# Fixed-Time Synchronization for Hybrid Coupled Dynamical Networks with Multilinks and Time-Varying Delays

Authors: Baolin Qiu; Lixiang Li; Haipeng Peng; Yixian Yang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This paper concerns the problem of fixed/finite-time synchronization of hybrid coupled dynamical networks. The considered dynamical networks with multilinks contain only one transmittal time-varying delay for each subnetwork, which makes us get hold of more interesting and practical points. Two kinds of delay-dependent feedback controllers with multilinks as well as appropriate Lyapunov functions are defined to achieve the goal of fixed-time synchronization and finite-time synchronization for the networks. Some novel and effective criteria of hybrid coupled networks are derived based on fixed-time and finite-time stability analysis. Finally, two numerical simulation examples are given to show the effectiveness of the results proposed in our paper.


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## Body

## 1. Introduction

The phenomenon of synchronization exists everywhere in the realistic living world. People’s applause begins with random rhythm, but it turns to the same rhythm at the end of a splendid drama. Every router of Internet periodically releases routing messages, respectively. However, the researchers have found that different router would eventually send routing messages in synchronous way, which led to the network traffic jam. Synchronization is common in every field and plays an important role. In fact, in physics, mathematics, and other fields, synchronous phenomenon of the coupled dynamical system has been studied widely in the past decade.We are involved in a variety of complex dynamical networks, such as communication networks, Internet, transportation networks, and electricity networks. And synchronous phenomenon of dynamical networks is common. Sometimes synchronization is good for people’s lives; sometimes synchronization may be harmful to people’s lives and may even cause huge losses or serious consequences. Therefore, researches on synchronization of complex networks are of great value and significance. In a recent decade, researches on synchronization control of complex dynamical networks attract many researchers’ attention [1, 2]. Meanwhile, lots of relevant achievements have been obtained. Researches on synchronization mainly concentrate on the following: complete synchronization [3, 4], projective synchronization [5–7], delay synchronization [8–10], finite-time synchronization [11–13], and so forth. Synchronization methods mainly contain the following: linear feedback method [14, 15], nonlinear feedback method [16], adaptive method [17], and so forth. According to the continuity of control law, synchronization techniques can be divided into continuous and noncontinuous synchronization control methods [18–20]. Compared with continuous control law, noncontinuous control law can reduce some control cost.Also, synchronization techniques can be divided into finite-time synchronization and infinite-time synchronization. Compared with the asymptotic stability and exponential stability, finite-time stability has relatively lower time complexity. Therefore, in the real world, obtaining synchronization in a setting time has more practical value. In secure communication, sending and receiving the information in a setting time can not only ensure the security of communication but also improve the efficiency and safety of communications [21]. However, achieving synchronization in finite time depends on the initial synchronization error value of the drive-response systems. The finite time for the unknown initial conditions of drive-response systems (the initial conditions of most of practical systems are hard to be achieved) is not fixed. Therefore, Polyakov defined the fixed-time stability theory. The time to obtain fixed synchronization is irrelevant to the initial synchronization error conditions of drive-response systems and only affected and controlled by the controller [22].In this paper, we study the hybrid coupled dynamical networks containing coupling delay and internal delay that are indeed common in real world, which makes the research work more meaningful. In previous studies, coupling delay is defined asD(xj(t-τ(t))-xi(t-τ(t))) [23–25]. In fact, as for the synchronization study of a pair of given drive-response systems, the time delay influences the variables that are transmitted from one system to another system being studied. Therefore, considering the coupling term of time delay as D(xj(t-τ(t))-xi(t)) is more reasonable [26].The concept of multilinks means that the edge number linking two nodes in complex networks is greater than one. Multilinked complex networks are ubiquitous in the real world, such as communication networks and transportation networks. Communication links of communication networks can be mail, telephone, QQ, letters, and so forth [27–29]. Compared with researches of single link networks [30–33], those of networks with multilinks are more practical and significant. There have been a lot of research achievements in this field so far [34–36]. However, to my best knowledge, few put their attention to the hybrid coupled networks with both one single time-varying delay coupling and multilinks. Few people study fixed-time synchronization and finite-time synchronization with the conditions of dynamical networks mentioned above. It is interesting to complete such study even to fill the gaps in this field.Motivated by the foregoing analysis, this paper studies fixed-time synchronization and finite-time synchronization for hybrid coupled dynamical networks with multilinks and one single time-varying delay coupling. Furthermore, we propose new model of multilinked complex dynamical networks with one single time-varying delay coupling term and design two suitable feedback controllers to ensure the synchronization. With the constructed Lyapunov functions, several novel criteria are derived to achieve fixed-time/finite-time synchronization of the complex dynamical networks described in this paper. Finally, we give two numerical simulation examples to verify the effectiveness and correctness of our proposed theoretical results.The rest of this paper is organized as follows. In Section2, some preliminaries are introduced, such as notations, some assumptions, and lemmas. In Sections 3 and 4, the main results proposed in this paper are introduced. In Section 5, two numerical simulation examples are provided to verify the effectiveness and correctness of our obtained theoretical results. Finally, we draw the conclusion.


