# New results of exponential synchronization of complex network with time-varying delays

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## Abstract

In this study, we elucidated the exponential synchronization of a complex network system with time-varying delay. Then the exponential synchronization control of several types of complex network systems with time-varying delay under no requirements of delay derivable were explored. The dynamic behavior of a system node shows time-varying delays. Thus, to derive suitable conditions for the exponential synchronization of different complex network systems, we designed a linear feedback controller for linear coupling functions, using the Lyapunov stability theory, Razumikhin theorem, and Newton–Leibniz formula. The exponential damping rates for the exponential synchronization of different complex network systems were then estimated. Finally, we validated our conclusions through a numerical simulation.

## Keywords

Exponential synchronization Complex network Time-varying delay Razumikhin theorem## 1 Introduction

Many problems in nature can be descrobed by complex network models, such as the World Wide Web (WWW), food chains, traffic networks, and social networks. Thus, complex networks have attracted widespread interest.

Synchronization is an important dynamic characteristic of a complex network. Many phenomena in nature are realized by the synchronization of complex networks. For example, Strogatz discovered the synchronized contractions of cardiac muscle cells [1]. Steinmetz et al. discovered that the attention selection modes of human beings and primates are closely related with the synchronous rates of their neurons [2]. Furthermore, synchronization technologies have been widely applied in practical life. Millerioux et al. achieved chaos secure transmission from the master–slave system [3]. Kunbert et al. suggested solving problems encountered in image processing through the use of synchronously generated autowaves [4]. Synchronization control has received considerable attention in the control field, and many control technologies have been introduced into the synchronization of networks. Adaptive, pulse, intermittent, and pinning controls have been used for the synchronous control of complex networks and thereby contributed to many outstanding works [5, 6, 7, 8].

Information transition in a network is often accompanied by time delay, which is a major cause of system instability. Therefore, studying the stability of a time-delay system has become a critical topic in control theory. Given that the time-delay system is an infinite-dimensional system, studying it is difficult, and no general research method for its study is currently available. Moreover, different problems require different control technologies. This condition often results in inconsistent research conclusions [9, 10]. Nevertheless, research on methods for the synchronous control of complex networks with time delay has achieved considerable progress [11, 12, 13, 14, 15, 16].

- (1)
The time-varying delay function is removed as a derivable in the study of synchronous control of common time-delay complex systems. The synchronous control problem in complex systems with time-varying delay is studied by using the Razumikhin theorem and the Newton–Leibniz formula.

- (2)
Several types of complex network systems with time-varying delay are nonlinear functions and contain time-varying delays, and the conditions for exponential synchronization are obtained on the basis of the hypothesis that the nonlinear function satisfies the Lipschitz function, whether or not the coupling function between the network nodes is linear. These conditions satisfy a linear matrix inequality and are easy to determine. The upper boundary of time-varying delay can easily be calculated with the MATLAB tool box.

### Notations

All the notations used in this paper are standard. For \(x \in \mathbb{R}^{n}\), let \(\Vert x \Vert \) denote the Euclidean vector norm, i.e., \(\Vert x \Vert = \sqrt{(x^{T}x)}\). For \(A \in \mathbb{R}^{N \times N}\), let \(\Vert P \Vert \) indicate the norm of *P* induced by the Euclidean vector norm, i.e., \(\Vert P\Vert = \sqrt{ \eta _{\max }(P^{T}P)}\), where \(\eta _{\max }(P^{T}P)\) is the maximum eigenvalue of \(P^{T}P\), *I* denotes the identity matrix with appropriate dimensions, \(\mathbb{R}^{m \times n}\) denotes the set of all \(m \times n\) real matrices, and \(\mathbb{R}^{n}\) is the *n*-dimensional Euclidean space. For symmetric matrices *A* and *B*, the notation \(A > B\) (\(A \ge B\)) means that the matrix \(A - B\) is positive definite (nonnegative); \(\operatorname{diag}\{\ldots\}\) is used to denote the block diagonal matrix; the superscript “*T*” represents the transpose; let \(A \otimes B\) be the Kronecker product of two matrices *A* and *B*.

## 2 Preliminaries

*N*dynamic nodes with linear couplings, which is described by

*i*th leader, \(f:\mathbb{R}^{n} \times \mathbb{R}^{n} \to \mathbb{R}^{n}\) is a continuously differential nonlinear vector function, \(c_{1} > 0\) and \(c_{2} > 0\) are the non-delay and time-varying delay coupling strengths, \(\varGamma _{1} = \operatorname{diag}(\gamma _{1}^{1},\gamma _{1}^{2},\ldots,\gamma _{1}^{n})\) and \(\varGamma _{2} = \operatorname{diag}( \gamma _{2}^{1},\gamma _{2}^{2},\ldots,\gamma _{2}^{n})\) are positive definite diagonal matrices, which represent the inner connection matrices between each pair of nodes; Meanwhile, \(A = (a_{ij}) \in \mathbb{R}^{N \times N}\), \(B = (b_{ij}) \in \mathbb{R}^{N \times N}\) are the non-delay and time-varying delay weight matrices. The entries are defined as follows: if is a link is present from the node

