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

Authors: Ruihan Chen; Tianfeng Zhao
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

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 order0<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.


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## Body

## 1. Introduction

Memristor was predicted as the fourth circuit element describing the relationship between magnetic flux and voltage by professor Chua [1] in 1971. This component was established successfully by HP Laboratories [2, 3] in 2008. Memristors are used instead of traditional resistive elements to simulate brain neuron synapses and build the memristor neural networks (MNNs) model [4–9] because they have memory characteristics. Now, it has been widely used in the field of information processing, associative memory, and image processing [10–12].Fractional calculus originated in the 17th 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 [14–16]. FMNNs can describe the memory properties of neurons more accurately and achieve many results in synchronization and stability [17–20]. The 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 [21–24]. 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. Therefore, it is very meaningful to study the synchronization problem in the case of parameter mismatch [25–28].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. The 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 fractional-orderα, 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, ℝn denotes a dimensional Euclidean space, and ℝm×n is the set of m×n real matrices. For any matrix B, BT denotes the transpose of B and I denotes the identity matrix. For any algebraic operations, matrices are assumed to have compatible dimensions. The notation ⋅ denotes the matrix 2-norm or the Euclidean vector norm. Let C−∞,0,ℝn denotes the family of continuous functions from −∞,0 to ℝn. λminA and λmaxA 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 χt as a continuous function except at some finite number of points tk at which χtk+=χtk and χtk− exist; then the set of piecewise right continuous function χt is defined as PCl=χ|χ:−τ,∞⟶ℝl.


## 1.1. Notations

ℝ is the space of real number, ℕ is the space of nonnegative integers, C is the space of complex number, ℝn denotes a dimensional Euclidean space, and ℝm×n is the set of m×n real matrices. For any matrix B, BT denotes the transpose of B and I denotes the identity matrix. For any algebraic operations, matrices are assumed to have compatible dimensions. The notation ⋅ denotes the matrix 2-norm or the Euclidean vector norm. Let C−∞,0,ℝn denotes the family of continuous functions from −∞,0 to ℝn. λminA and λmaxA 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 χt as a continuous function except at some finite number of points tk at which χtk+=χtk and χtk− exist; then the set of piecewise right continuous function χt is defined as PCl=χ|χ:−τ,∞⟶ℝl.


## 2. Preliminaries and System Description

### 2.1. 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. Therefore, we use Caputo fractional operator as the main tool in this paper.Definition 1.
(see [13]). The fractional integral of order α for a function ft is defined as(1)I0tαft=1Γα∫0tt−τα−1fτdτ,where t≥0, α>0, and Γα=∫0+∞e−ttα−1dt.Definition 2.
(see [13]). The Caputo fractional derivative of order α for a function ft is defined by(2)D0tαft=1Γn−α∫0tfnτt−τα−n+1dτ,where n−1<α<n; n is a positive integer. The Laplace transform of the Caputo fractional-order derivative is(3)ℒD0tαft=sαℒft−∑k=0n−1sα−k−1fk0,where α>0, n=α+1.Lemma 1.
LetZ1s and Z2s∈ℙℂ1 be jumping discontinuity at s=sk,s≥0,Z1sk+, and Z2sk+ exist, Z1sk=Z1sk−, Z2sk=Z2sk−. T he initial value Z1s=Z2s=h0≥0. If there are constants c1,c2>0, 0<α<1, and μ>0, such that(4)D0tα0Z1s≤c1Z1s+c2,s≠sk,Z1sk+≤μZ1sk−,k∈ℕ,(5)D0tαZ2s≥c1Z2s+c2,s≠sk,Z2sk+=μZ2sk−,k∈ℕ,and Z1s≤Z2s for s≤0, then Z1s≤Z2s for s≥0.Proof .
From (4) and (5), we have(6)Z1s≤h0Eαc1sα+∫0ss−tα−1×Eα,αc1s−tαc2dt,(7)Z2s≥h0Eαc1sα+∫0ss−tα−1×Eα,αc1s−tαc2dt.
We first prove(8)Z1s≤Z2s,0,t1.
AsZ1s,Z2s are continuous functions and Z1s≤Z2s for s≤0, according to (6) and (7), we have Z2s≥Z1s, and therefore (8) holds. Then, assume Z1s≤Z2s for t∈tk−1,tk,k≤c, and c∈ℕ. Then, we have Z1sc≤μZ1sc−≤μZ2sc−=Z2sc. In a similar manner to the proof of (8), we can have Z1s≤Z2s for s∈tc,tc+1. By mathematical induction, it is easy to conclude that Z1s≤Z2s for s≥0. This completes the proof.Lemma 2.
(see [28]). xt∈ℝn is a derivable and continuous function, for t≥t0; we have(9)Dt0tαxTtMxt≤2xTtMDt0tαxt,α∈0,1,where M∈ℝn×n is a positive definite matrix.
The Mittag–Leffler function is defined in the following.Definition 3.
(see [13]). Two-parameter Mittag–Leffler function is defined as(10)Eα,βt=∑k=0∞tkΓkα+β,where α>0, β>0, and t∈C. When β=0, we have(11)Eαt=∑k=0∞tkΓkα+1,where α>0, t∈C.Lemma 3.
(see [29]). t∈ℝ, t>0, and 0<α<1. Eαt is a monotone increasing function. For any integer N>1, and t≠0, argt>π/2, then we have the following asymptotic expansion:(12)Eα,βt≤1αt1−β/αet1/α−∑p=1Nt−pΓβ−αp+O1tN+1.
So, there existst0>0, when t>t0, and we have(13)Eαt≤1αet1/α,Eα,αt≤1αt1−α/αet1/α.