## 2. Preliminaries

Throughout this paper, we always have the functionsf(·) and g(·) to satisfy the Lipschitz condition, which mean the following assumptions [37].Assumption 1.
For anyx,y∈Rn, there exists a constant p(p≥0) such that(1)fx-fy≤px-y.Assumption 2.
For anyx,y∈Rn, there exists a constant q(q≥0) such that(2)gx-gy≤qx-y.
In order to prove the correctness of the results proposed in this paper, we need the following lemmas.Lemma 3 (see [38]).
Forx1,x2,x3,…,xN≥0 and p>1,0<q≤1, two inequalities hold as follows:(3)∑i=1Nxip≥N1-p∑i=1Nxip,∑i=1Nxiq≥∑i=1Nxiq.Lemma 4 (see [39], (chain rule)).
IfV(t):Rn→R and x(t) satisfy the following conditions,(1)
V(t) is C-regular,(2)
x(t) is absolutely continuous when it is on any compact subinterval of [0,+∞), then, the following holds:x(t) and V(x(t)):[0,+∞)→R are differentiable for a.e.t∈[0,+∞);(4)Vxt=υtx˙t,∀υt∈∂Vxt,where∂V(x(t)) is the Clarke generalized gradient of V at the point x(t).Lemma 5 (see [40]).
Let the functionV(t) be continuous and positive-definite, and V(t) makes the differential inequalities hold as follows:(5)V˙t≤-κVηt;∀t≥t0,Vt0≥0,where η∈(0,1) and κ is positive constant.
Then, for∀t0, V(t) can always satisfy the inequality as follows:(6)V1-ηt≤V1-ηt0-κ1-ηt-t0,t0≤t≤t1.Also, V(t)≡0,∀t≥t1. The settling time t1 is given as follows:(7)t1=t0+V1-ηt0κ1-η.Lemma 6 (see [41]).
Let the functionV(t) be continuous and positive-definite, and V(t) makes the differential inequalities hold as follows:(8)V˙t≤IVt-κVηt,∀t≥t0,V1-ηt0≤κI,where η∈(0,1) and κ and I are positive constant.
Then,V(t)≡0,∀t≥t1, where the settling time t1 is given as follows:(9)t1=t0+ln⁡1-I/κV1-ηt0Iη-1.Lemma 7 (see [22]).
Suppose that there exists a functionV:RnN→R+∪0 which is continuous radically unbounded such that the following hold:(1)
Ifx≠(0,…,0)T, then V(x)>0;V(x)=0⇒x=(0,…,0)T.(2)
Lete(t)=(e1T(t),…,eNT(t))T be any solution of error dynamical system. For some constants a,b>0,p>1,0<q<1, e(t) satisfies V˙(e(t))≤-aVp(e(t))-bVq(e(t)). Then, the following results hold:(10)Vet≡0,∀t≥t0,T0=1ap-1+1b1-q,whereT0 is the fixed settling time.


## 3. Finite-Time Synchronization for Hybrid Coupled Networks with Multilinks

In this section, we study finite-time synchronization for the multilinked complex dynamical networks with hybrid coupled terms and time-varying delays. By constructing Lyapunov function and designing appropriate controller, some useful criteria are derived to ensure that the multilinked complex dynamical networks proposed in this paper are synchronized.


### 3.1. Networks Model

The networks model is given as follows:(11)x˙it=fxit+gxit-τt+∑j=1,j≠iNGij0Dxjt-xit+∑j=1,j≠iNGij1Dτ1xjt-τ1t-xit+⋯+∑j=1,j≠iNGijmDτmxjt-τmt-xit,where i=1,2,…,N is the number of nodes and xi(t)=(xi1(t),xi2(t),…,xin(t))T∈Rn is the state vector of the ith node of the network; vector functions f,g:Rn→Rn are continuous differentiable; τ(t) is internal delay; τ(t)≤τ,τ is a positive constant; τk(t)(k=1,2,3,…,m) denotes the delay of the kth subnetwork correspondingly, and 0≤τk(t)≤τ; D=(Dij)n×n denotes the inner coupling matrix of the 0th subnetwork between node i and node j at the moment t; correspondingly, Dτk(k=1,2,3,…,m) denotes the inner coupling matrix of the kth subnetwork between node i and node j at the moment t-τk(t); Gk=(Gijk)n×n (k=0,1,2,…,m) denotes the configuration matrix of the kth subnetwork and satisfies the conditions as follows:(12)Giik=-∑j=1,j≠iNGijk,Gijk=Gjik≥0,i≠j,where k=0,1,2,…,m; Gijk>0 means there exists a connection between nodes i and j; otherwise, Gijk=0.Note 1.
We propose a new multilinked complex dynamical networks model with hybrid coupled terms and time-varying delays. Bringing in inner coupling matricesD&Dτk, our study of the finite/fixed-time synchronization for the hybrid coupled dynamical networks with both one single time-varying delay coupling and multilinks is more practical for the real world. To the best of our knowledge, this study is the first exploration in such a field.
Consider system (11) as the drive system; correspondingly, the response system is described as follows:(13)y˙it=fyit+gyit-τt+∑j=1,j≠iNGij0Dyjt-yit+∑j=1,j≠iNGij1Dτ1yjt-τ1t-yit+⋯+∑j=1,j≠iNGijmDτmyjt-τmt-yit+uit,where yi(t)=(yi1(t),yi2(t),…,yin(t))T∈Rn is the response state vector of the ith node for the response system; ui(t)(i=1,2,…,N) is the designed controller to achieve synchronization.
Assume thatC([-τ,0],Rn) is a Banach space which consists of continuous functions mapping the interval [-τ,0] into Rn with the norm ϕ=sup-τ≤θ≤0ϕ(θ). For the drive system (11) and the response system (13), their initial conditions are, respectively, given by xi(t)=ϕi(t)∈C([-τ,0],Rn) and yi(t)=φi(t)∈C([-τ,0],Rn). Assume that in each of model equation (11) and model equation (13) a unique solution exists in regard to the initial conditions mentioned.
In order to use networks model equations conveniently, we rewrite equation (11) and equation (13) with the comprehensive consideration of condition (12). The results are given as follows:(14)x˙it=fxit+gxit-τt+∑j=1NGij0Dxjt+∑j=1NGij1Dτ1xjt-τ1t+⋯+∑j=1NGijmDτmxjt-τmt-Gii1Dτ1xit-τ1t-xit-⋯-GiimDτmxit-τmt-xit,y˙it=fyit+gyit-τt+∑j=1NGij0Dyjt+∑j=1NGij1Dτ1yjt-τ1t+⋯+∑j=1NGijmDτmyjt-τmt-Gii1Dτ1yit-τ1t-yit-⋯-GiimDτmyit-τmt-yit+uit.
Let the synchronization errorsei(t)=yi(t)-xi(t),i=1,2,…,N. Then we have(15)e˙it=y˙it-x˙it=fyit-fxit+gyit-τt-gxit-τt+∑j=1NGij0Dejt+∑j=1NGij1Dτ1ejt-τ1t+⋯+∑j=1NGijmDτmejt-τmt-Gii1Dτ1eit-τ1t-eit-⋯-GiimDτmeit-τmt-eit+uit.


