# Projective Synchronization of Nonidentical Fractional-Order Memristive Neural Networks

Authors: Chong Chen; Zhixia Ding
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## 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.


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## Body

## 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:(1) In this paper, projective synchronization of NFMNN is studied by designing a new type of fractional-order integral sliding mode controller. (2) 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. (3) 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 Section2. 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 orderα for an integrable function f(t):0,+∞→R is defined as (1)0Itαft=1Γα∫0tt-ξα-1fξdξ,t≥0,where t≥0, α>0, and Γ(·) is the Gamma function which is defined as (2)Γx=∫0∞e-ttx-1dt,Rex>0.Definition 2 (see [41]).
The Caputo fractional-order derivative of orderα for a function f(t)∈Cn+1([0,+∞],R) is defined by (3)0CDtαft=1Γn-α∫0tfnξt-ξα-n+1dξ,where t≥0 and n-1<α<n∈N+. Particularly, when 0<α<1, (4)0CDtαft=1Γ1-α∫0tf′ξt-ξαdξ.A series of significant properties about fractional-order calculus are listed as follows [41].Property 3.
For any constantsv1 and v2, the linearity of Caputo fraction-order calculus is expressed by (5)0CDtαv1·ft+v2·gt=v1·0CDtαft+v2·0CDtαgt.Property 4.
Ifw(t)∈C1[0,T], T>0, then (6)0CDtα0CDtβwt=0CDtβ0CDtαwt=0CDtα+βwt,t∈0,T.where α>0, β>0, and α+β≤1.Lemma 5 (see [32]).
Ifw(t)∈C1([0,+∞),R), then the following inequality holds almost everywhere: (7)0CDtαwt≤sgnwt0CDtαwt,0<α<1.Lemma 6 (see [42]).
Letx(t)∈R be a continuous and derivable function. Then, for any t≥0, one have (8)120CDtαx2t≤xt0CDtαxt,α∈0,1.By applying Lemma6, the following formula can be obtained: (9)120CDtαxTtxt≤xTt0CDtαxt,where x(t)∈Rn, α∈(0,1), and t≥0.


