# Multiple Quasisynchronization of Uncertain Fractional-Order Delayed Neural Networks by Impulsive Control Mechanism

Authors: Biwen Li; Lin Xu
Journal: Complexity
 (2022)

---

## Abstract

We study the dynamical behavior of multiple quasi-synchronization of a type of fractional-order coupled neural networks (FCNNs) with delay and uncertain parameters. By utilizing the pinned pulse control strategy technique, we establish a new pulse controller, which realizes the multiple quasisynchronization of the system. Furthermore, we derive some new criteria of multiple quasisynchronization by using the comparison principle and mathematical analysis. Eventually, simulations are carried out with two examples to explicate the effectiveness of the conclusions.


---

## Body

## 1. Introduction

Fractional-order calculus is related to model memory, complexity, and heritability, so it has advantages over integer calculus (see [1, 2]). In many actual questions, we generally first consider the fractional dynamic system because it can better descript the actual problem than the integral order dynamic system. There have been many reports on fractional-order system dynamics. It plays an extremely significant effect in the modeling of engineering system, power system, and physical system (see [3, 4]). In Reference [3], Bao and Cao combined Caputo derivatives and fractional calculus inequalities and sufficiency condition for projection synchronization of fractional memristor-based neural networks (FMNNs) was theoretically derived. Xu et al. in Reference [5] studied a new fractional Hopfield neural network chaotic system and its application in image encryption. Li et al. in Reference [4] studied the application of neural network fractional-order PID in the control of piezoelectric stacks.Parameter uncertainty is caused by incomplete understanding of some knowledge of mathematical model, such as constitutive law and empirical quantity (see [6]). In various engineering discipline systems, the model parameters studied are often uncertain, so the parameter uncertainty needs to be considered when facing the actual system. Fortunately, in recent years, many scholars have considered the parameter uncertainty in the model. In Reference [7], the author designs an appropriate event triggering mechanism and controller to ensure the stability of randomly nonlinearity system of time lag with uncertain parameters. However, with the continuous maturity of technology, the task of designing a good controller for an uncertain fractional-order neural network so that the network can achieve the desired effect is still very arduous, and many problems need to be further studied.Synchronization is an extremely important dynamic behavior in complex dynamic networks, which has a wide range of applications in many fields. Therefore, many researchers have studied it (see [8]). Moreover, the synchronization behavior of fractional-order coupled neural networks (FCNNs) is also discussed. For example, Xu et al. designed a suitable controller in Reference [9], so that FCNNs with time-variable delays could realize the synchronization behavior in a finite time. Chen et al. [10] investigated the synchronization of FMNNs with time lag. It is a well-known fact that time delay often exists in plenty of complex networks; hence, it is very significant to premeditate time delay when studying FCNNs.In general, the coupled neural network is not synchronized without external force interference, so it is necessary to develop a controller to make it synchronize. Some scholars also use various control technologies, for example, impulse control, adaptive control, pinning control, and feedback control to achieve synchronization. However, if there are too many nodes in the network, the cost of applying controller to each node is too high and difficult to implement. Therefore, it is possible to try to control the network by only controlling the fixed part of the time and some nodes, so as to arrive the purpose of reducing the control cost. But, few authors have applied the pinning impulse control project to NNs (see Reference [11]). In Reference [11], Wang et al. theoretically derived a few sufficient conditions for pinning synchronization and robust synchronization of FCNNs through the pinning control strategy. Also, some networks can only achieve quasisynchronization due to external and internal interference, and there are few studies on quasisynchronization of NNs with couple (see [12, 13]). In Reference [12], Feng et al., based on the matrix-related knowledge theory and Lyapunov functional method, derived several simple sufficient optimality conditions for quasisynchronization of coupled memristor NNs theoretically, and a suitable controller is constructed to ensure the quasisynchronization of such networks. In Reference [13], Lv et al. introduced a type of activation function and a few sufficient conditions to guarantee that each subnetwork in the time-delay coupled neural network has multiple equilibrium states and made the network achieve dynamic and static multisynchronization by constructing an appropriate impulse controller and Lyapunov function. Based on the abovementioned phenomenon, this paper will research the multiple quasisynchronization issue of FCNNs with uncertain parameters and delay by pinning pulse control method.For as much as the above discussion, the major dedications of this article involve the following: (1) The multiple quasisynchronization problem of FCNNs with uncertain parameters and time delays is studied. (2) The concept of multiple quasisynchronization is proposed. (3) Aiming at the problem of multiple quasisynchronization, a new method combining pinning and pulse control is proposed.The rest of the main content of this article is as follows: Section2 mainly describes the prerequisites and models required in this article. The primary contribution is in Section 3. Section 4 gives two examples that demonstrate the validity of the conclusion. Finally, Section 5 gives the main conclusions of this article.