### 3.2. Finite-Time Synchronization between the Drive System and the Response System

Combining with the given lemmas, assumptions, and conditions, we study finite-time stability of the origin of the given error dynamical system, which denotes the finite-time synchronization for the multilinked complex dynamical networks (11) and (13).Definition 8 (the finite-time stability).
Through controlling of the designed controller, there always exists a positive constantt1, and the value of t1 depends on the value of the initial state vector ei(t)=Φi(t)∈C([-τ,0],Rn) for t∈[-τ,0]. If the following expressions hold(16)limt→t1⁡eit=0,0,…,0T,eit=0,0,…,0T,t>t1,i=1,2,…,N,then, the error dynamical system (15) is said to be finite-time stable, and this state of the error dynamical system is named as the finite-time stability. And t1 is called the settling time.
In order to obtain the target of finite-time synchronization for systems (11) and (13), we design the following controller as the control law of complex dynamical networks (13). The control law is given as follows:(17)uit=-n·sign⁡eitpieit1+qieit-τt1+∑k=1mξikeit-τkt1+k0nη/2eit1η,where k0>0, 0<η<1, and k=1,2,…,m; the parameters of pi,qi,ξik will be determined as follows.
The following theorem can be derived for the drive-response systems of models (11) and (13) with the given controller (17).Theorem 9.
Let Assumptions1 and 2 hold. If there exist the positive parameters pi,qi,ξik such that the following inequalities hold(18)pi≥α+2Gii0D+∑k′=1mGiik′Dτk′,qi≥β,ξik≥3GiikDτk,k=1,2,…,m,then, the drive system (11) and response system (13) can realize the synchronization in finite time. And the settling time t1 can be determined as t1=t0+V1-η(t0)/k0(1-η), where V(t0)=∑i=1N‍eit01/n.Proof.
In this paper, the Lyapunov function is constructed as follows:(19)Vt,et=∑i=1Neit1n=∑i=1NsignT⁡eiteitn.
Applying Lemma4 and the error system to the process of the proof, we calculate the derivative of V(t) step by step as follows:(20)
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The calculating result (20) contains many details. Firstly, in order to calculate conveniently later, we deal with the function ∑i=1NsignT⁡(ei(t))/nui(t) in (20). Then, we substitute the partial processing result into the previous result (20). We get(21)∑i=1NsignT⁡eitnuit=∑i=1NsignT⁡eitn-n·sign⁡eitpieit1+qieit-τt1+∑k=1mξikeit-τkt1+k0nη/2eit1η≤-∑i=1Npieit1+∑i=1Nqieit-τt1+∑i=1N∑k=1mξikeit-τkt1+∑i=1Nk0nη/2eit1η.
Secondly, we substitute result (21) into result (20). We get the following result:(22)V˙t≤∑i=1Nα+2Gii0D+∑k′=1mGiik′Dτk′-pieit1+∑i=1Nβ-qieit-τt1+∑i=1N∑k=1m3GiikDτk-ξikeit-τkt1-∑i=1Nk0nη/2eit1η≤-∑i=1Nk0nη/2eit1η≤-k0∑i=1Neit1nη=-k0Vηt.
According to Lemma5, the error dynamical system (15) can be stable in finite time. And the settling time is determined by t1=t0+V1-η(t0)/k0(1-η).
This completes the proof.Remark 10.
Lemma5 plays the role of an important theoretical basis for the previous proof, which supports the analysis of finite-time synchronization for hybrid coupled complex networks with multilinks and internal/coupling delay terms with single time-varying delays. Lemma 5 can be traced from [40].
According to Lemma6, the following corollary can be obtained.Corollary 11.
Let both Assumptions1 and 2 hold. The designed controller is described as follows:(23)uit=-n·sign⁡eitpieit1+qieit-τt1+∑k=1mξikeit-τkt1+k0nη/2eit1η,where k0>0 and 0<η<1.
And let the parameterspi,qi,ξik satisfy the following inequalities:(24)pi<α+2Gii0D+∑k′=1mGiik′Dτk′,qi≥β,ξik≥3GiikDτk,k=1,2,…,m,where pi>0, qi>0, and ξik>0. Besides, V1-η≤k0/nα+2Gii0D+∑k′=1m‍Giik′Dτk′-pi.
Then, we can obtain the conclusion that the complex system (11) and complex system (13) will realize synchronization in finite time.Remark 12.
According to Theorem9 and Corollary 11, some sufficient finite-time synchronization criteria of the multilinked hybrid coupled networks are derived for the drive-response systems of models (11) and (13) with the designed controller in this paper. The parameters pi>0, qi>0, and ξik>0(k=1,2,…,m) are critical to the finite-time synchronization of networks (11) and (13). Compared with the previous researches, the models of complex dynamical networks proposed in our paper are more general.