### 2.2. System Description

Now, we consider the following FMNNs as a drive system:(14)D0tαxit=−dixit+∑j=1nbijxjtfjxjtxi0=x0i,where i,j∈ℕ; α is the fractional order; xit represents the state variable of the ith neuron; di>0 denotes the self-feedback connection weight; x0=x01,x02,…,x0nT is the initial value of state vector; fj⋅ denote the activation functions; and bijxjt are memristive connection weights satisfy the following conditions:(15)bijxjt=bij∗,xjt≤ℸxj,bij∗∗,xjt>ℸxj,where the switching jumps ℸxj>0, bij∗, and bij∗∗ are all constants.The corresponding response system is described as(16)D0tαyit=−diyit+∑j=1nb˜ijyjtfjyjt,yi0=y0i,where θit is the impulsive controller, and the memristor-based connection weights b˜ijyjt satisfy the following condition:(17)b˜ijyjt=b˜ij∗,yjt≤ℸyj,b˜ij∗∗,yjt>ℸyj,where i,j∈ℕ and switching jumps ℸyj>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:(18)D0tαxit∈−dixit+∑j=1nKbijxjtfjxjt.Letb^ij=minbij∗,bij∗∗,bˇij=maxbij∗,bij∗∗, and cobij∗,bij∗∗=b^ij,bˇij. The set-valued maps are defined as(19)Kbijxjt=bij∗,xjt<ℸxj,cobij∗,bij∗∗,xjt=ℸxj,bij∗∗,xjt>ℸxj,and, besides, there exist measurable functions ηijxjt∈Kbijxjt such that(20)D0tαxit=−dixit+∑j=1nηijxjtfjxjt.Similarly, there exist measurable functionsηij˜yjt=Kb˜ijyjt such that(21)D0tαyit=−diyit+∑j=1nηij˜yjtfjyjt+θit.To ensure the uniqueness and existence of the solution of system (20) and (21), we make the following assumption.Assumption 1.
There exists a positive constantL´j such that(22)fjx−fjy≤L´jx−y,for all x,y∈ℝ,j∈ℕ.Lemma 4.
(see [30]). Let Assumption 1 be satisfied; then the following inequalities hold:(23)ηij˜yjtfjyjt−ηijxjtfjxjt≤b¯ijL´jejt+ΔbijL´jℸmax,where b¯ij=maxbij∗,bij∗∗,ℸmax=maxℸxj,ℸyj,Δbij=bij∗−bij∗∗.Remark 1.
To reduce the conservativeness of our results, the constraints on activation functionsfj±ℸxj=fj±ℸyj=0, gj±ℸxj=gj±ℸyj=0 are removed in this paper.
Denoteet=yt−xt as the synchronization error and the impulsive controller is designed as(24)θit=−aieit+∑k=1∞μk−1eitδt−tk,where ai are nonnegative constants, i∈ℕ; impulsive intensity μk≠0; δ⋅ denotes the Dirac impulsive function; and the time series ζ=t1,t2,… is a strictly increasing sequence of impulsive instants which satisfies tk−1<tk and limk⟶+∞tk=+∞.
With considering the impulsive controller (24), the controlled error neural networks could be rewritten as follows:(25)D0tαeit=−di+aieit+∑j=1nηij˜yjtfjyjt−ηijxjtfjxjt,t≠tk,Δeitk=eitk+=μkeitk−,k∈ℕ,where eit is right-hand continuous at t=tk and the initial conditions ei0=e0i∈C−∞,0,ℝn for i∈ℕ.