## 2. Preliminary Knowledge and Model Description

### 2.1. Fraction-Order Calculus

Firstly, existing definitions of fraction-order calculus are given, which can be seen in Reference [14], that are needed later.Define the Gamma functionΓ⋅ as below:(1)Γp=∫t0+∞tp−1exp−tdt,where p>0.Define the Caputo fractional derivativeDct0,tpg⋅ of the function gt as below:(2)cDt0,tpgt=1Γn−p∫t0tgnst−sp−n+1ds,where t0 is the initial time, t≥t0, p is the order, n−1<p<n,n∈Z+.Define the fractional integralIt0,tpg⋅ of the function gt as below:(3)It0,tpgt=1Γp∫t0tt−sp−1gsds,where t0 is the initial time, t≥t0.Define the Mittag-Leffler function with single parameterEp⋅ as below:(4)Eps=∑k=0+∞skΓkp+1,where p>0,s is a complex number.Define the Mittag-Leffler function with double parametersEp,p¯⋅ as below:(5)Ep,p¯s=∑k=0+∞skΓkp+q¯,where p>0,p¯>0,s is a complex number.


### 2.2. Model Description

A collection that makesZ+ a positive integer. The superscript T represents the transpose, and #ℑ is an element in the finite collection ℑ. Denote Rn is the set of n-dimensional real-valued vectors. R+ is the group of fixed non-negative numbers. For an arbitrary vector a∈Rn and the existence of a constant σ0>0, we record ℳa,σ0=xx−a<σ0 as a set of vectors, where the distance between x and a is less than σ0. The set of n×n real matrices is written as Rn×n. If a real matrix X>0, then X is a positive definite matrix. A⊗B represents the Kronecker product of matrices A and B. For any matrix A, λminA,λmaxA denotes its minimum eigenvalue and maximum eigenvalue, respectively, and the norm of A is defined as ‖A‖=λmaxATA1/2. In this article, we regard the following FCNNs with N same nodes, uncertainties, and time delays.(6)cDt0,tαxit=−P^+ΔPtxit+Q^+ΔQtfixit+R^+Δℛtfixit−τ+∑j=1NG^ijΓxjt+J,where i=1,2,…,N, and N≥2 represents the quantity of subnetworks; τ represents the time delay in transmission; xit=xi1t,xi2t,…,xintT is the state vector of i-th neuron; ΔPt,ΔQt,Δℛt are the norm-bounded parametric uncertainties; P^ is a diagonal matrix that expresses the self-feedback item of the j-th network, in which the diagonal elements are p1,p2,…,pn,pi>0. Q^=Qijn×n is the connection weight matrices and R^=rijn×n is the time lag join matrices, where i,j=1,2,…,n. fixit is the activation function; G^=G^ijN×N is the coupled matrix, when there exists a connection among the i-th node with the j-th node, i≠j,G^ij≠0, if not, G^ij=0, in which the diagonal elements are defined by G^ii=−∑j=1,j≠iNG^ij; Γ=diagγ1,γ2,…,γn is the internal coupling matrix; J=J1,J2,…,JN is the input vector.Next, we give several basic assumptions.(A1) The activation functionfi⋅ is continuous, for any vector x,y, exists ℒi>0, and the following formula holds:(7)fix−fiy≤ℒix−y.(A2)(8)ΔP=A^1C^1tℬ^1,ΔQ=A^2C^2tℬ^2,Δℛ=A^3C^3tℬ^3,where A^i,ℬ^ii=1,2,3 are constant matrices with the corresponding matching dimensions and C^iti=1,2,3 is an indeterminate matrix, where C^iTtC^it≤I (I is the unity matrix with the corresponding matching dimensions).Remark 1.
#Dk=κk means that the group Dk has κk nodes and κk≠0.
For any initial statext=x1Tt,x2Tt,…,xNTtT where xit∈C−τ,0,R,i=1,2,…,N, for any given initial value condition, there is a solution st, if all the node trajectories in the network satisfy the formula limt⟶∞‖xit−st‖=0,i=1,2,…,N, then this network is called complete synchronization. Furthermore, if the margin of error σ>0, exists T>0, for all xt and ∀t>T, xit−st<σ holds, then this network is called uniformly quasisynchronized.Definition 1 (see [15]).