### 3.3. Finite-Time Synchronization between Hybrid Coupled Dynamical Network and an Isolated Node

Suppose that we need multilinked hybrid coupled complex networks to synchronize to an isolated node in finite time. Then we will design the following complex networks model (25) as follows:(25)x˙it=fxit+gxit-τt+∑j=1NGij0Dxjt+∑j=1NGij1Dτ1xjt-τ1t+⋯+∑j=1NGijmDτmxjt-τmt-Gii1Dτ1xit-τ1t-xit-⋯-GiimDτmxit-τmt-xit+uit,where i=1,2,…,N.If the finite-time synchronization is obtained, that is,x1t=x2t=⋯=xN(t), the synchronized state model can be determined as follows:(26)s˙t=fst+gst-τt-Gii1Dτ1st-τ1t-st-⋯-GiimDτmst-τmt-st,where i=1,2,…,N. Apparently, all of x1(t),x2(t),…,xN(t) must not only satisfy x1(t)=x2(t)=…=xN(t) but also be invariant for different i while the finite-time synchronization is realized. But equation (26) does not meet the requirement. Therefore, we need the following assumption to be satisfied in order to complete the proof of this proposed supposition.Assumption 13.
For the configuration matricesGk(k=1,2,…,m) of multilinked hybrid coupled complex network, Giik=Gjjk=-ak, where k=0,1,2,…,m; i=1,2,…,N; j=1,2,…,N; and all of a1,a2,…,am are positive constants.
Useωi(t) to denote the synchronization errors, where ωi(t)=xi(t)-s(t),i=1,2,…,N. Combining with the given condition (12), ωi(t) can be described as follows:(27)ω˙it=x˙it-s˙t=fxit-fst+gxit-τt-gst-τt+∑j=1NGij0Dωjt+∑j=1NGij1Dτ1ωjt-τ1t+⋯+∑j=1NGijmDτmωjt-τmt-Gii1Dτ1ωit-τ1t-ωit-⋯-GiimDτmωit-τmt-ωit+uit.
The process of the proof for system (25) and isolated node (26) to obtain finite-time synchronization is the same as that for Theorem 9. Hence, we omit it.
Then, the following corollary can easily be obtained.Corollary 14.
Let all of Assumptions1, 2, and 13 hold. Suppose that there exist constants pi,qi,ξik satisfying the following inequalities:(28)pi≥α+2a0D+∑k′=1mak′Dτk′,qi≥β,ξik≥3akDτk,k=1,2,…,m,where pi>0, qi>0, and ξik>0.
The controller designed to obtain the finite-time synchronization is described as follows:(29)uit=-n·sign⁡ωitpiωit1+qiωit-τt1+∑k=1mξikωit-τkt1+k0nη/2ωit1η,where k0>0 and 0<η<1.
Then, the multilinked hybrid coupled complex networks (25) with control law (29) can realize the finite-time synchronization.


## 3.1. Networks Model

The networks model is given as follows:(11)x˙it=fxit+gxit-τt+∑j=1,j≠iNGij0Dxjt-xit+∑j=1,j≠iNGij1Dτ1xjt-τ1t-xit+⋯+∑j=1,j≠iNGijmDτmxjt-τmt-xit,where i=1,2,…,N is the number of nodes and xi(t)=(xi1(t),xi2(t),…,xin(t))T∈Rn is the state vector of the ith node of the network; vector functions f,g:Rn→Rn are continuous differentiable; τ(t) is internal delay; τ(t)≤τ,τ is a positive constant; τk(t)(k=1,2,3,…,m) denotes the delay of the kth subnetwork correspondingly, and 0≤τk(t)≤τ; D=(Dij)n×n denotes the inner coupling matrix of the 0th subnetwork between node i and node j at the moment t; correspondingly, Dτk(k=1,2,3,…,m) denotes the inner coupling matrix of the kth subnetwork between node i and node j at the moment t-τk(t); Gk=(Gijk)n×n (k=0,1,2,…,m) denotes the configuration matrix of the kth subnetwork and satisfies the conditions as follows:(12)Giik=-∑j=1,j≠iNGijk,Gijk=Gjik≥0,i≠j,where k=0,1,2,…,m; Gijk>0 means there exists a connection between nodes i and j; otherwise, Gijk=0.Note 1.
We propose a new multilinked complex dynamical networks model with hybrid coupled terms and time-varying delays. Bringing in inner coupling matricesD&Dτk, our study of the finite/fixed-time synchronization for the hybrid coupled dynamical networks with both one single time-varying delay coupling and multilinks is more practical for the real world. To the best of our knowledge, this study is the first exploration in such a field.
Consider system (11) as the drive system; correspondingly, the response system is described as follows:(13)y˙it=fyit+gyit-τt+∑j=1,j≠iNGij0Dyjt-yit+∑j=1,j≠iNGij1Dτ1yjt-τ1t-yit+⋯+∑j=1,j≠iNGijmDτmyjt-τmt-yit+uit,where yi(t)=(yi1(t),yi2(t),…,yin(t))T∈Rn is the response state vector of the ith node for the response system; ui(t)(i=1,2,…,N) is the designed controller to achieve synchronization.
Assume thatC([-τ,0],Rn) is a Banach space which consists of continuous functions mapping the interval [-τ,0] into Rn with the norm ϕ=sup-τ≤θ≤0ϕ(θ). For the drive system (11) and the response system (13), their initial conditions are, respectively, given by xi(t)=ϕi(t)∈C([-τ,0],Rn) and yi(t)=φi(t)∈C([-τ,0],Rn). Assume that in each of model equation (11) and model equation (13) a unique solution exists in regard to the initial conditions mentioned.
In order to use networks model equations conveniently, we rewrite equation (11) and equation (13) with the comprehensive consideration of condition (12). The results are given as follows:(14)x˙it=fxit+gxit-τt+∑j=1NGij0Dxjt+∑j=1NGij1Dτ1xjt-τ1t+⋯+∑j=1NGijmDτmxjt-τmt-Gii1Dτ1xit-τ1t-xit-⋯-GiimDτmxit-τmt-xit,y˙it=fyit+gyit-τt+∑j=1NGij0Dyjt+∑j=1NGij1Dτ1yjt-τ1t+⋯+∑j=1NGijmDτmyjt-τmt-Gii1Dτ1yit-τ1t-yit-⋯-GiimDτmyit-τmt-yit+uit.
Let the synchronization errorsei(t)=yi(t)-xi(t),i=1,2,…,N. Then we have(15)e˙it=y˙it-x˙it=fyit-fxit+gyit-τt-gxit-τt+∑j=1NGij0Dejt+∑j=1NGij1Dτ1ejt-τ1t+⋯+∑j=1NGijmDτmejt-τmt-Gii1Dτ1eit-τ1t-eit-⋯-GiimDτmeit-τmt-eit+uit.