### 2.3. The Definition of Quasi-Synchronization and Average Impulsive Interval

Now, the definitions of quasi-synchronization and average impulsive interval are given as follows.Definition 4.
(see [24]). If there exists a compact set Θ for any ei0∈C−∞,0,ℝn, with the error et converging to Θ=et≤ϱ as t goes to infinity, then system (14) will achieve quasi-synchronization with error bound ϱ>0.Definition 5.
(see [26]). Consider the impulsive sequence ζ=t1,t2,… in the time interval t,T. ωt,T is represented as the number of impulsive times, if there exist positive numbers N0 and Ta, such that(26)T−tTa−N0≤ωt,T≤T−tTa+N0,∀T≥t≥0,and then, Ta is greater than the average impulsive interval of the impulsive sequence ζ.


## 2.1. 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. Therefore, we use Caputo fractional operator as the main tool in this paper.Definition 1.
(see [13]). The fractional integral of order α for a function ft is defined as(1)I0tαft=1Γα∫0tt−τα−1fτdτ,where t≥0, α>0, and Γα=∫0+∞e−ttα−1dt.Definition 2.
(see [13]). The Caputo fractional derivative of order α for a function ft is defined by(2)D0tαft=1Γn−α∫0tfnτt−τα−n+1dτ,where n−1<α<n; n is a positive integer. The Laplace transform of the Caputo fractional-order derivative is(3)ℒD0tαft=sαℒft−∑k=0n−1sα−k−1fk0,where α>0, n=α+1.Lemma 1.
LetZ1s and Z2s∈ℙℂ1 be jumping discontinuity at s=sk,s≥0,Z1sk+, and Z2sk+ exist, Z1sk=Z1sk−, Z2sk=Z2sk−. T he initial value Z1s=Z2s=h0≥0. If there are constants c1,c2>0, 0<α<1, and μ>0, such that(4)D0tα0Z1s≤c1Z1s+c2,s≠sk,Z1sk+≤μZ1sk−,k∈ℕ,(5)D0tαZ2s≥c1Z2s+c2,s≠sk,Z2sk+=μZ2sk−,k∈ℕ,and Z1s≤Z2s for s≤0, then Z1s≤Z2s for s≥0.Proof .
From (4) and (5), we have(6)Z1s≤h0Eαc1sα+∫0ss−tα−1×Eα,αc1s−tαc2dt,(7)Z2s≥h0Eαc1sα+∫0ss−tα−1×Eα,αc1s−tαc2dt.
We first prove(8)Z1s≤Z2s,0,t1.
AsZ1s,Z2s are continuous functions and Z1s≤Z2s for s≤0, according to (6) and (7), we have Z2s≥Z1s, and therefore (8) holds. Then, assume Z1s≤Z2s for t∈tk−1,tk,k≤c, and c∈ℕ. Then, we have Z1sc≤μZ1sc−≤μZ2sc−=Z2sc. In a similar manner to the proof of (8), we can have Z1s≤Z2s for s∈tc,tc+1. By mathematical induction, it is easy to conclude that Z1s≤Z2s for s≥0. This completes the proof.Lemma 2.
(see [28]). xt∈ℝn is a derivable and continuous function, for t≥t0; we have(9)Dt0tαxTtMxt≤2xTtMDt0tαxt,α∈0,1,where M∈ℝn×n is a positive definite matrix.
The Mittag–Leffler function is defined in the following.Definition 3.
(see [13]). Two-parameter Mittag–Leffler function is defined as(10)Eα,βt=∑k=0∞tkΓkα+β,where α>0, β>0, and t∈C. When β=0, we have(11)Eαt=∑k=0∞tkΓkα+1,where α>0, t∈C.Lemma 3.
(see [29]). t∈ℝ, t>0, and 0<α<1. Eαt is a monotone increasing function. For any integer N>1, and t≠0, argt>π/2, then we have the following asymptotic expansion:(12)Eα,βt≤1αt1−β/αet1/α−∑p=1Nt−pΓβ−αp+O1tN+1.
So, there existst0>0, when t>t0, and we have(13)Eαt≤1αet1/α,Eα,αt≤1αt1−α/αet1/α.