For an arbitrary complex network withN nodes, D1,D2,…,Dm is a set of disjoint nodes, that is, ∑k=1mDk=1,2,…,N, Dk=lk1,lk2,…, Dk∩Du=∅ for k≠u. The network is called multiple quasisynchronization with the error vector δ=δ1,δ2,…,δmT>0 under any initial value conditions, if there exist a series of reference solutions s1t,s2t,…,smt and for any constant κ>0 small enough, T exists, for ∀t>T, the nodes xit∈ℳskt,σk,i∈Dk holds, in which ℳskt,σk≠ℳsut,σ¯u,u≠k.Remark 2.
It can be seen from Definition1 that skt is the reference trajectory for all nodes in set #Dk.
The target trajectorysi¯t satisfies the following formula:(9)cDt0,tαsi¯t=−P^+ΔPtsi¯t+Q^+ΔQtfi¯si¯t+R^+Δℛtfi¯si¯t−τ+J,where if xit∈ℳskt,σk, then si¯t=skt, i=1,2,…,N,k=1,2,…,m.
Now let’s noteε¯it=xit−si¯t is the error signal, where i=1,2,…,N, and devise a new pinned pulse controller as shown below:(10)uit=∑h=1+∞θkε¯itδt−th,i∈ℑkth,#ℑkth=ωk,0,i∉ℑkth,where δ⋅ and θk represents the Dirac impulsive function and impulsive gain, respectively, and thh=0,1,2,… indicates the instant of the pulse that satisfies 0=t1<t2<⋯<th<⋯,limth⟶+∞th=+∞. The node set on t=th is represented by ∑k=1mℑkth=ℑ1th,ℑ2th,…,ℑmth⊂D1,D2,…,Dm⊂1,2,…,N, and make 0<ωk≤κk,k=1,2,…,m, namely, ℑkth is a subset of Dk, and ℑkth represents the set of pinned nodes at t=th. Assume the error vector ε¯i1≥ε¯i2≥⋯≥ε¯in. Under the pulse controller (10), the system of errors can be described:(11)cDt0,tαε¯it=−P^+ΔPtε¯it+Q^+ΔQtf˜iε¯it+R^+Δℛtf˜iε¯it−τ+∑j=1NG^ijΓε¯it,t≠thε¯ith+=1+θkε¯ith−,i∈∑k=1mℑkth,ε¯ith+=ε¯ith−,i∉∑k=1mℑkth,where h=0,1,2,…,ε¯ith+=limt⟶th+ε¯it,ε¯ith−=limt⟶th−ε¯ith,f˜iε¯it=fixit−fi¯si¯t,f˜iε¯it−τ=fixit−τ−fi¯si¯t−τ, and f˜i0=0.
The initial value condition of the above error system (11) is as follows:(12)ε¯is=ϕis,s∈−τ,0,where ϕis∈C−τ,0,Rn and i=1,2,…,N.Lemma 1 (see [16]).
Ifxt∈Rn is a vector-valued function that is differentiable and continuous for t, next for any α∈0,1 and t≥t0, we have the following relationship:(13)cDt0,tαxTtPxt≤2xTtPcDt0,tαxt,where P∈Rn×n is a constant matrix that is symmetric and positive definite.Lemma 2 (see [17]).
Letℛ^,W^, and S^t be the real matrices corresponding matching dimensions, then if there is S^TtS^t≤I, then there is the following equation:(14)ℛ^TS^TtW^T+W^S^tℛ^≤1ξℛ^Tℛ^+ξW^W^T,where ξ>0 is the constant.Lemma 3 (see [17]).
Letℛ^ and W^ be the real matrices corresponding matching dimensions, then(15)ℛ^TW^+W^TR^≤ξℛ^Tℛ^+1ξW^TW^,where ξ>0 is the constant.Lemma 4 (see [18]).
For arbitrary vectorx1,x2∈Rn and Q∈Rn×n which is a positive definite matrix, we have the below inequalities hold:(16)2x1Tx2≤x1TQ−1x1+x2TQx2.Lemma 5 (see [19]).
For positive definite matrixR, vector xi with proper dimensionality and symmetric matrix W, then we have the following:(17)λminR−1WxiTRxi≤xiTWxi≤λmaxR−1WxiTRxi,where λmax⋅,λmin⋅ denote the maximum and minimum eigenvalues, respectively, and R−1 stands for the inverse of a matrix R.Definition 2.
The definition of the pinning rateηk at t=th is as follows:(18)∑i∈ℑkthε¯iTth−ε¯ith−∑i∈Dkε¯iTth−ε¯ith−=ηk,here the pinning rate ηk is related to time and impulse instants, and we can also determine the lower bound of the pinning rate ηk.Lemma 6 (see [20]).
Consider the following system, where the system has a time delay:(19)cDt0,tαVkt≤−K1Vkt+K2Vkt−τ,t>0,i∈Dk,Vks=Φks,s∈−τ,0,and the linear fractional-order delay differential system is as follows:(20)cDt0,tαWkt=−K1Wkt+K2Wkt−τ,t>0,i∈Dk,Wkt=Φks,s∈−τ,0,where except for the point tk,k=1,2,…, Vkt,Wkt∈Rn is continuous everywhere, and Φks≥0 is continuous in −τ,0. If K1>0,K2>0, then Vkt≤Wkt,t∈0,+∞.