## 3.2. Finite-Time Synchronization between the Drive System and the Response System

Combining with the given lemmas, assumptions, and conditions, we study finite-time stability of the origin of the given error dynamical system, which denotes the finite-time synchronization for the multilinked complex dynamical networks (11) and (13).Definition 8 (the finite-time stability).
Through controlling of the designed controller, there always exists a positive constantt1, and the value of t1 depends on the value of the initial state vector ei(t)=Φi(t)∈C([-τ,0],Rn) for t∈[-τ,0]. If the following expressions hold(16)limt→t1⁡eit=0,0,…,0T,eit=0,0,…,0T,t>t1,i=1,2,…,N,then, the error dynamical system (15) is said to be finite-time stable, and this state of the error dynamical system is named as the finite-time stability. And t1 is called the settling time.
In order to obtain the target of finite-time synchronization for systems (11) and (13), we design the following controller as the control law of complex dynamical networks (13). The control law is given as follows:(17)uit=-n·sign⁡eitpieit1+qieit-τt1+∑k=1mξikeit-τkt1+k0nη/2eit1η,where k0>0, 0<η<1, and k=1,2,…,m; the parameters of pi,qi,ξik will be determined as follows.
The following theorem can be derived for the drive-response systems of models (11) and (13) with the given controller (17).Theorem 9.
Let Assumptions1 and 2 hold. If there exist the positive parameters pi,qi,ξik such that the following inequalities hold(18)pi≥α+2Gii0D+∑k′=1mGiik′Dτk′,qi≥β,ξik≥3GiikDτk,k=1,2,…,m,then, the drive system (11) and response system (13) can realize the synchronization in finite time. And the settling time t1 can be determined as t1=t0+V1-η(t0)/k0(1-η), where V(t0)=∑i=1N‍eit01/n.Proof.
In this paper, the Lyapunov function is constructed as follows:(19)Vt,et=∑i=1Neit1n=∑i=1NsignT⁡eiteitn.
Applying Lemma4 and the error system to the process of the proof, we calculate the derivative of V(t) step by step as follows:(20)
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The calculating result (20) contains many details. Firstly, in order to calculate conveniently later, we deal with the function ∑i=1NsignT⁡(ei(t))/nui(t) in (20). Then, we substitute the partial processing result into the previous result (20). We get(21)∑i=1NsignT⁡eitnuit=∑i=1NsignT⁡eitn-n·sign⁡eitpieit1+qieit-τt1+∑k=1mξikeit-τkt1+k0nη/2eit1η≤-∑i=1Npieit1+∑i=1Nqieit-τt1+∑i=1N∑k=1mξikeit-τkt1+∑i=1Nk0nη/2eit1η.
Secondly, we substitute result (21) into result (20). We get the following result:(22)V˙t≤∑i=1Nα+2Gii0D+∑k′=1mGiik′Dτk′-pieit1+∑i=1Nβ-qieit-τt1+∑i=1N∑k=1m3GiikDτk-ξikeit-τkt1-∑i=1Nk0nη/2eit1η≤-∑i=1Nk0nη/2eit1η≤-k0∑i=1Neit1nη=-k0Vηt.
According to Lemma5, the error dynamical system (15) can be stable in finite time. And the settling time is determined by t1=t0+V1-η(t0)/k0(1-η).
This completes the proof.Remark 10.
Lemma5 plays the role of an important theoretical basis for the previous proof, which supports the analysis of finite-time synchronization for hybrid coupled complex networks with multilinks and internal/coupling delay terms with single time-varying delays. Lemma 5 can be traced from [40].
According to Lemma6, the following corollary can be obtained.Corollary 11.
Let both Assumptions1 and 2 hold. The designed controller is described as follows:(23)uit=-n·sign⁡eitpieit1+qieit-τt1+∑k=1mξikeit-τkt1+k0nη/2eit1η,where k0>0 and 0<η<1.
And let the parameterspi,qi,ξik satisfy the following inequalities:(24)pi<α+2Gii0D+∑k′=1mGiik′Dτk′,qi≥β,ξik≥3GiikDτk,k=1,2,…,m,where pi>0, qi>0, and ξik>0. Besides, V1-η≤k0/nα+2Gii0D+∑k′=1m‍Giik′Dτk′-pi.
Then, we can obtain the conclusion that the complex system (11) and complex system (13) will realize synchronization in finite time.Remark 12.
According to Theorem9 and Corollary 11, some sufficient finite-time synchronization criteria of the multilinked hybrid coupled networks are derived for the drive-response systems of models (11) and (13) with the designed controller in this paper. The parameters pi>0, qi>0, and ξik>0(k=1,2,…,m) are critical to the finite-time synchronization of networks (11) and (13). Compared with the previous researches, the models of complex dynamical networks proposed in our paper are more general.


