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

Authors: Xiaofan Li; Yuan Ge; Hongjian Liu; Huiyuan Li; Jian-an Fang
Journal: Complexity
 (2020)

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## Abstract

This 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. Then, 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.


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## Body

## 1. 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 [7–10].In the past few years, dynamic behaviors of neural networks (NNs) have gained many attentions [11–21], 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]. The 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 [31–34]. 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. Then, 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.


## 2. Preliminaries and Model Description

The following preliminaries on fractional calculus are recalled.Definition 1 (see [37]).
Given an arbitrary integrable functionχt, its Riemann–Liouville fractional integral with fractional order α>0 is defined as(1)Iαχt=1Γα∫t0tt−sα−1χsds,where t≥t0, Γα=∫0+∞sα−1e−sds.Definition 2 (see [37]).
Given an arbitrary differentiable functionχt, its Caputo fractional derivative is defined as(2)Dtαt0Cχt=1Γk−α∫t0tt−sk−α−1χksds,where the fractional order α>0, t≥t0,α∈k−1,k,k is a positive integer.
Consider an FOMNN as follows:(3)Dtα0Crlt=−alrlt+∑m=1nblmrmtgmrmt+Jl,where al>0 is the self-inhibition, Dtα0Crlt refers to the Caputo fractional derivative, 0<α<1, rlt refers to the state, gmrmt refers to the activation function, Jl is an external input or bias, and blmrmt is the memristor connection weight. The initial value of (3) is r0=r10,r20,…,rn0T∈ℝn and l∈ℕ.
The memristor connection weightblmrmt switches among different numbers, which can be simply modeled as follows:(4)blmrmt=blm∗,rmt≤ϒl,blm∗∗,rmt>ϒl,where the switching jump ϒl>0 and blm∗ and blm∗∗ are constant numbers.
Letb¯lm=minblm∗,blm∗∗, b¯lm=maxblm∗,blm∗∗, and(5)coblmrmt=blm∗,rmt|<ϒl,blm∗∗,rmt|>ϒl,b¯lm,b¯lm,rmt|=ϒl..
Then, according to the theories of differential inclusion and set-valued map, for (3), we have(6)Dtα0Crlt∈−alrlt+∑m=1ncoblmrmtgmrmt+Jl.
Or equivalently, there existsb⌣lmrmt∈coblmrmt, such that(7)Dtα0Crlt=−alrlt+∑m=1nb⌣lmrmtgmrmt+Jl.
The drive-response synchronization is considered. The corresponding response system of the drive system (3) is(8)Dtα0Cwlt=−alwlt+∑m=1nblmwmtgmwmt+Jl+ul,where ul denotes the state feedback controller, and the initial condition ws=w10,w20,…,wn0T and l∈ℕ. Similarly,(9)Dtα0Cwlt∈−alwlt+∑m=1ncoblmwmtgmwmt+Jl+ul.
Or equivalently, there existsb⌣lmwmt∈coblmwmt, such that(10)Dtα0Cwlt=−alwlt+∑m=1nb⌣lmwmtgmwmt+Jl+ul.Assumption 1.
For∀x,y∈ℝ, and l∈ℕ, the function gl·: ℝ⟶ℝ is monotone nondecreasing and satisfies gl0=0, gl.≤M˜l, and glx−gly|≤ηl|x−y, where M˜landηl, are positive constants.
Next, let the synchronization errorelt=rlt−wlt. For (3) and (8), the following synchronization error dynamics system can be obtained as(11)Dtα0Celt=−alelt+∑m=1nb⌣lmrmtgmrmt−b⌣lmwmtgmwmt−ul,=−alelt+∑m=1nb⌣lmrmtζ˜memt+∑m=1nb⌣lmrmt−b⌣lmwmtgmwmt−ul,where ζ˜memt=gmrmt−gmwmt. According to Assumption 1, we can know that ζ˜mem is also monotone nondecreasing, bounded, and sgnemtζ˜memt=ζ˜memt≤ηmemt.Definition 3 (see [37]).
The drive FOMNN (3) is globally synchronized with the response FOMNN (8), if there are two positive constants ν1 and ν2 such that(12)etl≤ℳet0Eα−v1tαv2,where t0 denotes the initial time, ℳet is locally Lipschitz on ℝn, ℳ0=0, and ℳet>0.Lemma 1 (see [38]).
LetVt,et be a continuous function: ℝ+×ℝn⟶ℝ+ and satisfy Dtα0CVt,et≤ϱVt,et, where 0<α<1 and ϱ is a constant. Then, the following inequality holds:(13)Vt,et≤V0,e0Eαϱtα.Lemma 2 (see [39]).
Ifz3>1,z4>1 and z3−1z4−1=1, the following inequality holds:(14)z1z2≤κz3z3z1z3+1z4κz4z2z4,where κ is an arbitrary positive number and z1,z2∈ℝ2.