## 2.2. System Description

Now, we consider the following FMNNs as a drive system:(14)D0tαxit=−dixit+∑j=1nbijxjtfjxjtxi0=x0i,where i,j∈ℕ; α is the fractional order; xit represents the state variable of the ith neuron; di>0 denotes the self-feedback connection weight; x0=x01,x02,…,x0nT is the initial value of state vector; fj⋅ denote the activation functions; and bijxjt are memristive connection weights satisfy the following conditions:(15)bijxjt=bij∗,xjt≤ℸxj,bij∗∗,xjt>ℸxj,where the switching jumps ℸxj>0, bij∗, and bij∗∗ are all constants.The corresponding response system is described as(16)D0tαyit=−diyit+∑j=1nb˜ijyjtfjyjt,yi0=y0i,where θit is the impulsive controller, and the memristor-based connection weights b˜ijyjt satisfy the following condition:(17)b˜ijyjt=b˜ij∗,yjt≤ℸyj,b˜ij∗∗,yjt>ℸyj,where i,j∈ℕ and switching jumps ℸyj>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:(18)D0tαxit∈−dixit+∑j=1nKbijxjtfjxjt.Letb^ij=minbij∗,bij∗∗,bˇij=maxbij∗,bij∗∗, and cobij∗,bij∗∗=b^ij,bˇij. The set-valued maps are defined as(19)Kbijxjt=bij∗,xjt<ℸxj,cobij∗,bij∗∗,xjt=ℸxj,bij∗∗,xjt>ℸxj,and, besides, there exist measurable functions ηijxjt∈Kbijxjt such that(20)D0tαxit=−dixit+∑j=1nηijxjtfjxjt.Similarly, there exist measurable functionsηij˜yjt=Kb˜ijyjt such that(21)D0tαyit=−diyit+∑j=1nηij˜yjtfjyjt+θit.To ensure the uniqueness and existence of the solution of system (20) and (21), we make the following assumption.Assumption 1.
There exists a positive constantL´j such that(22)fjx−fjy≤L´jx−y,for all x,y∈ℝ,j∈ℕ.Lemma 4.
(see [30]). Let Assumption 1 be satisfied; then the following inequalities hold:(23)ηij˜yjtfjyjt−ηijxjtfjxjt≤b¯ijL´jejt+ΔbijL´jℸmax,where b¯ij=maxbij∗,bij∗∗,ℸmax=maxℸxj,ℸyj,Δbij=bij∗−bij∗∗.Remark 1.
To reduce the conservativeness of our results, the constraints on activation functionsfj±ℸxj=fj±ℸyj=0, gj±ℸxj=gj±ℸyj=0 are removed in this paper.
Denoteet=yt−xt as the synchronization error and the impulsive controller is designed as(24)θit=−aieit+∑k=1∞μk−1eitδt−tk,where ai are nonnegative constants, i∈ℕ; impulsive intensity μk≠0; δ⋅ denotes the Dirac impulsive function; and the time series ζ=t1,t2,… is a strictly increasing sequence of impulsive instants which satisfies tk−1<tk and limk⟶+∞tk=+∞.
With considering the impulsive controller (24), the controlled error neural networks could be rewritten as follows:(25)D0tαeit=−di+aieit+∑j=1nηij˜yjtfjyjt−ηijxjtfjxjt,t≠tk,Δeitk=eitk+=μkeitk−,k∈ℕ,where eit is right-hand continuous at t=tk and the initial conditions ei0=e0i∈C−∞,0,ℝn for i∈ℕ.


## 2.3. The Definition of Quasi-Synchronization and Average Impulsive Interval

Now, the definitions of quasi-synchronization and average impulsive interval are given as follows.Definition 4.
(see [24]). If there exists a compact set Θ for any ei0∈C−∞,0,ℝn, with the error et converging to Θ=et≤ϱ as t goes to infinity, then system (14) will achieve quasi-synchronization with error bound ϱ>0.Definition 5.
(see [26]). Consider the impulsive sequence ζ=t1,t2,… in the time interval t,T. ωt,T is represented as the number of impulsive times, if there exist positive numbers N0 and Ta, such that(26)T−tTa−N0≤ωt,T≤T−tTa+N0,∀T≥t≥0,and then, Ta is greater than the average impulsive interval of the impulsive sequence ζ.