## 3.3. Finite-Time Synchronization between Hybrid Coupled Dynamical Network and an Isolated Node

Suppose that we need multilinked hybrid coupled complex networks to synchronize to an isolated node in finite time. Then we will design the following complex networks model (25) as follows:(25)x˙it=fxit+gxit-τt+∑j=1NGij0Dxjt+∑j=1NGij1Dτ1xjt-τ1t+⋯+∑j=1NGijmDτmxjt-τmt-Gii1Dτ1xit-τ1t-xit-⋯-GiimDτmxit-τmt-xit+uit,where i=1,2,…,N.If the finite-time synchronization is obtained, that is,x1t=x2t=⋯=xN(t), the synchronized state model can be determined as follows:(26)s˙t=fst+gst-τt-Gii1Dτ1st-τ1t-st-⋯-GiimDτmst-τmt-st,where i=1,2,…,N. Apparently, all of x1(t),x2(t),…,xN(t) must not only satisfy x1(t)=x2(t)=…=xN(t) but also be invariant for different i while the finite-time synchronization is realized. But equation (26) does not meet the requirement. Therefore, we need the following assumption to be satisfied in order to complete the proof of this proposed supposition.Assumption 13.
For the configuration matricesGk(k=1,2,…,m) of multilinked hybrid coupled complex network, Giik=Gjjk=-ak, where k=0,1,2,…,m; i=1,2,…,N; j=1,2,…,N; and all of a1,a2,…,am are positive constants.
Useωi(t) to denote the synchronization errors, where ωi(t)=xi(t)-s(t),i=1,2,…,N. Combining with the given condition (12), ωi(t) can be described as follows:(27)ω˙it=x˙it-s˙t=fxit-fst+gxit-τt-gst-τt+∑j=1NGij0Dωjt+∑j=1NGij1Dτ1ωjt-τ1t+⋯+∑j=1NGijmDτmωjt-τmt-Gii1Dτ1ωit-τ1t-ωit-⋯-GiimDτmωit-τmt-ωit+uit.
The process of the proof for system (25) and isolated node (26) to obtain finite-time synchronization is the same as that for Theorem 9. Hence, we omit it.
Then, the following corollary can easily be obtained.Corollary 14.
Let all of Assumptions1, 2, and 13 hold. Suppose that there exist constants pi,qi,ξik satisfying the following inequalities:(28)pi≥α+2a0D+∑k′=1mak′Dτk′,qi≥β,ξik≥3akDτk,k=1,2,…,m,where pi>0, qi>0, and ξik>0.
The controller designed to obtain the finite-time synchronization is described as follows:(29)uit=-n·sign⁡ωitpiωit1+qiωit-τt1+∑k=1mξikωit-τkt1+k0nη/2ωit1η,where k0>0 and 0<η<1.
Then, the multilinked hybrid coupled complex networks (25) with control law (29) can realize the finite-time synchronization.


