# Finite-Time Synchronization of Complex Dynamical Networks with Nondelayed and Delayed Coupling by Continuous Function Controller

Authors: Xie Chengrong; Xing Yu; Xia Qing; Dongbing Tong; Yuhua Xu
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

This paper investigates the finite-time synchronization of complex dynamical networks with nondelayed and delayed coupling. By designing a simple continuous function controller, sufficient criteria for finite-time synchronization of dynamical networks with nondelayed and delayed coupling are obtained. As a special case, the continuous function controller designed in this paper may be the simplest and easy to implement for the finite-time synchronization of dynamical networks without delay. Finally, numerical simulations are given to verify the effectiveness of the conclusions presented in this paper.


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## Body

## 1. Introduction

As synchronization behavior of complex systems is a common phenomenon in the field of nature and engineering technology, synchronization control of complex systems has been attracted people to study [1–8]. However, in many fields, people begin to realize that finite-time control of complex systems may be more practical than infinite-time control of complex systems [9–14]. For example, when the brain is stimulated by external signals, the brain’s nervous system will respond accordingly, and the response output signals of the central nervous system and the sensory nervous system should be consistent. That is to say, the driving neural network and the response neural network move under different starting conditions. By properly controlling the neural network, the trajectories of the two neural networks will coincide in a finite-time, which is the finite-time control of the neural dynamic network. Since finite-time of system control mainly depends on the initial state of complex systems, the initial state of complex systems may be random, which is not conducive to solving practical problems. Subsequently, Polyakov proposed a finite-time control method independent of the initial state of complex systems, that is, so-called fixed-time control of complex systems, which compensated for the limitation of the finite-time control method [15, 16]. Therefore, studying the finite-time control problem of complex networks, on the one hand, it expands synchronization control theory of complex networks; on the other hand, the finite-time control method may save the control cost better and may have greater practical value in the stability control application of complex systems.At present, based on the research of finite-time control in complex networks, the authors have obtained some interesting results, such as finite-time control via optimal method [17], finite-time control via intermittent feedback [18], finite-time control via linear and nonlinear control method [19], finite-time control with nonlinear dynamics and uncertainties [20], finite-time control with switched discontinuous systems, and so on [21]. However, in these discussions, the designed controller usually contains symbolic functions. As the designed controller contains symbolic function, which is a discontinuous function, a flutter phenomenon may occur during the control process. The occurrence of the flutter phenomenon will obviously affect the finite-time control effect of complex networks. In order to eliminate the flutter phenomenon, some authors have proposed finite-time controller with continuous functions. For example, in [22], the authors effectively synchronize two complex networks by designing continuous function controllers, eliminating the adverse effects caused by symbolic function controllers, that is, the controller is ri=−liwit−k1wim1/m2t−k2wim3/m4t, where li>0, k1>0, k2>0, m1,m2,m3,andm4 are positive odd integers satisfying, m1>m2,m3<m4, and wi means the error. In [23], a simpler continuous function controller for fixed-time synchronization of complex networks with nondelayed coupling was proposed, that is, the controller is ri=−k1wim1/m2t−k2wim3/m4t. Therefore, designing simple and effective finite-time controllers is still an interesting topic.In addition, in the network, time-delay is inevitable. For example, in the process of transmission and response, the physical limitations of propagation speed and the existence of network congestion often lead to delay in complex networks. The generation of time-delay may affect the stability of the whole network, such as the oscillation and instability of the system. Therefore, it is undoubtedly of great application value and theoretical significance to study the synchronization control of the time-delay coupled network. Although the finite-time synchronization of time-delay complex networks has been discussed in the existing literatures [24, 25], finite-time controllers designed in their papers were generally complex and contained symbolic functions, which may cause chattering during network synchronization. In general, an important criterion for studying finite-time control of complex systems is that the controller is as simple as possible and easy to implement. As far as we know, finite-time synchronization of dynamical networks with delayed coupling was less considered by using a simple continuous function controller. In this paper, we attempt to design a continuous finite-time controller to realize finite-time control of complex systems with delayed, and we also make the effort to propose the simplest finite-time continuous function controller for complex networks with nondelayed coupling.This paper is organized as follows: Section2 gives the model and preliminaries. Section 3 puts forward some finite-time synchronization criteria for networks. Section 4 gives an illustrative example. The conclusions are given in Section 5.