## 3. Main Results

In this part, we explore the quasi-synchronization of system (25) by impulsive controller (24). For two different impulsive intensities −α<μk≤α and μk≠0 and μk≤−α or μk>α, we have Theorems 1 and 2.Theorem 1.
When impulsive intensity−α<μk≤α and μk≠0, if there exist diagonal matrices Q>0,P>0, scalars β>0, and d=diagd1,d2,…,dn, B¯=bij¯n×n, L´=diagL1´,L2´,…,Ln´, π=NTPN, N=ΔBL´ℸ˜max, ΔB=Δbijn×n, ℸ˜max=diagℸmax,ℸmax,…,ℸmax, ι=maxk∈ℕμk2, if(27)1−2Qd−2Qa+QB¯L´+L´B¯TQ+P−1Q2−βQ<02Υ=ln1/αTa+β1/α<0are satisfied, then error systems (25) achieve quasi-synchronization with error bound β1/α−1/αι/α−N0π/λminQ⋅−Y at the convergence rate h/2, where h is the unique solution of the equation h+lnι/α/Ta+β1/α=0.Proof.
Construct the following Lyapunov function:(28)Vt=∑i=1nQieit2.
Fort∈tk−1,tkk=1,2,…, according to Lemmas 2 and 4, we get(29)D0tαVt≤2∑i=1neitQisigneitD0tαVeit=2∑i=1neitQisigneit−di+aieit+∑j=1nηij˜yjtfjyjt−ηijxjtfjxjt≤−2∑i=1neitQidi+aieit+2∑i=1n∑j=1neitQiηij˜yjtfjyjt−ηijxjtfjxjt≤−2∑i=1neitQidi+aieit+2∑i=1n∑j=1neitQib¯ijL´jejt+ΔbijL´jℸmax=−2etTQd+aet+2etTQ×B¯L´et+2etTQN≤−2etTQd+aet+2etTQ×B¯L´et+etTP−1Q2et+NTPN=etT−2Qd−2Qa+QB¯L´+L´B¯TQ+P−1Q2−βQet+βetTQet+NTPN.
Based on (4) and (5) of Theorem 1, we get(30)D0tαVt≤β∑i=1nQieit2+π=βVt+π.
Fort=tk,k∈N, according to (25), it yields(31)Vtk+=∑i=1NeiTtk+Qieitk+=μk2Vtk−.
LetΛt be the unique solution of the following delayed impulsive comparison system for any γ>0:(32)D0tαΛt=βΛt+π+γ,Λtk+=μk2Λtk−,k=1,2,…,Λt=λmaxQe02.
Whent>0, we have vt≤Λt according to Lemma 1.
Fort∈0,t1, using the Laplace transform on (32), one gets(33)Λt=EαβtαΛ0+∫0tt−sα−1Eα,αβt−sα×π+γds.
By using Lemma3, we have(34)Eαβtα≤1αeβ1/αt,Eα,αβt−sα≤β1/α−1αt−s1−αeβ1/αt−s.
Then, we can transform (33) as(35)Λt≤1αeβ1/αtΛ0+∫0tβ1/α−1αeβ1/αt−s×π+γds.
Fort∈t1,t2, one obtains from (35) and the second equation of (32) that(36)Λt1+≤ιαeβ1/αt1Λ0+∫0t1ιβ1/α−1αeβ1/αt1−s×π+γds,Λt≤1αeβ1/αt−t1Λt1++∫t1tβ1/α−1αeβ1/αt−s×π+γds=ια2eβ1/αtΛ0+ια2∫0t1β1/α−1eβ1/αt−s×π+γds+1α∫t1tβ1/α−1eβ1/αt−s×π+γds.
Fort∈t2,t3,(37)Λt2+≤ι2α2eβ1/αt2Λ0+ι2α2∫0t1β1/α−1eβ1/αt2−s×π+γds+ια∫t1t2β1/α−1eβ1/αt2−s×π+γds,Λt≤1αeβ1/αt−t2Λt2++∫t2tβ1/α−1αeβ1/αt−s×π+γds=ι2α3eβ1/αtΛ0+ι2α3∫0t1β1/α−1eβ1/αt−s×π+γds+ια2∫t1t2β1/α−1eβ1/αt−s×π+γds+1α∫t2tβ1/α−1eβ1/αt−s×π+γds.
By induction, we can derive that, fort∈tk,tk+1,(38)Λt≤∏0≤tk≤tιαeβ1/αt1αΛ0+∫0t∏s≤tk≤tιαeβ1/αt−s×β1/α−1απ+γds.
According to Definition5, we have(39)eβ1/αt−s∏s≤tk≤tια≤eβ1/αt−sιαt−s/Ta−N0≤ια−N0eln1/α/Ta+β1/αt−s.
Substituting (39) into (38) yields(40)Λt≤ια−N0λmaxQ∑i=1Nei021αelnι/α/Ta+β1/αt+∫0tια−N0elnι/α/Ta+β1/αt−sβ1/α−1α×π+γds≤Ωelnι/α/Ta+β1/αt+∫0tβ1/α−1αelnι/α/Ta+β1/αt−s×ια−N0π+γds,where Ω=ι/α−N01/αλmaxQsupe02>0. Define Δh=h+lnι/α/Ta+β1/α. Note that Δh is a continuous function, Δ0=lnι/α/Ta+β1/α<0, and Δ˙h=1>0. Above all, Δh=0 has a unique solution for h>0. Since γ>0,h>0,0<ι<1, we obtain(41)Λt≤ια−N0λmaxQe02<Ωe−ht+β1/α−1/αι/α−N0π+γ/−lnι/α/Ta+β1/α,for t≤0, which implies that (41) holds for all t>0. Assume that if (41) does not hold for all t>0 , then there at least exists a time instant t∗>0,(42)Λt∗≥Ωe−ht∗+β1/α−1/αι/α−N0π+γ−lnι/α/Ta+β1/α,but, for t<t∗, we have(43)Λt<Ωe−ht+β1/α−1/αι/α−N0π+γ−lnι/α/Ta+β1/α.