## 2.1. Fraction-Order Calculus

Firstly, existing definitions of fraction-order calculus are given, which can be seen in Reference [14], that are needed later.Define the Gamma functionΓ⋅ as below:(1)Γp=∫t0+∞tp−1exp−tdt,where p>0.Define the Caputo fractional derivativeDct0,tpg⋅ of the function gt as below:(2)cDt0,tpgt=1Γn−p∫t0tgnst−sp−n+1ds,where t0 is the initial time, t≥t0, p is the order, n−1<p<n,n∈Z+.Define the fractional integralIt0,tpg⋅ of the function gt as below:(3)It0,tpgt=1Γp∫t0tt−sp−1gsds,where t0 is the initial time, t≥t0.Define the Mittag-Leffler function with single parameterEp⋅ as below:(4)Eps=∑k=0+∞skΓkp+1,where p>0,s is a complex number.Define the Mittag-Leffler function with double parametersEp,p¯⋅ as below:(5)Ep,p¯s=∑k=0+∞skΓkp+q¯,where p>0,p¯>0,s is a complex number.


## 2.2. Model Description

A collection that makesZ+ a positive integer. The superscript T represents the transpose, and #ℑ is an element in the finite collection ℑ. Denote Rn is the set of n-dimensional real-valued vectors. R+ is the group of fixed non-negative numbers. For an arbitrary vector a∈Rn and the existence of a constant σ0>0, we record ℳa,σ0=xx−a<σ0 as a set of vectors, where the distance between x and a is less than σ0. The set of n×n real matrices is written as Rn×n. If a real matrix X>0, then X is a positive definite matrix. A⊗B represents the Kronecker product of matrices A and B. For any matrix A, λminA,λmaxA denotes its minimum eigenvalue and maximum eigenvalue, respectively, and the norm of A is defined as ‖A‖=λmaxATA1/2. In this article, we regard the following FCNNs with N same nodes, uncertainties, and time delays.(6)cDt0,tαxit=−P^+ΔPtxit+Q^+ΔQtfixit+R^+Δℛtfixit−τ+∑j=1NG^ijΓxjt+J,where i=1,2,…,N, and N≥2 represents the quantity of subnetworks; τ represents the time delay in transmission; xit=xi1t,xi2t,…,xintT is the state vector of i-th neuron; ΔPt,ΔQt,Δℛt are the norm-bounded parametric uncertainties; P^ is a diagonal matrix that expresses the self-feedback item of the j-th network, in which the diagonal elements are p1,p2,…,pn,pi>0. Q^=Qijn×n is the connection weight matrices and R^=rijn×n is the time lag join matrices, where i,j=1,2,…,n. fixit is the activation function; G^=G^ijN×N is the coupled matrix, when there exists a connection among the i-th node with the j-th node, i≠j,G^ij≠0, if not, G^ij=0, in which the diagonal elements are defined by G^ii=−∑j=1,j≠iNG^ij; Γ=diagγ1,γ2,…,γn is the internal coupling matrix; J=J1,J2,…,JN is the input vector.Next, we give several basic assumptions.(A1) The activation functionfi⋅ is continuous, for any vector x,y, exists ℒi>0, and the following formula holds:(7)fix−fiy≤ℒix−y.(A2)(8)ΔP=A^1C^1tℬ^1,ΔQ=A^2C^2tℬ^2,Δℛ=A^3C^3tℬ^3,where A^i,ℬ^ii=1,2,3 are constant matrices with the corresponding matching dimensions and C^iti=1,2,3 is an indeterminate matrix, where C^iTtC^it≤I (I is the unity matrix with the corresponding matching dimensions).Remark 1.
#Dk=κk means that the group Dk has κk nodes and κk≠0.
For any initial statext=x1Tt,x2Tt,…,xNTtT where xit∈C−τ,0,R,i=1,2,…,N, for any given initial value condition, there is a solution st, if all the node trajectories in the network satisfy the formula limt⟶∞‖xit−st‖=0,i=1,2,…,N, then this network is called complete synchronization. Furthermore, if the margin of error σ>0, exists T>0, for all xt and ∀t>T, xit−st<σ holds, then this network is called uniformly quasisynchronized.Definition 1 (see [15]).