## 2. Model and Preliminaries

Consider the following dynamical network with delay coupling:(1)μ˙it=gμit+∑j=1Nbijμjt+∑j=1Ncijμjt−τ,where μit=μi1t,μi2t,…,μintT∈Rn is the state vector, i=1,2,…,N, N denotes natural number, and g⋅ is the activation function. B=bijn×n and C=cijn×n denote connection weight matrix. τ means time-delay.The controlled dynamic network with delay coupling is represented as follows:(2)ϑ˙it=gϑit+∑j=1Nbijϑjt+∑j=1Ncijϑjt−τ+ri,where ϑit=ϑi1t,ϑi2t,…,ϑintT∈Rn is the state vector and i=1,2,…,N, ri means the controller.Assumed the errors are expressed aswi=ϑi−μi, then(3)w˙it=gϑit−gμit+∑j=1Nbijwjt+∑j=1Ncijwjt−τ+ri.Assumption 1.
Assume the functiong⋅ satisfies the following condition:(4)gϑit−gμit≤γϑit−μit,γ∈R+.Lemma 1 (see [22]).
For any vectorsμ,ϑ∈Rm and positive definite matrix Ω∈Rm×m, the following matrix inequality holds:(5)2μTϑ≤μTΩμ+ϑTΩ−1ϑ.Lemma 2 (see [24]).
Consider the dynamical system as follows:(6)μ˙=gμt,g0=0,μ∈Rn,μ0=μ0.Suppose a continuous and positive definiteVμ satisfies(7)υ˙μ≤l1υμ−l2υξμ,υ1−ξμ<l2l1,l1>0,l2>0,0<ζ<1.Then, the origin of the dynamical system (6) is finite-time stable, and(8)Tμμ0≤1l1ξ−1ln1−l1l2υ1−ξμ0.Lemma 3 (see [26]).
ForΘi≥0, i=1,2,…,n, 0<λ≤1, κ>1, then(9)∑i=1nΘiλ≥∑i=1nΘiλ,∑i=1nΘiκ≥n1−κ∑i=1nΘiκ.