## 4. Fixed-Time Synchronization for Hybrid Coupled Networks with Multilinks

In this section, combining with the given lemmas, assumptions, conditions, and so on, we study the fixed-time stability of the origin for the given error dynamical system (15), which denotes the fixed-time synchronization for the multilinked complex dynamical networks (11) and (13).So, we need to define what the fixed-time stability is, and its definition is described as follows.Definition 15 (the fixed-time stability).
Through controlling of the designed controller, there always exists a positive and fixed constantT0, and the value of T0 is irrelevant to the initial conditions. If the following expressions hold(30)limt→T0⁡eit=0,0,…,0T,eit≡0,0,…,0T,t>T0,i=1,2,…,N,then, the error dynamical system (15) is said to be fixed-time stable; the state of the error dynamical system is called the fixed-time stability. And T0 is named as the fixed settling time.Note 2.
Through analyzing finite-time stability and fixed-time stability, we can easily find that finite-time synchronization depends on the initial conditions of the complex networks; on the contrary, fixed-time synchronization is irrelevant to the initial conditions of the complex networks. Therefore, it is meaningful to study fixed-time synchronization, especially when the initial conditions are hard to be achieved.
The control law designed to obtain the fixed-time synchronization is given as follows:(31)uit=-n·sign⁡eitpieit1+qieit-τt1+∑k=1mξikeit-τkt1+aeit1pnp/2+beit1qnq/2,where a,b>0, p>1, 0<q<1, k=1,2,…,m, and the parameters pi, qi, and ξik will be determined later on.Theorem 16.
Let Assumptions1 and 2 hold. If there exist the positive parameters pi,qi,ξik(k=1,2,…,m) such that the following inequalities hold(32)pi≥α+2Gii0D+∑k′=1mGiik′Dτk′,qi≥β,ξik≥3GiikDτk,k=1,2,…,m,then, according to Lemma 7, the drive system (11) and response system (13) can realize synchronization in fixed-time T0. And the fixed settling time T0 can be determined as T0=1/aN1-p(p-1)+1/b(1-q).The following is the process of the proof.Proof.
The designed Lyapunov function is given as follows:(33)Vt,et=∑i=1Neit1n=∑i=1NsignT⁡eiteitn,where e(t)=(e1T(t),e2T(t),…,enT(t))T.
Applying Lemma4 and the error system to the process of the proof, we calculate the derivative of V(t) step by step as follows:(34)V˙t=∑i=1NsignT⁡eite˙itn⋮≤∑i=1Nα+2Gii0D+∑k′=1mGiik′Dτk′eit1+∑i=1Nβeit-τt1+∑i=1N∑k=1m3GiikDτkeit-τkt1+∑i=1NsignT⁡eitnuit.
The front part of the proof here is the same as that of Theorem9. Hence, we omit the front part of proof and step to the calculating result (34) directly. The calculating result (34) contains many details. Firstly, in order to calculate conveniently later, we deal with the function ∑i=1N‍signT⁡(ei(t))/nui(t) in (34). Then substitute the partial processing result into the previous result (34). We get(35)∑i=1NsignT⁡eitnuit=∑i=1NsignT⁡eitn-n·sign⁡eitpieit1+qieit-τt1+∑k=1mξikeit-τkt1+aeit1pnp/2+beit1qnq/2≤-∑i=1Npieit1+∑i=1Nqieit-τt1+∑i=1N∑k=1mξikeit-τkt1+a∑i=1Neit1pnp/2+b∑i=1Neit1qnq/2.
Substituting the calculating result (35) into (34), continue to calculate as follows:(36)
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Taking expressions (32) and Lemma 3 into consideration, we can get the following calculating result step by step:(37)V˙t≤-a∑i=1Neit1pnp/2-b∑i=1Neit1qnq/2V˙t≤-aN1-p∑i=1Neit1np-b∑i=1Neit1nq≤-aN1-pVpt-bVqt.
Through comprehensive consideration, the given error dynamical system (15) is fixed-time stable. According to Lemma 7, the fixed settling time can be calculated as follows:(38)T0=1aN1-pp-1+1b1-q.
The proof is completed here.Remark 17.
So far, lots of achievements for finite-time synchronization of complex networks have been obtained, while, to the best of our knowledge, there is little attention drawn to the fixed-time synchronization of the multilinked hybrid coupled complex networks with time-varying delays or few published papers in this field. That is why this study can draw our attention so much.Corollary 18.
Let all of Assumptions1, 2, and 13 hold. Suppose that there exist positive constants pi, qi, and ξik satisfying the following inequalities:(39)pi≥α+2a0D+∑k′=1mak′Dτk′,qi≥β,ξik≥3akDτk,k=1,2,…,m,where pi>0, qi>0, and ξik>0.
The controller designed to obtain the fixed-time synchronization is given as follows:(40)uit=-n·sign⁡ωitpiωit1+qiωit-τt1+∑k=1mξikωit-τkt1+aωit1pnp/2+bωit1qnq/2,where a,b>0, p>1, and 0<q<1.
Then, the multilinked hybrid coupled complex networks (25) with control law (40) can realize fixed-time synchronization. The fixed settling time can be calculated by T0=1/aN1-p(p-1)+1/b(1-q).Proof.
The process of the proof for system (25) to obtain fixed-time synchronization is the same as that for Theorem 16. Hence, we omit it.