For an arbitrary complex network withN nodes, D1,D2,…,Dm is a set of disjoint nodes, that is, ∑k=1mDk=1,2,…,N, Dk=lk1,lk2,…, Dk∩Du=∅ for k≠u. The network is called multiple quasisynchronization with the error vector δ=δ1,δ2,…,δmT>0 under any initial value conditions, if there exist a series of reference solutions s1t,s2t,…,smt and for any constant κ>0 small enough, T exists, for ∀t>T, the nodes xit∈ℳskt,σk,i∈Dk holds, in which ℳskt,σk≠ℳsut,σ¯u,u≠k.Remark 2.
It can be seen from Definition1 that skt is the reference trajectory for all nodes in set #Dk.
The target trajectorysi¯t satisfies the following formula:(9)cDt0,tαsi¯t=−P^+ΔPtsi¯t+Q^+ΔQtfi¯si¯t+R^+Δℛtfi¯si¯t−τ+J,where if xit∈ℳskt,σk, then si¯t=skt, i=1,2,…,N,k=1,2,…,m.
Now let’s noteε¯it=xit−si¯t is the error signal, where i=1,2,…,N, and devise a new pinned pulse controller as shown below:(10)uit=∑h=1+∞θkε¯itδt−th,i∈ℑkth,#ℑkth=ωk,0,i∉ℑkth,where δ⋅ and θk represents the Dirac impulsive function and impulsive gain, respectively, and thh=0,1,2,… indicates the instant of the pulse that satisfies 0=t1<t2<⋯<th<⋯,limth⟶+∞th=+∞. The node set on t=th is represented by ∑k=1mℑkth=ℑ1th,ℑ2th,…,ℑmth⊂D1,D2,…,Dm⊂1,2,…,N, and make 0<ωk≤κk,k=1,2,…,m, namely, ℑkth is a subset of Dk, and ℑkth represents the set of pinned nodes at t=th. Assume the error vector ε¯i1≥ε¯i2≥⋯≥ε¯in. Under the pulse controller (10), the system of errors can be described:(11)cDt0,tαε¯it=−P^+ΔPtε¯it+Q^+ΔQtf˜iε¯it+R^+Δℛtf˜iε¯it−τ+∑j=1NG^ijΓε¯it,t≠thε¯ith+=1+θkε¯ith−,i∈∑k=1mℑkth,ε¯ith+=ε¯ith−,i∉∑k=1mℑkth,where h=0,1,2,…,ε¯ith+=limt⟶th+ε¯it,ε¯ith−=limt⟶th−ε¯ith,f˜iε¯it=fixit−fi¯si¯t,f˜iε¯it−τ=fixit−τ−fi¯si¯t−τ, and f˜i0=0.
The initial value condition of the above error system (11) is as follows:(12)ε¯is=ϕis,s∈−τ,0,where ϕis∈C−τ,0,Rn and i=1,2,…,N.Lemma 1 (see [16]).
Ifxt∈Rn is a vector-valued function that is differentiable and continuous for t, next for any α∈0,1 and t≥t0, we have the following relationship:(13)cDt0,tαxTtPxt≤2xTtPcDt0,tαxt,where P∈Rn×n is a constant matrix that is symmetric and positive definite.Lemma 2 (see [17]).
Letℛ^,W^, and S^t be the real matrices corresponding matching dimensions, then if there is S^TtS^t≤I, then there is the following equation:(14)ℛ^TS^TtW^T+W^S^tℛ^≤1ξℛ^Tℛ^+ξW^W^T,where ξ>0 is the constant.Lemma 3 (see [17]).
Letℛ^ and W^ be the real matrices corresponding matching dimensions, then(15)ℛ^TW^+W^TR^≤ξℛ^Tℛ^+1ξW^TW^,where ξ>0 is the constant.Lemma 4 (see [18]).
For arbitrary vectorx1,x2∈Rn and Q∈Rn×n which is a positive definite matrix, we have the below inequalities hold:(16)2x1Tx2≤x1TQ−1x1+x2TQx2.Lemma 5 (see [19]).
For positive definite matrixR, vector xi with proper dimensionality and symmetric matrix W, then we have the following:(17)λminR−1WxiTRxi≤xiTWxi≤λmaxR−1WxiTRxi,where λmax⋅,λmin⋅ denote the maximum and minimum eigenvalues, respectively, and R−1 stands for the inverse of a matrix R.Definition 2.
The definition of the pinning rateηk at t=th is as follows:(18)∑i∈ℑkthε¯iTth−ε¯ith−∑i∈Dkε¯iTth−ε¯ith−=ηk,here the pinning rate ηk is related to time and impulse instants, and we can also determine the lower bound of the pinning rate ηk.Lemma 6 (see [20]).
Consider the following system, where the system has a time delay:(19)cDt0,tαVkt≤−K1Vkt+K2Vkt−τ,t>0,i∈Dk,Vks=Φks,s∈−τ,0,and the linear fractional-order delay differential system is as follows:(20)cDt0,tαWkt=−K1Wkt+K2Wkt−τ,t>0,i∈Dk,Wkt=Φks,s∈−τ,0,where except for the point tk,k=1,2,…, Vkt,Wkt∈Rn is continuous everywhere, and Φks≥0 is continuous in −τ,0. If K1>0,K2>0, then Vkt≤Wkt,t∈0,+∞.