Denotelnι/α/Ta+β1/α=−r; combining (40)and (43), we have(44)Λt∗≤Ωelnι/α/Ta+β1/αt∗+∫0t∗β1/α−1αelnι/α/Ta+β1/αt∗−s×ια−N0π+γds<e−rt∗Ω+β1/α−1/αι/α−N0π+γr+∫0t∗β1/α−1α×ersια−N0π+γds≤e−rt∗Ω+β1/α−1/αι/α−N0π+γr+ert∗−1β1/α−1/αι/α−N0π+γr=Ωe−ht∗+β1/α−1/αι/α−N0π+γr.
Obviously, inequality (44) contradicts (42), which implies (41) holds for all t>0. Letting γ⟶0, we have(45)λminQet2≤Vt≤Λt<Ωe−ht+β1/α−1/αι/α−N0π−lnι/α/Ta+β1/α.
Thus, according to (45), the error converges into(46)Θ=et∈ℝn|et≤β1/α−1/αι/α−N0πλminQ⋅−Y,when t⟶+∞. The proof for condition 1 is completed.
When impulsive intensityμk≤−α or μk>α, we have Theorem 2.Theorem 2.
When impulsive intensityμk≤−α or μk>α, if there exist diagonal matrices Q>0,P>0, scalars β>0, and d=diagd1,d2,…,dn, B¯=bij¯n×n, L´=diagL´1,L´2,…,L´n, π=NTPN, N=ΔBL´ℸ˜max, ΔB=Δbijn×n, ℸ˜max=diagℸmax,ℸmax,…,ℸmax, ι=maxk∈ℕμk2, if(47)1−2Qd−2Qa+QB¯L´+L´B¯TQ+P−1Q2−βQ<02Υ¯=lnι/αTa+β1/α−β1/α+1<0are satisfied, then error systems (25) achieve quasi-synchronization with error bound β1/α−1/αι/αN0π/λminQ⋅−Y¯ at the convergence rate h′/2, where h′ is the unique solution of the transcendental equation h′+lnι/α/Ta+β1/α−β1/α+1=0.Proof .
The same as the proof in Theorem1, we obtain(48)eβ1/αt−s∏s≤tk≤tια≤eβ1/αt−sιαt−s/Ta+N0≤ιαN0elnι/α/Ta+β1/αt−s.
Substituting (48) into (40) yields(49)Λt≤ιαN0λmaxQ∑i=1Nei021αelnι/α/Ta+β1/αt+∫0tιαN0elnι/α/Ta+β1/αt−sβ1/α−1α×π+γds≤Ω¯elnι/α/Ta+β1/αt+∫0tβ1/α−1αelnι/α/Ta+β1/αt−s×ιαN0π+γds,where Ω¯=ι/αN01/αλmaxQsupe02>0. Define Δh′=h′+lnι/α/Ta+β1/α−β1/α+1. Note that Δ0=lnι/α/Ta+β1/α−β1/α+1<0 and Δ˙h′=1>0. So, function Δh′=0 has a unique solution h′>0.
Similar to (41), we can prove the following inequality is true for all t>0:(50)Λt<Ω¯e−h′t+β1/α−1/αι/αN0π+γ−lnι/α/Ta+β1/α+β1/α+1.
Lettingγ⟶0, we have(51)λminQet2≤Vet≤Λt<Ω¯e−h′t+β1/α−1/αι/αN0π−lnι/α/Ta+β1/α+β1/α+1.
Then, according to (51), the error converges into(52)Θ¯=et∈ℝn|et≤β1/α−1/αι/αN0πλminQ⋅−Υ¯,with the convergence rate h′/2. So far, all proofs are over.Remark 2.
In impulsive dynamical networks or impulsive control mechanism, the impulsive effectμk 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 μk 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 μk is talked over under the value ranges: (1) −α<μk≤α and μk≠0, (2) μk≤−α or μk>α, where μk only have a relation with variable α. The range of gain μk 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<μk≤α and μk≠0, the impulsive effect μk plays a positive role in the quasi-synchronization. In this case, the feedback control gain ai could be set as zero in order to save control costs and manufacture. However, for the case μk≤−α or μk>α, the impulsive effect μk 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 μk=−1, it implies eitk+=−eitk−, which is impossible for impulsive control. If μk=0, eitk+=eitk=0, which means that there is no impulsive control effect.