### 2.2. System Description

In this section, we consider a class of FMNN as the master system, which is described by the following equation:(10)0CDtαxit=-cixit+∑i=1naijxjtfjxjt+Ii, where α∈(0,1), x(t)=(x1(t),x2(t),…,xn(t))T∈Rn, xi(t) denotes the state variable associated with the ith neuron, i∈N+, t≥0; ci>0 is self-regulation parameters of neurons; Ii is a constant of the external input; fi(t) denotes nonlinear activation function; aij(xj(t)) express connection memristive weights, defined by(11)aijxjt=aij∗xjt>Tjcoaij∗,aij∘xjt=Tjaij∘xjt<Tj, where i∈N+, Tj>0 denotes switching of memristor, and aij∗ and aij∘ are any constants. And the matrix form of the master system is given by(12)0CDtαxt=-Cxt+Afxt+I, where C=diag(c1,c2,…,cn), A=(aij(xj(t)))n×n, and I=(I1,I2,…,In)T.Similarly, the corresponding slave system is described as(13)0CDtαyt=-Dyt+Bgyt+H+ut, where u(t) is the control input; D=diag(d1,d2,…,dn), di>0; H=(H1,H2,…,Hn)T is the external input vector; B=(bij(yj(t)))n×n, bij(yj(t)) are expressed as(14)bijyjt=bij∗yjt>Tjcobij∗,bij∘yjt=Tjbij∘yjt<Tj.Assuming that the measurement output of system (12) depends on the instantaneous state, the form is as follows:(15)zt=Mxt, where z(t)∈Rm and M∈Rm×n 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 functionx(t)=(x1(t),x2(t),…,xn(t))T is a solution of system (12) on [0,+∞) with initial condition x(0)=(x1(0),x2(0),…,xn(0))T, if x(t) is absolutely continuous on any compact interval of [0,+∞), x(0)=(x1(0),x2(0),…,xn(0))T and(16)0CDtαxit∈-cixit+∑j=1ncoaijxjtfjxjt+Ii,for t≥0, i∈N+, where 0<α<1, or there exist θij[xj(t)]∈co[aij(xj(t))] such that(17)0CDtαxit=-cixit+∑j=1nθijxjtfjxjt+Ii,where (18)coaijxjt=aij∗xjt>Tjcoaij∗,aij∘xjt=Tjaij∘xjt<Tj,t∈0,+∞,i∈N+.And in the same way, we can obtain(19)0CDtαyit∈-diyit+∑j=1ncobijyjtgjyjt+Hi+uit, where t≥0, i∈N+, and 0<α<1, or there exist ωij[yj(t)]∈co[bij(yj(t))] such that(20)0CDtαyit=-diyit+∑j=1nωijyjtgjyjt+Hi+uit,where(21)cobijyjt=bij∗yjt>Tjcobij∗,bij∘yjt=Tjbij∘yjt<Tj,t∈0,+∞,i∈N+.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.j∈N+, there exist Lipschitz constants Fj>0, Gj>0; for any v1≠v2∈R, the following conditions are always established: (22)fjv1-fjv2≤Fjv1-v2,and (23)gjv1-gjv2≤Gjv1-v2.Assumption 9.
f j ( x ) and gj(x) are bounded on R. Meanwhile, fj(±Tj)=gj(±Tj)=0.Definition 10 (see [32]).
If there exists a nonzero constantξ for any solution x(t) and y(t) of master system (12) and slave system (13), respectively, one obtains(24)limt→+∞yt-ξxt=0,then the globally asymptotically projective synchronization of the master system (12) and slave system (13) can be realized, where · denotes the Euclidean norm and ξ denotes the projective coefficient.Remark 11.
Ifξ=-1, the global antisynchronization of system (12) and (13) can be attainable. If ξ=1, system (12) and (13) can achieve globally asymptotically complete synchronization. If ξ=0, there is the following form: limt→+∞∥y(t)∥=0, 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(25)Eα,βx=∑l=0∞xlΓlα+β,α>0,β>0,x∈C.For β=0, we have Eα(x)=Eα,1(x) defined as (26)Eαx=∑l=0∞xlΓlα+1α>0,x∈C.Definition 13 (see [25]).
Ifx(t)=x∗ is an equilibrium point of the system (12), and there are two constants Ξ>0 and Υ>0 for any solution x(t) of FMNN (12) with initial value x(0), one has (27)xt-x∗≤Ξx0-x∗Eα-Υtα,t≥0.In particular, x∗=0, and one has (28)xt≤Ξx0Eα-Υtα,t≥0.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]).
Forx∗ is an equilibrium point of the FMNN (12), if there is a Lyapunov function V(t,x(t)):[0,+∞)×D∈Rn and class-κ function ϑi(i=1,2,3) satisfying (29)ϑ1xt≤Vt,xt≤ϑ2xt,0CDtαVt,xt≤-ϑ3xt,where α∈(0,1) and the origin is included in a domain of D⊆Rn, so the equilibrium point of the FMNN (12) is asymptotically stable.


## 2.1. Caputo Fractional-Order Calculus

Definition 1 (see [41]).
The fractional-order integral of orderα for an integrable function f(t):0,+∞→R is defined as (1)0Itαft=1Γα∫0tt-ξα-1fξdξ,t≥0,where t≥0, α>0, and Γ(·) is the Gamma function which is defined as (2)Γx=∫0∞e-ttx-1dt,Rex>0.Definition 2 (see [41]).
The Caputo fractional-order derivative of orderα for a function f(t)∈Cn+1([0,+∞],R) is defined by (3)0CDtαft=1Γn-α∫0tfnξt-ξα-n+1dξ,where t≥0 and n-1<α<n∈N+. Particularly, when 0<α<1, (4)0CDtαft=1Γ1-α∫0tf′ξt-ξαdξ.A series of significant properties about fractional-order calculus are listed as follows [41].Property 3.
For any constantsv1 and v2, the linearity of Caputo fraction-order calculus is expressed by (5)0CDtαv1·ft+v2·gt=v1·0CDtαft+v2·0CDtαgt.Property 4.
Ifw(t)∈C1[0,T], T>0, then (6)0CDtα0CDtβwt=0CDtβ0CDtαwt=0CDtα+βwt,t∈0,T.where α>0, β>0, and α+β≤1.Lemma 5 (see [32]).
Ifw(t)∈C1([0,+∞),R), then the following inequality holds almost everywhere: (7)0CDtαwt≤sgnwt0CDtαwt,0<α<1.Lemma 6 (see [42]).
Letx(t)∈R be a continuous and derivable function. Then, for any t≥0, one have (8)120CDtαx2t≤xt0CDtαxt,α∈0,1.By applying Lemma6, the following formula can be obtained: (9)120CDtαxTtxt≤xTt0CDtαxt,where x(t)∈Rn, α∈(0,1), and t≥0.