## 3. Main Result

We will derive several synchronization standards in this section. Under the action of the pinning impulsive controller,Dk, i∈Dk,Dk∈D1,D2,…,Dm, system (7) and reference trajectory skt∈s1t,s2t,…,smt to achieve multiple quasisynchronization.Theorem 1.
Letξi>0i=1,2,3. For any i∈Dk,k=1,2,…,m, under the pinning impulsive control (10), system (7) can achieve multiple quasisynchronization if Assumptions (A1) and (A2) hold, there exist symmetric matrices Mi∈Rn×n>0i=1,2 and positive definite matrix P∈Rn×n and such that(21)G^00PΓ<0,(22)ω1=PP^+P^TP+ξ1PA^1A^1TP+1ξ1ℬ^1Tℬ^1−PQ^M1−1Q^TP−ℒiTM1ℒi−ξ2PA2^A2^TP−1ξ2ℒiTℬ^2Tℬ^2ℒi−PR^M2−1R^TP−ξ3PA^3A^3TP≥K1P>0,(23)ω2=ℒiTM2ℒi+1ξ3ℒiTℬ^3Tℬ^3ℒi≤K2P,(24)ω3=1+θk2ηkλmaxPλminP+1−ηkλmaxPλminP≤ρk∈0,1,where K1>0,K2>0,K1>2K2 and λmaxPκ/λminP<σk. Define the Lyapunov function as follows:(25)Vkt=∑i∈Dkε¯iTtPε¯it.Fort∈th−1,th,h=0,1,2,…, from Lemma 1 we obtained the following:(26)cDt0,tαVkt≤∑i∈Dk2¯εiTtPcDt0,tαε¯it=∑i∈Dk2¯εiTtP−P^+ΔPtε¯it+Q^+ΔQtf˜iε¯it+R^+Δℛtf˜iε¯it−τ+∑j∈DkG^ijΓε¯jt=2∑i∈Dkε¯iTtP−P^+ΔPtε¯it+2∑i∈Dkε¯iTtPQ^+ΔQtf˜iε¯it+2∑i∈Dkε¯iTtPR^+Δℛtf˜iε¯it−τ+2∑i∈Dkε¯iTtP∑j∈DkG^ijΓε¯jt.By Assumptions (A1) and (A2) and Lemmas1-4, we obtain the following:(27)∑i∈Dk2ε¯iTtP−P^+ΔPtε¯it≤∑i∈Dkε¯iTt−PP^+P^TP+PA^1C^1tℬ^1+ℬ^1TC^1TtA^1TPε¯it≤∑i∈Dkε¯iTt−PP^+P^TP+ξ1PA^1A^1TP+1ξ1ℬ^1Tℬ^1ε¯it,(28)∑i∈Dk2ε¯iTtPQ^+ΔQtf˜iε¯it≤∑i∈Dk2ε¯iTtPQ^f˜iε¯it+∑i∈Dk2ε¯iTtPA^2C^2tℬ^2f˜iε¯it≤∑i∈Dkε¯iTtPQ^M1−1Q^TPε¯it+∑i∈Dkf˜iTε¯itM1f˜iε¯it+∑i∈Dkε¯iTtPA^2C^2tℬ^2f˜iε¯it+∑i∈Dkf˜iTε¯itℬ^2TC^2TtA^2TPε¯it≤∑i∈Dkε¯iTtPQ^M1−1Q^TPε¯it+∑i∈Dkε¯iTtℒiTM1ℒiε¯it+∑i∈Dk1ξ2ε¯iTtℒiTℬ^2Tℬ^2ℒiε¯it+∑i∈Dkξ2ε¯iTtPA^2A^2TPε¯it=∑i∈DkeiTtPQ^M1−1Q^TP+LiTM1Li+1ξ2LiTB^2TB^2Li+ξ2PA^2A^2TPeit,(29)∑i∈Dk2ε¯iTtPR^+Δℛtf˜iε¯it−τ≤∑i∈Dk2ε¯iTtPR^f˜iε¯it+∑i∈Dk2ε¯iTtPA^3C^3tℬ^3f˜iε¯it−τ≤∑i∈Dkε¯iTtPR^M2−1R^TPε¯it+∑i∈Dkf˜iTε¯it−τM2f˜iε¯it−τ+∑i∈Dkε¯iTtPA^3C^3tℬ^3f˜iε¯it−τ+∑i∈Dkf˜iTε¯it−τℬ^3TC^3TtA^3TPε¯it≤∑i∈Dkε¯iTtPR^M2−1R^TPε¯it+∑i∈Dkε¯iTt−τℒiTM2ℒiε¯it−τ+∑i∈Dk1ξ3ε¯iTt−τℒiTℬ^3Tℬ^3ℒiε¯it−τ+∑i∈Dkξ3ε¯iTtPA^3A^3TPε¯it=∑i∈Dkε¯iTtPR^M2−1R^TP+ξ3PA^3A^3TPε¯it+∑i∈Dkε¯iTt−τℒiTM2ℒi+1ξ3ℒiTℬ^3Tℬ^3ℒiε¯it−τ.Form (21), we have the following:(30)∑i∈Dk2ε¯iTtP∑j∈DkG^ijΓε¯jt=2ε¯TtG^⊗PΓε¯t≤0.Substituting (27)-(30) into (27282930) yields the following equation:(31)cDt0,tαVkt≤∑i∈Dk−ε¯iTtω1ε¯it+∑i∈Dkε¯iTt−τω2ε¯it−τ+2ε¯TtG^⊗PΓε¯t.From (21)-(24), we have the following equation:(32)cDt0,tαVkt≤∑i∈Dk−ε¯iTtK1Pε¯it+ε¯iTt−τK2Pε¯it−τ≤−K1Vkt+K2Vkt−τ.Whent=th, from (11) and (24), Lemma 5, and Definition 2, we obtain the following:(33)Vkth+=∑i∈Dkε¯iTth+Pε¯ith+=∑i∈ℑkthε¯iTth+Pε¯ith++∑i∉ℑkthε¯iTth+Pε¯ith+=∑i∈ℑkth1+θk2ε¯iTth−Pε¯ith−+∑i∉ℑkthε¯iTth−Pε¯ith−≤1+θk2λmaxP∑i∈ℑkthε¯iTth−ε¯ith−+λmaxP∑i∉ℑkthε¯iTth−ε¯ith−≤1+θk2λmaxP∑i∈Dkε¯iTthε¯ith+1−ηkλmaxP∑2i∈Dkε¯iTthε¯ith≤ω3Vkth≤ρkVkth.Now, consider the following system with time lag:(34)cDt0,tαWkt=−K1Wkt+K2Wkt−τ,t>0,Wkt=Φks,s∈−τ,0.Iflimt⟶∞Wkt=0,Φks≥0, afterward through Lemma 6, we can have limt⟶∞Vkt=0,Φks≥0.Next, we will prove that whenK1>2K2K1>0,K2>0, there is limt⟶∞Wkt=0,Φks≥0.In order to distinguish the subsystem subscript in this article from the original imaginary unit, here, we change the original inherent imaginary uniti˜. The characteristic (34) can be changed to the following form according to Corollary 3 in Reference [21].