## 4. 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 α=0.9,fjs=tanhs,j∈ℕ. Choose d1=3.4,d2=1.5,ℸxj=1L´j=1,(53)b11x1=1.3,x1≤1,1.6,x1>1,b12x2=−2,x2≤1,−2.45,x2>1,b21x1=−0.3,x1≤1,−0.6,x1>1,b22x2=2.3,x2≤1,2.6,x2>1.
The switching jumps and memristive connection weights of the response system (16) are the same as system (14). The initial conditions of the drive-response system are x0i=1.5,1T, y0i=0.5,−0.5T. The average impulsive interval is less than Ta=0.02 and the positive constant N0=1.Case 1: when−0.9<μk≤0.9 and μk≠0, for Theorem 1, considering disadvantageous impulsive effects here, set ai=8i∈N, μk=−0.7, so we get ι=μk2=0.49. Denoting β=7, then we have Υ=lnι/α/Ta+β1/α=−21.7099<0. By solving conditions (4) and (5) in Theorem 1, we drive P=diag15.0648,14.9541, Q=diag22.5648,21.4627, N=ΔBL´ℸ˜max=0.75,0.6T, π=NTPN=13.8574 and the corresponding error bound could be calculated as(54)|et≤β1/α−1/αι/α−N0πλminQ⋅−Υ=0.2677.In Figure1, 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 et derived from the theorem. It implies the systems achieve quasi-synchronization with a given error bound.Case 2: whenμk>0.9, for Theorem 2, impulsive effect μk=1.3, so ι=1.12=1.69; we set the control gain ai=0,i∈ℕ. As a similar process in Example 1, we have Υ¯=lnι/α/Ta+β1/α−β1/α+1=−28.6372<0 and(55)|et≤β1/α−1/αι/αN0πλminQ⋅−Y¯=0.2356.
The chaotic attractors of the drive-response system with initial conditions are shown in Figure3. In Figure 4, one can find that the error is under the bound 0.1647, which is less than the error bound et derived from the theorem. It can be obviously seen that the drive-response systems achieved quasi-synchronization.Figure 1
Chaotic attractor of the drive-response system whenμk=−0.7.Figure 2
The synchronization erroret when μk=−0.7.Figure 3
Chaotic attractor of the drive-response system whenμk=1.3.Figure 4
The synchronization erroret when μk=1.3.


## 5. 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.



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