## 3. Main Results

Based on the preparation of the second section, the section will discuss the finite-time synchronization of two complex networks.Theorem 1.
If Assumption1 holds, two complex dynamical networks (1) and (2) can be synchronized by the following simpler finite-time controller:(10)ri=−kwiTt−τwit−τwit2wit−hiwiξt,and(11)T≤1lξ−1ln1−21−ξ/2lλminH⊗Iυ1−ξ/2w0,where k=N/2λmaxΩ−1, l=γ+λmaxB⊗I+Nδ2/2λmaxΩ⊗I, H=diagh1,…,hN, hi>0, ξ=n1/n2, n1 and n2 are positive odd integers, n1<n2, and Ω is the positive definite matrix.Proof.
Consider the following function:(12)υt=12wTtwt,where w=w1T,w2T,…,wnTT.
So,(13)υ˙t=∑i=1NwiTtgϑit−gμit+∑j=1Nbijwjt+∑j=1Ncijwjt−τ+ri≤∑i=1NγwiTtwit+∑i=1NwiTt∑j=1Nbijwjt+∑i=1NwiTt∑j=1Ncijwjt−τ−∑i=1NwiTtkwiTt−τwit−τwit2wit−λminH⊗IwTtwξt.
In equation (13), obviously,(14)wTtwξt=∑i=1N∑j=1nwij2t1+ξ/2≥∑i=1N∑j=1nwij2t1+ξ/2≥wTtwt1+ξ/2=21+ξ/2υt1+ξ/2∑i=1NwiTt∑j=1Ncijwjt−τ≤Nδ22∑i=1NwitTΩwit+N2∑j=1Nwjt−τTΩ−1wjt−τ,where cij≤δ∈R+.
So,(15)υ˙t≤∑i=1NγwiTtwit+∑i=1NwiTt∑j=1Nbijwjt+Nδ22∑i=1NwitTΩwit+N2∑j=1Nwjt−τTΩ−1wjt−τ−∑i=1NkwiTt−τwit−τwit2wiTtwit−λminH⊗I21+ξ/2υt1+ξ/2≤∑i=1NγwiTtwit+∑i=1NwiTt∑j=1Nbijwjt+Nδ22∑i=1NwitTΩwit+N2λmaxΩ−1∑j=1Nwjt−τTwjt−τ−k∑i=1NwiTt−τwit−τ−λminH⊗I21+ξ/2υt1+ξ/2≤∑i=1NγwiTtwit+∑i=1NwiTt∑j=1Nbijwjt+Nδ22∑i=1NwitTΩwit+N2λmaxΩ−1−k∑j=1Nwjt−τTwjt−τ−∑i=1NlwiTtwit+∑i=1NlwiTtwit−λminH⊗I21+ξ/2υt1+ξ/2.
Whenk=N/2λmaxΩ−1, l=γ+λmaxB⊗I+Nδ2/2λmaxΩ⊗I, we have(16)υ˙t≤∑i=1NlwiTtwit−λminH⊗I21+ξ/2υt1+ξ/2=2lυt−λminH⊗I21+ξ/2υt1+ξ/2.
Then, the origin of the error systems (3) is finite-time stable, and T satisfies(17)T≤1lξ−1ln1−21−ξ/2lλminH⊗Iυ1−ξ/2w0.
The proof is completed.Remark 1.
At present, few continuous function controllers are designed for finite-time synchronization of delay dynamical networks. Therefore, the continuous function finite-time controller in this paper may be more practical.Remark 2.
From the proof of Theorem1, when the delay is time-varying, Theorem 1 still holds.Remark 3.
Ifcij=0, the coupled dynamical networks (1)-(2) can reduce to complex dynamical networks with the nondelayed coupling dynamical network. In this case, the finite-time controller (10) is not available. In this case, since the network does not contain delay, and the following Corollary 1 can be obtained.Remark 4.
The matrixΩ is a positive definite matrix given arbitrarily. To guarantee the optimality of result, the matrix Ω should have minimum eigenvalue. As the convergence time is related to the matrix Ω, the control strength k is also related to the matrix Ω. From their expressions, we can see that the smaller the eigenvalue of the matrix Ω, the shorter the convergence time, but the greater the control strength k, and how to choose the matrix Ω to guarantee the optimality of result will be the topic for further consideration in the future.
WhenC=0, the complex networks (1)-(2) changed into the following complex dynamical networks with nondelayed coupling:(18)μ˙it=gμit+∑j=1Nbijμjt,(19)ϑ˙it=gϑit+∑j=1Nbijϑjt+ri.Corollary 1.
If Assumption1 holds, two complex dynamical networks with nondelayed coupling (18) and (19) can be synchronized by the simplest finite-time controller:(20)ri=−hiwiξt,and the time is bounded as(21)T≤1lξ−1ln1−21−ξ/2lλminH⊗Iυ1−ξ/2w0,where l=γ+λmaxB⊗I, ξ=n1/n2, and n1 and n2 are positive odd integers, n1<n2.
The proof of Corollary1 is similar to that of Theorem 1, so we omit their reasoning process here.Remark 5.
Compared with the existing literature [15–18] on finite-time synchronization of complex networks without delay, the controller (20) designed in this paper may be the simplest one. Generally speaking, simple controllers may have better application in practical engineering.Remark 6.
From (11) and (21), synchronization time of complex networks is the same, that is to say, the delay term of complex networks does not affect the synchronization time of complex networks.Remark 7.
In [9, 10, 15, 24, 25], the authors proposed continuous function controllers without symbolic functions, but the synchronization of delay networks was not discussed in their paper.


## 4. Numerical Simulation Example

Considering the following chaos system [27],(22)ϑ˙i=−1010028−1000−8/3ϑi1ϑi2ϑi3+0−ϑi1ϑi3ϑi1ϑi2.It is well known that the chaos system is bounded, and our analyses show(23)gϑi−gμi2≤100.5571wi2.If we use four nodes to describe the coupled complex network, then(24)μ˙it=gμit+∑j=14bijμjt,bij4×4=−41211−63223−61121−4.Let initial values of the state variable are rand [0, 1] andγ=115, ξ=0.8, and hi=200. Figure 1 shows that the driver network and the response network cannot be synchronized without a controller. Figure 2 shows finite-time synchronization of two networks under the controller. Numerical simulations illustrate the validity of Corollary 1.Figure 1
Error evolution of two networks.Figure 2
Synchronization evolution of two networks.


## 5. Conclusion

The paper has studied synchronization of two complex dynamical networks with nondelayed and delayed coupling by the finite-time controller. The designed finite-time controller was continuous function. Especially, the continuous function controller designed might be the simplest for finite-time synchronization of complex dynamical networks without delay. Finally, the numerical simulation verified the validity of the theoretical results. As the results are sufficient conditions in this paper, which implies that there is still room for further improvement, which will be our next research topic.



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