## 2.2. System Description

In this section, we consider a class of FMNN as the master system, which is described by the following equation:(10)0CDtαxit=-cixit+∑i=1naijxjtfjxjt+Ii, where α∈(0,1), x(t)=(x1(t),x2(t),…,xn(t))T∈Rn, xi(t) denotes the state variable associated with the ith neuron, i∈N+, t≥0; ci>0 is self-regulation parameters of neurons; Ii is a constant of the external input; fi(t) denotes nonlinear activation function; aij(xj(t)) express connection memristive weights, defined by(11)aijxjt=aij∗xjt>Tjcoaij∗,aij∘xjt=Tjaij∘xjt<Tj, where i∈N+, Tj>0 denotes switching of memristor, and aij∗ and aij∘ are any constants. And the matrix form of the master system is given by(12)0CDtαxt=-Cxt+Afxt+I, where C=diag(c1,c2,…,cn), A=(aij(xj(t)))n×n, and I=(I1,I2,…,In)T.Similarly, the corresponding slave system is described as(13)0CDtαyt=-Dyt+Bgyt+H+ut, where u(t) is the control input; D=diag(d1,d2,…,dn), di>0; H=(H1,H2,…,Hn)T is the external input vector; B=(bij(yj(t)))n×n, bij(yj(t)) are expressed as(14)bijyjt=bij∗yjt>Tjcobij∗,bij∘yjt=Tjbij∘yjt<Tj.Assuming that the measurement output of system (12) depends on the instantaneous state, the form is as follows:(15)zt=Mxt, where z(t)∈Rm and M∈Rm×n 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 functionx(t)=(x1(t),x2(t),…,xn(t))T is a solution of system (12) on [0,+∞) with initial condition x(0)=(x1(0),x2(0),…,xn(0))T, if x(t) is absolutely continuous on any compact interval of [0,+∞), x(0)=(x1(0),x2(0),…,xn(0))T and(16)0CDtαxit∈-cixit+∑j=1ncoaijxjtfjxjt+Ii,for t≥0, i∈N+, where 0<α<1, or there exist θij[xj(t)]∈co[aij(xj(t))] such that(17)0CDtαxit=-cixit+∑j=1nθijxjtfjxjt+Ii,where (18)coaijxjt=aij∗xjt>Tjcoaij∗,aij∘xjt=Tjaij∘xjt<Tj,t∈0,+∞,i∈N+.And in the same way, we can obtain(19)0CDtαyit∈-diyit+∑j=1ncobijyjtgjyjt+Hi+uit, where t≥0, i∈N+, and 0<α<1, or there exist ωij[yj(t)]∈co[bij(yj(t))] such that(20)0CDtαyit=-diyit+∑j=1nωijyjtgjyjt+Hi+uit,where(21)cobijyjt=bij∗yjt>Tjcobij∗,bij∘yjt=Tjbij∘yjt<Tj,t∈0,+∞,i∈N+.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.j∈N+, there exist Lipschitz constants Fj>0, Gj>0; for any v1≠v2∈R, the following conditions are always established: (22)fjv1-fjv2≤Fjv1-v2,and (23)gjv1-gjv2≤Gjv1-v2.Assumption 9.
f j ( x ) and gj(x) are bounded on R. Meanwhile, fj(±Tj)=gj(±Tj)=0.Definition 10 (see [32]).
If there exists a nonzero constantξ for any solution x(t) and y(t) of master system (12) and slave system (13), respectively, one obtains(24)limt→+∞yt-ξxt=0,then the globally asymptotically projective synchronization of the master system (12) and slave system (13) can be realized, where · denotes the Euclidean norm and ξ denotes the projective coefficient.Remark 11.
Ifξ=-1, the global antisynchronization of system (12) and (13) can be attainable. If ξ=1, system (12) and (13) can achieve globally asymptotically complete synchronization. If ξ=0, there is the following form: limt→+∞∥y(t)∥=0, 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(25)Eα,βx=∑l=0∞xlΓlα+β,α>0,β>0,x∈C.For β=0, we have Eα(x)=Eα,1(x) defined as (26)Eαx=∑l=0∞xlΓlα+1α>0,x∈C.Definition 13 (see [25]).
Ifx(t)=x∗ is an equilibrium point of the system (12), and there are two constants Ξ>0 and Υ>0 for any solution x(t) of FMNN (12) with initial value x(0), one has (27)xt-x∗≤Ξx0-x∗Eα-Υtα,t≥0.In particular, x∗=0, and one has (28)xt≤Ξx0Eα-Υtα,t≥0.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]).
Forx∗ is an equilibrium point of the FMNN (12), if there is a Lyapunov function V(t,x(t)):[0,+∞)×D∈Rn and class-κ function ϑi(i=1,2,3) satisfying (29)ϑ1xt≤Vt,xt≤ϑ2xt,0CDtαVt,xt≤-ϑ3xt,where α∈(0,1) and the origin is included in a domain of D⊆Rn, so the equilibrium point of the FMNN (12) is asymptotically stable.