## 3. Main Results

In this section, two state feedback controllers are provided for achieving synchronization of FOMNNs.The following two state feedback controllers are provided:(15)ult=λlelt,(16)ult=σlelt+γlsignelt,where λl,σl and γl denote the control parameters, l∈ℕ.Theorem 1.
By holding Assumption1, the drive FOMNN (3) is globally synchronized with the response FOMNN (8) via the controller (15), if there are two positive constants ς1 and ς2 so that(17)2al+λl−∑m=1nb´lmηmς1+b´mlηlς1+blm∗∗−blm∗ηmς2+bml∗∗−bml∗ηlς2>0,l∈ℕ.Proof.
Consider the Lyapunov function,(18)Vt,et=∑l=1n12el2t.
Also, by calculating the Caputo fractional derivative along the trajectory of (11) with 0<α<1, we get(19)Dtα0CVt,et≤∑l=1nelt0CDlαelt,=∑l=1nelt−alelt+∑m=1bb⌣lmrmtζ˜memt+∑m=1nb⌣lmrmt−b⌣lmwmtgmwmt−λlelt,≤∑l=1n−al+λlel2t+∑l=1n∑m=1nb´lmeltζ˜memt+∑l=1n∑m=1nb⌣lmrmt−b⌣lmwmtgmwmtelt.
By utilizing Lemma2, we get(20)∑l=1n∑m=1nb´lmeltζ˜memt≤∑l=1n∑m=1nb´lmeltηmemt≤∑l=1n∑m=1nb´lmηm12eltς11/22+12emtς1−1/22,=∑l=1n∑m=1nb´lmηmς12el2t+∑l=1n∑m=1nb´lmηm2ς1em2t,=∑l=1n∑m=1nb´lmηmς12+b´mlηl2ς1el2t.
Similarly,(21)∑l=1n∑m=1nb⌣lmrmt−b⌣lmwmtgmwmtelt≤∑l=1n∑m=1nblm∗∗−blm∗eltηmemt≤∑l=1n∑m=1nblm∗∗−blm∗ηmς22+bml∗∗−bml∗ηl2ς2el2t.
Substituting (20), (21) into (19),(22)Dtα0CVt,et≤∑l=1n−al+λlel2t+∑l=1n∑m=1nb´lmηmς12+b´mlηl2ς1e12t+∑l=1n∑m=1nblm∗∗−blm∗ηmς22+bml∗∗−bml∗ηl2ς2el2t,=∑l=1n−a1−λl+∑m=1nb´lmηmς12+b´mlηl2ς1+blm∗∗−blm∗ηmς22+bml∗∗−bml∗ηl2ς2el2t≤−min1≤l≤n2al+λl−∑m=1nb´lmηmς1+b´mlηlς1+blm∗∗−blm∗ηmς2+bml∗∗−bml∗ηlς2Vt,et.
According to (17), we choose a constant ϱ>0 so that(23)−min1≤l≤n2al+λl−∑m=1nb´lmηmς1+bml′ηlς1+blm∗∗−blm∗ηmς2+bml∗∗−bml∗ηlς2≤−ϱ<0.
From (22) and (23), we get(24)Dtα0CVt,et≤−ϱVt,et.
By utilizing Lemma1, we can know(25)Vt,et≤V0,e0Eα−ϱtα,namely,(26)et2≤rs−ws2Eα−ϱtα1/2.
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). The proof of Theorem 1 is completed.Theorem 2.
By holding Assumption1, the drive FOMNN (3) is globally synchronized with the response FOMNN (8) via controller (16), if there are positive constants ρl so that(27)∑m=1nρlblm∗∗−blm∗M˜m−ρlγl≤0,(28)al+σl−∑m=1nρmb´mlηlρl>0.Proof.
Firstly, we can know that, ont∈0,+∞, elt is a differentiable and continuous function. Therefore, delt/dt is piecewise continuous, and lims⟶t+dels/ds exits for ∀t∈ℝ+. Then, we choose the Lyapunov function:(29)Vt,et=∑l=1nρlelt.
Now, by calculating the Caputo fractional derivative along the trajectory of (11) with 0<α<1, we get(30)0CDtαVt+,et+≤∑l=1nρlsigneltDtα0Celt,=∑l=1nρlsignelt−alelt+∑m=1nb⌣lmrmtζ˜memt+∑m=1nb⌣lmrmt−b⌣lmwmtgmwmt−σlelt−γlsignelt≤∑l=1n−ρlal+σlelt+∑m=1nρlb´lmζ˜memt+∑m=1nρlb⌣lmrmt−b⌣lmwmtgmwmt−ρlγl≤∑l=1n−ρlal+σlelt+∑m=1nρlb´lmη˜memt+∑m=1nρlblm∗∗−blm∗∗M˜m−ρlγl.