## 5. Numerical Simulations

We provide two simulation examples to illustrate the correctness and effectiveness of the results obtained by our paper in this section.Example 1.
This example considers the drive system (41) which is the 3-linked hybrid coupled complex networks and consists of 3 nodes. Each of the nodes is 2-dimensional.(41)x˙it=fxit+gxit-τt+∑j=1,j≠i3Gij0Dxjt-xit+∑j=1,j≠i3Gij1Dτ1xjt-τ1t-xit∑j=1,j≠i3Gij2Dτ2xjt-τ2t-xit,i=1,2,3,where(42)fxit=-1001xi1txi2t+1+π810.21+π8f1xi1tf2xi2t,gxit-τt=2π81.30.10.12π81.3f1xi1t-τtf2xi2t-τt.
Andfi(x)=x+1-x-1/2,i=1,2.
The time delays are chosen as follows:(43)τt=et1+et,τ1t=1.51-cos⁡2t,τ2t=1+sin⁡t.
The configuration matrices are chosen as follows:(44)G0=-0.30.20.10.2-0.60.40.10.4-0.5,G1=-0.80.50.30.5-0.70.20.30.2-0.5,G2=-0.60.40.20.4-0.90.50.20.5-0.7.
The inner coupling matrices are chosen as follows:(45)D=1021,Dτ1=31.221,Dτ2=1.1412.5.
Correspondingly, the response system is given as follows:(46)y˙it=fyit+gyit-τt+∑j=1,j≠i3Gij0Dyjt-yit+∑j=1,j≠i3Gij1Dτ1yjt-τ1t-yit+∑j=1,j≠i3Gij2Dτ2yjt-τ2t-yit+uit,i=1,2,3.
The initial conditions for simulation are given as follows:(47)x1t=7-t-1,x2t=3-5+t,x3t=2+2t-3,y1t=t+1-8,y2t=52,y3t=-1-6,t∈-5,0.
We define the convergence errors aseit1=yi1-xi1+yi2-xi2,i=1,2,3.
We divide simulation Example1 into three cases to illustrate our obtained results in more detail and in more normalized manner. Let α=3.9 and β=1.Case 1.
We study the state trajectories ofxij(t) and yij(t)(i=1,2,3;j=1,2) when the drive-response systems are without control inputs, which are showed in Figure 1. Moreover, the trajectories of eit1,i=1,2,3 without control inputs can be seen in Figure 2.Figure 1
The state trajectories of system (41) and system (46) without control inputs.Figure 2
The convergence errors between system (41) and system (46) without control inputs.Case 2.
We simulate the result of Theorem9 when inequalities in (18) are satisfied. We choose the values of the parameters as follows: p1=12, q1=1, ξ11=10, and ξ12=9; p2=14, q2=1, ξ21=9, and ξ22=14; p3=12, q3=1, ξ31=6, and ξ32=11; k0=2; η=1/2. Let t0=0. According to Theorem 9, we can calculate that the value of the settling time t1 is 4.4496. The simulation of finite-time synchronization for the drive-response systems (41) and (46) in settling time t1 with control inputs (17) is showed in Figure 3.Figure 3
The convergence errors between system (41) and system (46) under controller (17).Case 3.
We simulate the result of Theorem16 when inequalities in (32) are satisfied. We choose the values of the parameters as follows: p1=12, q1=1, ξ11=10, and ξ12=9; p2=14, q2=1, ξ21=9, and ξ22=14; p3=12, q3=1, ξ31=6, and ξ32=11; a=1, b=1, p=2, and q=1/2. According to Theorem 16, we can calculate that the value of the fixed settling time T0 is 5. The simulation of fixed-time synchronization for the drive-response systems (41) and (46) in fixed settling time T0 with control inputs (31) is showed in Figure 4.Figure 4
The convergence errors between system (41) and system (46) under controller (31).Example 2.
This example considers the drive system (48) which is the 3-linked hybrid coupled complex network and consists of 3 nodes. Each of the nodes is 2-dimensional. The system which is similar to (41) is given as follows:(48)x˙it=fxit+gxit-τt+∑j=1,j≠i3Gij0Dxjt-xit+∑j=1,j≠i3Gij1Dτ1xjt-τ1t-xit∑j=1,j≠i3Gij2Dτ2xjt-τ2t-xit+uit,i=1,2,3,where ui(t) denotes suitable control inputs.
And we will use the samef(·), g(·), fi(x), D, Dτ1, Dτ2, τ(t), τ1(t), and τ2(t) given in Example 1.
The configuration matrices are chosen as follows:(49)G0=-0.30.150.150.15-0.30.150.150.15-0.3,G1=-0.40.20.20.2-0.40.20.20.2-0.4,G2=-0.50.250.250.25-0.50.250.250.25-0.5.
According to (26), the synchronized state model is given as follows:(50)s˙t=fst+gst-τt--0.4Dτ1st-τ1t-st--0.5Dτ2st-τ2t-st.
The initial conditions for simulation are given as follows:(51)st=8-t-1,x1t=3-5+t,x2t=2+2t-3,x3t=t+1-8,t∈-5,0.In Example2, we define the convergence errors as ωit1=xi1-s1+|xi2-s2|,i=1,2,3. And we divide simulation Example 2 into three cases to illustrate our obtained results in more detail and in more normalized manner.Case 1.
We study the state trajectories ofxij(t) and sj(t)(i=1,2,3;j=1,2) when systems (48) and (50) are without control inputs, which are showed in Figure 5. Moreover, the trajectories of ωit1,i=1,2,3, without control inputs can be seen in Figure 6.Figure 5
The state trajectories of systems (48) and (50) without control inputs.Figure 6
The convergence errors between system (48) and system (50) without control inputs.Case 2.
We simulate the result of Corollary14. We choose the values of the parameters as follows: p1=10, q1=1, ξ11=5, and ξ12=8; p2=10 and q2=1, ξ21=5, and ξ22=8; p3=10, q3=1, ξ31=5, and ξ32=8; k0=2 and η=1/2. The simulation of finite-time synchronization for systems (48) and (50) in settling time t1 (through calculating, t1=4.6819) with control inputs (29) is showed in Figure 7.Figure 7
The convergence errors of systems (48) and (50) under controller (29).Case 3.
We simulate the result of Corollary18 when inequalities in (39) are satisfied. We choose the values of the parameters as follows: p1=10, q1=1, ξ11=5, and ξ12=8; p2=10, q2=1, ξ21=5, and ξ22=8; p3=10, q3=1, ξ31=5, and ξ32=8; a=1, b=1, p=2, and q=1/2. According to Corollary 18, we can calculate that the value of the fixed settling time T0 is 5. The simulation of fixed-time synchronization for the drive-response systems (48) and (50) in fixed settling time T0 with control inputs (40) is showed in Figure 8.Figure 8
The convergence errors between systems (48) and (50) under controller (40).


## 6. Conclusions

In this paper, we study the finite-time synchronization of new multilinked hybrid coupled complex dynamical networks with one single time-varying delay coupling for the first time. Furthermore, we study the fixed-time synchronization of new multilinked hybrid coupled complex dynamical networks with one single time-varying delay coupling for the first time. Two suitable feedback controllers are designed in order to obtain the finite/fixed-time synchronization. Several synchronization criteria are proposed through strict calculating and theorem derivation. Finally, we use two appropriate numerical examples to illustrate the correctness and effectiveness of the results proposed in this paper. The problem of fixed-time synchronization and finite-time synchronization for the hybrid coupled dynamical networks whose coupling terms contain only one transmittal time-varying delay comes naturally for multilinked systems which simulate the real world in closer and more realistic way. It is the first time to study this exact problem and the significance of solving the problem that make our work interesting and meaningful.



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*Source: 8435349-2017-01-18.xml*