## 3. Main Results

The synchronization error is defined ase(t)=y(t)-ξx(t). From the master system (12) and slave system (13), the error system is described by(30)0CDtαet=-Det+B∗get+ξxt-B^∗gξxt+ξC-Dxt+B^∗gξxt-ξA∗fxt+H-ξI+ut, where B∗=(ωij[yj(t)])n×n, B^∗=(ωij[ξxj(t)])n×n, and A∗=(θij[xj(t)])n×n. 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(31)St=et+0ItαCet-B∗get+ξxt-B^∗gξxt+NMyt-ξzt=et+0Itα-B∗get+ξxt-B^∗gξxt+C+NMet, where S(t)=(S1(t),S2(t),…,Sn(t))T; C, B∗, and B^∗ are defined in the master system (12) and the slave system (20), respectively; N∈Rn×m is denoted as the gain matrix which can be selected appropriately; M∈Rm×n 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 S(t)=0 and S˙(t)=0.Based on Property4, one has (32)S˙t=0CDt1-α0CDtαSt=0,which implies that(33)0CDtαSt=0,Assume that Assumptions8 and 9 hold; expression (31) can be written as(34)0CDtαSt=0CDtαet-B∗gyt-B^∗gξxt+C+NMet.Substituting (30) into (34) and combining (33), then(35)0CDtαSt=-D+C+NMet+ξC-Dxt+B^∗gξxt-ξA∗fxt+H-ξI+ut=0.By means of (35), an equivalent control law can be written as(36)uqt=--D+C+NMet-ξC-Dxt-H-B^∗gξxt+ξA∗fxt+ξI.Combining (30) with (36), the dynamics of sliding mode can be written as(37)0CDtαet=-C+NMet+B∗get+ξxt-B^∗gξxt, and obviously, e∗=0 is an equilibrium point of the sliding mode dynamics (37).On account of the sliding mode control theory, a reaching law is denoted as(38)up=-δsgnSt,where(39)sgnSit=1,Sit>00,Sit=0-1,Sit<0,i=1,2,…,n∈N+,and δ>0 is the switching gain. Lastly, the sliding mode controller u(t) is designed as(40)ut=uqt+upt=--D+C+NMet-ξC-Dxt+ξA∗fxt-B^∗gξxt-H+ξI-σsgnSt.Remark 16.
As mentioned in [45, 46], some undesirable dynamic properties may be produced owing to the discontinuous function sgn(S(t)) existing in the sliding mode control system (40). To deal with the problem of the harmful chattering, the sgn(S(t)) is replaced by a continuous function tanh⁡(S(t)). Hence, the sliding mode control (40) is altered as follows:(41)ut=uqt+upt=--D+C+NMet-ξC-Dxt+ξA∗fxt-B^∗gξxt-H+ξI-σtanh⁡St.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 S(t)=0 based on the sliding model controller (40).Proof.
Constructing the positive definite function as a Lyapunov function candidate(42)V=12STtSt, in view of Lemma 6 and Definition 2, expression (42) can be written as(43)0CDtαVt=0CDtα12STtSt≤STt0CDtαSt=∑i=1nSit·-σ·sgnSit=-σSt1, where ·1 denotes the 1-norm.
Based on Lemma15, when σ>0 holds, the trajectories of system can asymptotically converge to S(t)=0, 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.i∈N+, one has(44)-ci+∑j=1n∑k=1mNjkMki+∑j=1nbji♡Gi<0, where ci>0 is the constant in the master system (17); Njk denotes the gain of system (31); Mki is the coefficient of the system (15); |bji♡|=max⁡{|bji∗|,|bji∘|}; and Gi is the Lipschitz constants of function g(·) in the Assumption 8.Theorem 19.