From (27), it follows that(31)Dtα0CVt+,et+≤∑l=1n−ρlal+σlelt+∑m=1nρlb´lmηmemt≤∑l=1n−ρlal+σlelt+∑l=1n∑m=1nρmb´mlηlelt=∑l=1n−al+σlρlelt+∑l=1n∑m=1nρmb´mlηlρlρlelt≤−min1≤l≤nal+σl−∑m=1nρmb´mlηlρlVt,et.
According to (28), we choose a constant ϱ>0 so that(32)−min1≤l≤nal+σl−∑m=1nρmb´mlηlρl≤−ϱ<0.
From (31) and (32),(33)Dtα0CVt,et≤−ϱVt,et.
By utilizing Lemma1, we can know(34)Vt,et≤V0,e0Eα−ϱtα,that is,(35)∑l=1nρlelt≤∑l=1nρlrl0−wl0Eα−ϱtα,and then,(36)∑l=1nelt≤max1≤l≤nρlmin1≤l≤nρl∑l=1nrl0−wl0Eα−ϱtα=∑l=1nmax1≤l≤nρlmin1≤l≤nρlrl0−wl0Eα−ϱtα.
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). The proof of Theorem 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 Assumption1, the drive FOMNN (3) is globally synchronized with the response FOMNN (8) via controller (15), if(37)al+λl−∑m=1nblm∗∗−blm∗ηmς2+bml∗∗−bml∗ηlς2−λmaxΞ>0,l∈ℕ,where(38)Ξ=0,b´lmηmn×n,b´lmηmn×nT,0,2n×2n.Proof.
Firstly, according to the matrix theory, we can easily know that−θ is also the eigenvalue of Ξ if θ is its eigenvalue. Therefore, the maximum eigenvalues of matrix Ξ is greater than zero, that is, λmaxΞ>0.
Consider the Lyapunov function,(39)Vt,et=∑l=1nel2t.
Then, by calculating the Caputo fractional derivative along the trajectory of (11) with 0<α<1, we get(40)0CDtαVt,et≤∑l=1n2eltDtα0Celt,=∑l=1n2elt−alelt+∑m=1nb⌣lmrmtζ˜memt+∑m=1nb⌣lmrmt−b⌣lmwmtgmwmt−λlelt≤∑l=1n−2al+λlel2t+∑l=1n∑m=1n2b´lmeltζ˜memt+∑l=1n∑m=1n2b⌣lmrmt−b⌣lmwmtgmwmtelt≤∑l=1n−2al+λlel2t+etT,ζ˜′etTTΞetT,ζ˜′etT+∑l=1n∑m=1nblm∗∗−blm∗ηmς2+bml∗∗−bml∗ηlς2el2t≤∑l=1n−2al−2λl+∑m=1nblm∗∗−blm∗ηmς2+bml∗∗−bml∗ηlς2el2t++2λmaxΞ∑l=1nel2t≤−2min1≤l≤nal+λl−∑m=1nblm∗∗−blm∗ηmς2+bml∗∗−bml∗ηlς2−λmaxΞVt,et,where et=e1t,e2t,…,ent,ζ˜′et=1/η1ζ˜1e1t,1/η2ζ˜2e2t,…,1/ηnζ˜nent.
According to (37), we choose a constant ϱ>0 so that(41)−min1≤l≤nal+λl−∑m=1nblm∗∗−blm∗ηmς2+bml∗∗−bml∗ηlς2−λmaxΞ≤−ϱ<0.
From (40) and (41), we can get(42)Dtα0CVt,et≤−2ϱVt,et.
By utilizing Lemma1, we can know(43)Vt,et≤V0,e0Eα−2ϱtα,namely,(44)∑l=1ne2t≤∑l=1nr10−w102Eα−2ϱtα,and then,(45)∑l=1ne2t21/2≤∑l=1nrl0−wl022Eα−2ϱtα1/2.
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). The proof of Theorem 3 is completed.Remark 1.
Free-weighting parametersς1 and ς2 are introduced in synchronization criterion (17) of Theorem 1. Also, free-weighting parameters ς1 and ς2 can be used to reduce the conservativeness of the synchronization criterion.Remark 2.
The 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.