(35)υ¯kα+K1−K2e−υ¯kτ=0,If K1>2K2 and (35) has no pure imaginary roots, so the zero solution of equation (34) is globally Lyapunov asymptotically stable, namely, limt⟶∞Wkt=0,Φks≥0.Next, we will use contradiction analysis to show that (35) does not have pure imaginary roots. Then, suppose (35) has pure imaginary roots υ¯k, and υ¯k=vki˜=vkcosπ/2+i˜sinπ/2, where vk is a real number. If vk≤0, then υ¯k=vki˜=vkcosπ/2−i˜sinπ/2, and if vk>0, then υ¯k=vki˜=vkcosπ/2+i˜sinπ/2.By substitutingυ¯k=vki˜ into (35), we can obtain the following:(36)vki˜α+K12=K2e−τvki˜2,that is,(37)vk2α+2K1cosαπ2vkα+K12=K2cosτvk2+K2sinτvk2≤2K22.Let(38)hkxk=xk2+2K1cosαπ2xk+K12−K2cosτvk2+K2sinτvk2.So,(39)hk0=K12−K2cosτvk2+K2sinτvk2≥K12−2K22.BecauseK1>2K2K1>0,K2>0, so hk0>0. We know that hk is a second-order polynomial, so we have hkvkα>0, which contradicts (37). That is, (37) has no solution, which means that (35) has no pure imaginary roots, namely, limt⟶∞Vkt=0.Therefore, there existsTk, and for arbitrary κ>0 and for all t>Tk, we have the following equation:(40)Vkt<λmaxPκ,t>Tk,where Vkt=∑i∈Dkε¯iTtPε¯it, so we have the following equation:(41)λminPε¯it2<λmaxPκ,t>Tk,that is,(42)ε¯it≤λmaxPκλminP<σk,where i∈Dk and ∑k=1mTk=T1,T2,…,TN. So there exits T=maxT1,T2,…,TN, for ∀t>T and any small positive number σk>0, such that 0<xit−skt<σk,k=1,2,…,m.IfΔPt=0,ΔQt=0,Δℛt=0, (2.1) will degrade into(43)cDt0,tαxit=−P^xit+Q^fixit+R^fixit−τ+∑j=1NG^ijΓxjt+J.Corollary 1.
For anyi∈Dk,k=1,2,…,m, under the pinning impulsive control (10), system (43) can achieve multiple quasisynchronization if Assumptions (A1) and (A2) hold, there exist symmetric matrices Mi∈Rn×n>0i=1,2, and positive definite matrix P∈Rn×n such that(44)G^00PΓ<0,ω1=PP^+P^TP−PQ^M1−1Q^TP−ℒiTM1ℒi−PR^M2−1R^TP≥K1P>0,ω2=ℒiTM2ℒi≤K2P,ω3=1+θk2ηkλmaxPλminP+1−ηkλmaxPλminP≤ρk∈0,1,where K1>0, K2>0, K1>2K2, and λmaxPκ/λminP<σk.Remark 3.
We give general theoretical results for multiple quasisynchronization of FCNNs with uncertain terms and delays in Theorem1 Among existing references, the quasisynchronization problem of FCNNs with uncertainty is rarely discussed. Moreover, unlike the analytical method of Reference [22], the model (7) in this article is a fractional-order system instead of the integer-order model in Reference [22]. The analysis and processing method of fractional-order system is unlike that of integer-order system, so it cannot be applied directly.Remark 4.
Multiple quasisynchronization is the extension of quasisynchronization. Whenm=1 in reference trajectory indicates that there is only one reference track, next, the multiple quasisynchronization is reduced to quasisynchronization.Remark 5.
Corollary1 gives the sufficient conditions for multiple quasisynchronization of DFCNN when the uncertainty term is zero.Remark 6.
In Reference [23], the author solved the synchronization problem through adaptive control method. In this article to reduce the control cost and realize the pinned pulse control, only some nodes need to be controlled to be in the bounded field where they share the reference trajectory.Remark 7.
Compared with Reference [20], the difference of this article is that we consider parameter uncertainty satisfying bounded conditions in the model, and the advantage is that the model considered in this article is more practical in practical systems and applications. In particular, our model is fractional-order, and parameter uncertainties and coupling terms are taken into account in the model.