In the case of Assumptions8, 9, and 18, system (37) is globally asymptotically stable.Proof.
Firstly, expression (37) can be written as(45)0CDtαeit=-cieit-∑j=1n∑k=1mNikMkjejt+∑j=1nωijyjtgjyjt-ωijξxjtgjξxjt.
Constructing the functionV(t) as a Lyapunov function candidate can be written as(46)Vt=∑i=1neit.
According to Lemma5, the following inequality is obtained:(47)0CDtαVt=0CDtα∑i=1neit≤∑i=1nsgneit0CDtαeit.
And from expression (45) to (47), one has(48)0CDtαVt≤∑i=1nsgneit-cieit-∑j=1n∑k=1mNikMkjejt+∑j=1nωijyjtgjyjt-ωijξxjtgjξxjt=-ci∑i=1neit+∑i=1nsgneit∑j=1n∑k=1mNikMkjejt+∑i=1nsgneit∑j=1nωijyjtgjyjt-ωijξxjtgjξxjt}.
Based on Assumptions8 and 9, expression (48) can be written as(49)0CDtαVt≤-ci∑i=1neit+∑i=1neit∑j=1n∑k=1mNjkMki+∑i=1n∑j=1nejtb♡ijGj=∑i=1neit-ci+∑j=1n∑k=1mNjkMki+b♡ijGj; from Assumption 18, |bji♡|=max{|bji∗|,|bji∘|}; and let φi=ci-∑j=1n∑k=1m|NjkMki|-b♡ijGj>0, φ=min{φi}>0; one has(50)0CDtαVt≤-φi∑i=1neit≤-φVt;based on Lemma 15, an equilibrium point e∗=0 is the Mittag-Leffler stable, which means that system (37) is globally Mittag-Leffler stable, i.e., limt→∞e(t)=0. This completes the proof.Remark 20.
According to Theorem19, when e(t)=y(t)-ξx(t), the following form can be obtained: (51)yt-ξxt→0t→+∞;this means that the master system (17) and the slave system (20) achieve globally projective synchronization on account of the controller (40).
Ifξ=-1, the control input (40) is designed as(52)ut=uqt+upt=--D+C+NMet+C-Dxt-A∗fxt-B^∗g-xt-H-I-σsgnSt, where the sliding surface S(t) is designed by (31). A special case of Theorems 17 and 19 is obtained as follows.Corollary 21.
Under Assumptions8–18, the master system (17) and the slave system (20) are the globally asymptotic antisynchronization based on the controller (52).
Ifξ=1, the control input (40) is redesigned as(53)ut=uqt+upt=--D+C+NMet-C-Dxt+A∗fxt-B^∗gxt-H+I-σsgnSt.Corollary 22.
Assume that Assumptions8–18 hold, the master system (17) and slave system (20) are the global asymptotically complete synchronization on account of the controller (53).
Ifξ=0, the sliding surface (31) is rewritten as(54)St=yt+0Itα-B∗gyt-B^∗g0+C+NMyt, and the control input (40) is redesigned as(55)ut=uqt+upt=--D+C+NMyt-B^∗g0-H-σsgnSt.Corollary 23.
Under Assumptions8–18, 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.,(56)t0CDtαyt=-Cyt+Aˇ∗fyt+I+ut, where Aˇ∗=(θij[yj(t)])n×n, the sliding mode surface (31) is denoted as(57)St=et+0Itα-Aˇ∗fet+ξxt-A^∗fξxt+C+NMet, where A^∗=(θij[ξxj(t)])n×n; the control input (40) is rewritten as(58)ut=uqt+upt=-NMet+ξA∗fxt-A^∗fξxt+ξ-1I-σsgnSt.Similar to Theorems17 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:(59)0CDtαxt=-Cxt+Afxt+I0CDtαyt=-Dyt+Bgyt+H+ut, where α=0.98; I=H=[0,0,0]T; the activation function f(x(t))=tanh⁡(|x(t)|-1) and g(y(t))=sin⁡(|y(t)|-1); A=(aij)n×n and