13), the error system is described by where , , and . Based on Definition 10, the problem of projective synchronization for NFMNN is transformed into the problem of asymptotic stability of error systems (30).

In this part, a fractional-order integral sliding mode controller is designed to deal with the problem of projective synchronization of NFMNN. The sliding mode surface is defined as where ; , , and are defined in the master system (12) and the slave system (20), respectively; is denoted as the gain matrix which can be selected appropriately; is the coefficient matrix in (15).

According to the theory of sliding mode control [44], for the sake of the error system (30) operating on the sliding mode, the sliding surface and its derivative must satisfy and .

Based on Property 4, one has which implies that

Assume that Assumptions 8 and 9 hold; expression (31) can be written as

Substituting (30) into (34) and combining (33), then

By means of (35), an equivalent control law can be written as

Combining (30) with (36), the dynamics of sliding mode can be written as and obviously, is an equilibrium point of the sliding mode dynamics (37).

On account of the sliding mode control theory, a reaching law is denoted as whereand is the switching gain. Lastly, the sliding mode controller is designed as

*Remark 16. *As mentioned in [45, 46], some undesirable dynamic properties may be produced owing to the discontinuous function existing in the sliding mode control system (40). To deal with the problem of the harmful chattering, the is replaced by a continuous function . Hence, the sliding mode control (40) is altered as follows:

Theorem 17. *Assume that the sliding model surface is denoted by (31) and Assumption 9 holds; the trajectories of error system (30) can be asymptotically driven on the switching surface based on the sliding model controller (40).*

*Proof. *Constructing the positive definite function as a Lyapunov function candidate in view of Lemma 6 and Definition 2, expression (42) can be written as where denotes the 1-norm.

Based on Lemma 15, when holds, the trajectories of system can asymptotically converge to , which means that the trajectories of the error system are driven to the designed sliding mode surface and stayed on it at any subsequent moment. This obtains the proof.

In order to analyze the stability of the sliding motion, the following assumption is given as follows.

*Assumption 18. *For a.a. , one has where is the constant in the master system (17); denotes the gain of system (31); is the coefficient of the system (15); ; and is the Lipschitz constants of function in the Assumption 8.

Theorem 19. *In the case of Assumptions 8, 9, and 18, system (37) is globally asymptotically stable.*

*Proof. *Firstly, expression (37) can be written asConstructing the function as a Lyapunov function candidate can be written asAccording to Lemma 5, the following inequality is obtained:And from expression (45) to (47), one hasBased on Assumptions 8 and 9, expression (48) can be written as from Assumption 18, ; and let , ; one hasbased on Lemma 15, an equilibrium point is the Mittag-Leffler stable, which means that system (37) is globally Mittag-Leffler stable, i.e., . This completes the proof.

*Remark 20. *According to Theorem 19, when , the following form can be obtained: this means that the master system (17) and the slave system (20) achieve globally projective synchronization on account of the controller (40).

If , the control input (40) is designed as where the sliding surface is designed by (31). A special case of Theorems 17 and 19 is obtained as follows.

Corollary 21. *Under Assumptions 8–18, the master system (17) and the slave system (20) are the globally asymptotic antisynchronization based on the controller (52).**If , the control input (40) is redesigned as*

Corollary 22. *Assume that Assumptions 8–18 hold, the master system (17) and slave system (20) are the global asymptotically complete synchronization on account of the controller (53).**If , the sliding surface (31) is rewritten as and the control input (40) is redesigned as*

Corollary 23. *Under Assumptions 8–18, the slave system (20) is the globally asymptotically stabilized to the origin based on the controller (54).*

If the activation function and the parameters of slave system (20) are the same as master system (17), i.e., where , the sliding mode surface (31) is denoted as where ; the control input (40) is rewritten as

Similar to Theorems 17 and 19, the following inference can be given.

Corollary 24. *The master system (17) and the slave system (56) are the globally projective synchronization based on the controller (58) if the Assumptions 8–18 hold.*

*Remark 25. *Chen, Zeng et al. [25] discussed the synchronization of FMNN based on the linear feedback controller. Bao & Cao [33] discussed the projective synchronization of identical fractional-order memristive neural networks by using adaptive feedback controller. But their results were about the same system. In fact, the parameters and functions of systems are variant due to environmental changes. Thus, our systems are general. In addition, the above control scheme cannot obtain projective synchronization of NFMNN. However, the fractional-order sliding mode controller can solve such problems in this paper. In a word, our system is general and our results achieve a valuable improvement.

#### 4. Numerical Simulations

In this section, the results obtained in this paper are proved to be effectiveness and correctness by a numerical example.

*Example 1. *Consider the following three-dimensional FMNN as the master system and slave system: where ; ; the activation function and ; and ; , ;

The measured output is defined as , where the parameter . Next, the simulation diagram is given as follows.

Figure 1 describes phase diagram of master system (12) under the condition of initial value . Under the condition of initial value , the phase trajectories of slave system (13) is showed in Figure 2 without the controller (41).

When the projective coefficient and the gain matrix , Figures 3 and 4 imply that the master-slave system (59) is globally asymptotic antisynchronization from Corollary 21 under the control scheme (52).

Similarly, based on the control scheme (52) Figures 5 and 6 depict the simulation results for with the gain matrix , which indicate the master-slave (59) is globally asymptotically complete synchronization form Corollary 22.

In the following, the slave system of the globally asymptotically stabilized to the origin is considered based on the control scheme (54). In this case, and . The Corollary 23 is illustrated in Figure 7.

Similarly, when and the gain matrix , the master-slave system (59) is globally asymptotically projective synchronization on the control scheme (40) which are showed in Figures 8 and 9.

When and the gain matrix , Figures 10 and 11 show that the master-slave system is globally asymptotically projective synchronization under the control scheme (40).

#### 5. Conclusion

In this paper, the problem of projective synchronization of NFMNN is studied. By designing a new type of fractional-order sliding mode controller and analyzing the reachability of sliding mode surface and the stability of sliding motion, the effective criteria for projective synchronization of NFMNN are obtained. Furthermore, the complete synchronization and antisynchronization of NFMNN, the stability of FMNN, and the projective synchronization of the identical FMNN are special cases in this paper. In the future, we will consider the lag projective synchronization of NFMNN with delays.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Authors’ Contributions

Chong Chen carried out the main results and completed the corresponding proof. Zhixia Ding provides suggestions for the revision of the full text details. All authors read and approved the final manuscript.

#### Acknowledgments

This research is supported by National Science Foundation of China (61703312 and 61703313).