## 4. Examples

In this section, we give two numerical simulation examples to illustrate the abovementioned theoretical values.Example 1.
We design the FCNNs with uncertain terms and delays (7), where fixit=tanhxit, i=1,2, m=1,2, time-delay τ=1, and α=0.96, and the parameter matrix of the network is as follows:(45)P^=5.2005.2,Q^=4.8−2−32.5,R^=0.25001.5,ΔP=cost1111,ΔQ=sint1111,Δℛ=cost1111,G^=−50.20.4−1,J=00.
Two corresponding reference trajectories (9), where k=1,2,α=0.96,τ=1,fi¯si¯t=tanhsi¯t.
We selectΓ=diag1,1, pulse gain θ1=−0.2, θ2=−0.6, η1=0.7, η2=0.4, ρ1=0.84, and ρ2=0.74. It can be proved that the conditions (21)-(24) in Theorem 1 are established and can be obtained by the following calculation:(46)P=1.00170.00220.00221.0945,at the same time, λmaxP/λminP=10/9, σ1=0.08, and σ2=0.12, after that we can get ω1=173.0088−0.0003−0.000321.0120,ω2=2.4908004.4362,ω3=0.83110.3271 through the abovementioned formulas (22)-(24). Then, the uncertain fractional-order neural network can realize multiple quasisynchronization, and the convergence of its error signals e1mt, e2mt, and m=1,2, under the fixed pulse controller is shown in Figure 1.Figure 1
Trajectory ofe11, e12, e21, and e22 of the system in Example 1 under the impulsive controller.Example 2.
We design the FCNNs with uncertain terms and delays (6), where fixit=tanhxit, i=1,2, m=1,2, time delay τ=1, and α=0.98, and the parameter matrix of the network is as follows:(47)P^=6006,Q^=5−5−53.8,R^=0.25001.5,ΔP=sint1111,ΔQ=cost1111,Δℛ=sint1111,G^=−50.20.4−1,J=00.
Two corresponding reference trajectories (9), where k=1,2,α=0.98,τ=1,fi¯si¯t=tanhsi¯t.
We selectΓ=diag1,1, pulse gain θ1=−0.3, θ2=−0.7, η1=0.5, η2=0.6, ρ1=0.83, and ρ2=0.51. It can be proved that conditions (21)-(24) in Theorem 1 are established and can be obtained by the following calculation:(48)P=1.00170.00220.00221.0945,at the same time, λmaxP/λminP=10/9, σ1=0.06, and σ2=0.18. After that we can get ω1=135.2047−0.0012−0.001274.7350,ω2=3.0918001.1362,ω3=0.82780.5044 through the abovementioned formulas (22)-(24). Then, the uncertain fractional-order neural network can realize multiple quasisynchronization, and the convergence of its error signals e1mt, e2mt, m=1,2, and under the fixed pulse controller is shown in Figure 2.Figure 2
Trajectory ofe11,e12,e21, and e22 of the system in Example 2 under the impulsive controller.


## 5. Conclusion

In this article, the multiple quasisynchronization problem of FCNNs with uncertainty and time-delay is studied. Firstly, our main theoretical method is to construct an impulse controller to control some nodes and then divide the nodes into several disjoint subsets, so as to make the system achieve multiple quasisynchronization. Secondly, using the relevant knowledge of the comparison principle and the method of constructing the Lyapunov function, we obtain the sufficient conditions for the system to realize multiple quasisynchronization. Finally, two examples are given to carry out simulation operations to demonstrate the validity of the theoretical results in this article.



---

*Source: 8742772-2022-10-17.xml*