B=(bij)n×n; i, j=1,2,3; (60)C=1.50001.50001.5,D=0.070000.080000.05,a11=-5x1<1-3x1≥1,a12=5x2<16x2≥1,a13=5x3<14.5x3≥1,a21=-9x1<1-7x1≥1,a22=2x2<13x2≥1,a23=-1x3<13x1≥1,a31=-1x1<11x1≥1,a32=-9x2<1-8x2≥1,a33=2x3<13x1≥1,b11=0.45y1<10.33y1≥1,b12=-0.22y2<1-0.29y2≥1,b13=-0.2y3<1-0.25y3≥1,b21=-0.5y1<1-0.6y1≥1,b22=0.19y2<10.25y2≥1,b23=-0.15y3<10.28y3≥1,b31=-0.11y1<10.11y1≥1,b32=0.7y2<10.6y2≥1,b33=-0.21y1<1-0.275y1≥1.The measured outputz(t) is defined as z(t)=Mx(t), where the parameter M=[1,1,1]. Next, the simulation diagram is given as follows.Figure1 describes phase diagram of master system (12) under the condition of initial value x(0)=(1.1,1.2,-1.5)T. Under the condition of initial value y(0)=(-0.5,1.5,-2.5)T, the phase trajectories of slave system (13) is showed in Figure 2 without the controller (41).Figure 1
Phase plot of master system with initial conditionx(0)=(1.1,1.3,-1.5)T.Figure 2
Phase plot of slave system with initial conditiony(0)=(-0.5,1.5,-2.5)T without the controller u(t).When the projective coefficientξ=-1 and the gain matrix N=(0.5,0.3,0.1)T, Figures 3 and 4 imply that the master-slave system (59) is globally asymptotic antisynchronization from Corollary 21 under the control scheme (52).Figure 3
State trajectories of master-slave systems withξ = -1.Figure 4
Select 10 random initial conditions and state trajectories of the synchronization errorse1(t), e2(t), and e3(t) with ξ =-1.Similarly, based on the control scheme (52) Figures 5 and 6 depict the simulation results for ξ=1 with the gain matrix N=(0.3,0,0)T, which indicate the master-slave (59) is globally asymptotically complete synchronization form Corollary 22.Figure 5
State trajectories of master-slave systems withξ = 1.Figure 6
Select 10 random initial conditions and state trajectories of the synchronization errorse1(t), e2(t), and e3(t) with ξ=1.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, ξ=0 and N=(-0.1,0.3,-0.2)T. The Corollary 23 is illustrated in Figure 7.Figure 7
State trajectories of slave systems withξ = 0.Similarly, whenξ=-2 and the gain matrix N=(0.4,-0.1,0.1)T, the master-slave system (59) is globally asymptotically projective synchronization on the control scheme (40) which are showed in Figures 8 and 9.Figure 8
State trajectories of master-slave systems withξ = -2.Figure 9
Select 10 random initial conditions and state trajectories of the synchronization errorse1(t), e2(t), and e3(t) with ξ=-2.Whenξ=3 and the gain matrix N=(-0.1,-0.1,-0.1)T, Figures 10 and 11 show that the master-slave system is globally asymptotically projective synchronization under the control scheme (40).Figure 10
State trajectories of master-slave systems withξ = 3.Figure 11
Select 10 random initial conditions and state trajectories of the synchronization errorse1(t), e2(t), and e3(t) with ξ=3.


## 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.



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