## 4. Numerical Simulation

Now, we provide a numerical simulation to illustrate the effectiveness of results.We consider the FOMNN (8) as the response system and the drive FOMNN as follows:(46)Dtα0Cr1t=−r1t+b11r1tg1r1t+b12r2tg2r2t+J1,Dtα0Cr1t=−1.2r2t+b21r1tg1r1t+b22r2tg2r2t+J2,where t∈0,+∞,J1=J2=0, α=0.98,glrlt=tanhrlt,l=1,2, and(47)b11r1t=2,r1t≤1,2.2,r1t>1,b12r2t=−1.2,r2t≤1,−0.9,r2t>1,b21r1t=1.8,r1t≤1,2.1,r1t>1,b22r2t=1.7,r2t≤1,2.2,r2t>1.Then, we can obtainb´11=2.2,b´12=1.2,b´21=2.1,b´22=2.2,η1=η2=1, and M˜1=M˜2=1. Let r10=1.1,r20=−0.8,w10=−0.9,w20=0.6; Figure 1 depicts the chaotic behavior, error and state trajectories of the derive FOMNN (46), and the response FOMNN (8) without an external controller, which are not synchronous.Figure 1
The chaotic behavior, error signal and state trajectories of the derive FOMNN (46), and the response FOMNN (8) without a controller.
(a)(b)(c)(d)Now, according to (17) in Theorem 1, we can get(48)21+λ1−3.4ς1−4.3ς1−0.5ς2−0.5ς2>0,(49)21.2+λ2−4.3ς1−3.4ς1−0.8ς2−0.8ς2>0.Next, we chooseς1=ς2=1 for convenient calculation, then we can obtain λ1>3.35, λ2>3.45, and choose λ1=3.4,λ2=3.5. The condition of Theorem 1 is satisfied, that is, the drive FOMNN (46) and the response FOMNN (8) are globally synchronized via the controller u1t=3.4e1t,u2t=3.5e2t. Let r10=1.1,r20=−0.8,w10=−0.9,w20=0.6, and then, the chaotic behaviors, error and state trajectories of the drive FOMNN (46), and the response FOMNN (8) via the controller u1t=3.4e1t,u2t=3.5e2t are shown in Figure 2.Figure 2
The chaotic behaviors, error and state trajectories of the derive FOMNN (46), and the response FOMNN (8) with the state feedback controller u1t=3.4e1t,u2t=3.5e2t.
(a)(b)(c)(d)Similarly, according to (27) in Theorem 2 and by selecting ρ1=ρ2=1, we can get γ1≥0.5,γ2≥0.7 and choose γ1=0.5,γ2=0.7. Moreover, according to (28) in Theorem 2, we can get σ1≥3.3,σ2≥2.2 and choose σ1=3.4,σ2=2.3. The conditions of Theorem 2 are satisfied, that is, the drive FOMNN (46) and the response FOMNN (8) are globally synchronized via the controller u1t=3.4e1t+0.5signe1t,u2t=2.3e2t+0.7signe1t. Let r10=−1.9,r20=−1.8,w10=1.2,w20=−0.6; then, the chaotic behaviors, error and state trajectories of the drive FOMNN (46), and the response FOMNN (8) via the controller u1t=3.4e1t+0.5signe1t,u2t=2.3e2t+0.7signe2t are shown in Figure 3.Figure 3
The chaotic behaviors, error and state trajectories of the derive FOMNN (46), and the response FOMNN (8) with the state feedback controller u1t=3.4e1t+0.5signe1t,u2t=2.3e2t+0.7signe2t.
(a)(b)(c)(d)Moreover, in Theorem3, system parameters matrix Ξ can be easily obtained as(50)Ξ=002.21.2002.12.22.21.2002.12.200.By easily calculating, the maximal eigenvalue of system parameters matrixΞ is λmaxΞ=3.7875. Next, according to (37) in Theorem 3, choosing ς1=ς2=1, we can obtain λ1>3.7875 and λ2>3.9875 and choose λ1=3.8,λ2=4. The condition of Theorem 3 is satisfied, that is, the drive FOMNN (46) and the response FOMNN (8) are globally synchronized via the controller u1t=3.8e1t,u2t=4e2t. Let r10=1.5,r20=−3,w10=−1.3,w20=2.6; then, the chaotic behaviors, error and state trajectories of the drive FOMNN (46), and the response FOMNN (8) via the controller u1t=3.8e1t,u2t=4e2t are shown in Figure 4.Figure 4
The chaotic behaviors, error and state trajectories of the derive FOMNN (46), and the response FOMNN (8) with the state feedback controller u1t=3.8e1t,u2t=4e2t.
(a)(b)(c)(d)


## 5. 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. Then, 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. 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.



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