*i*to the node

*j*(\({i \ne j}\)), then \(a_{ij} \ne 0\), \(b_{ij} \ne 0\); otherwise \(a_{ij} = 0\), \(b_{ij} = 0\), and the diagonal elements of matrices

*A*and

*B*are defined as \(a_{ii} = - \sum_{j = 1,j \ne i}^{N} a_{ij}\), \(b_{ii} = - \sum_{j = 1,j \ne i}^{N} b_{ij}\). Given that the networks in this paper are direct, \(a_{ij} \ne a_{ji}\) and \(b_{ij} \ne b_{ji}\) (\(i,j = 1,2,\ldots,N\)). The coupling time-varying delay \(\tau (t)\) is a bound function and a positive constant

*τ*satisfying \(0 \le \tau (t) \le \tau \), \(\forall t > 0\), \(u_{i} \in \mathbb{R}^{N \times N}\) is the control input of node

*i*. In this paper, we aim to design a suitable controller

*i*th node.

Let \(\tilde{f}(x(t),x(t - \tau (t))) = f(x(t),x(t - \tau (t))) - f(s(t),s(t - \tau (t)))\), and \(K = \operatorname{diag}(k_{1},k_{2},\ldots,k_{n})\).

Therefore, if the error system (6) is exponentially stable, then the complex network (1) is exponentially synchronized.

The following assumptions and lemmas are needed for the derivation of our main results.

### Definition 1

(*see* [9])

*ξ*,

*α*are present, so every error vector \(e(t,\phi )\) of the system satisfies

### Lemma 1

(*Cauchy inequality* [9])

*For any vector*\(x,y \in \mathbb{R}^{n}\)

*and positive definite matrix*\(W \in \mathbb{R}^{N \times N}\),

*the following matrix inequality holds*:

### Lemma 2

(*Razumikhin stability theorem* [28])

*Assume that*\(\upsilon ,u,\omega :\mathbb{R}^{ +} \to \mathbb{R}^{ +} \)

*are nondecreasing*,

*and*\(u(v)\), \(\upsilon (v)\)

*are positive for*\(v \ge 0\), \(\upsilon (0) = u(0) = 0\),

*and*\(q > 1\).

*If a function*\(V(t,e): \mathbb{R}^{ +} \times \mathbb{R}^{ +} \to \mathbb{R}^{ +} \)

*exists*,

*so*

- (i)
\(u( \Vert e \Vert ) \le V(t,e) \le \upsilon ( \Vert e \Vert )\), \(t \in \mathbb{R}^{ +}\), \(e \in \mathbb{R}^{n}\),

- (ii)
\(\dot{V}(t,e(t)) \le - \omega ( \Vert e \Vert )\),

*if*\(V(t + v,e(t + v)) \le qV(t,e(t))\), \(\forall v \in [ - \tau ,0]\), \(t \ge 0\),

*then the error system is asymptotically stable*.

### Lemma 3

(*Schur complement* [29])

*The matrix*$\left[\begin{array}{cc}{S}_{11}& {S}_{12}\\ {S}_{21}& {S}_{22}\end{array}\right]<0$,

*where*\(S_{11} = S_{11}^{T}\), \(S_{12} = S_{21}^{T}\)

*is equivalent to any one of the following conditions*:

- (1)
\(S_{22} < 0\), \(S_{11} - S_{12}S_{22}^{ - 1}S_{12}^{T} < 0\);

- (2)
\(S_{11} < 0\), \(S_{22} - S_{12}^{T}S_{11}^{ - 1}S_{12} < 0\).

### Assumption 1

(*see* [30])

### Remark 1

In some studies on complex network synchronization problems, the effects of time delay were not considered. For example Jin-Liang Wang discussed the output synchronization and \(H_{\infty } \) output synchronization of multi-weighted complex network in [31] but did not consider the network model with time delay. Moreover, many studies on complex network often require derivable or bounded time delay [20, 21, 22, 23, 24, 25, 26, 27]. In the proposed model, the requirements for time delay (\(\tau (t)\)) are reduced, and the time delay does not need to be differentiated.

## 3 Main result

### 3.1 Synchronization control of a general complex dynamical network with dynamic behavior including time-varying delays and linear time-varying coupling delay consisting of *N* identical nodes with linear couplings

### Theorem 1

*The complex network system*(1)

*is exponentially synchronized if positive numbers*

*β*,

*δ*,

*λ*, \(L_{1}\), \(L_{2}\)

*and a positive matrix*

*P*

*are present and the following inequality equation holds*:

*where*\(\gamma _{1} = [4\lambda ^{ - 1} + 4\lambda ^{ - 1}\tau + 2K](P + \beta I) + \delta I\).

*Moreover*,

*the error vector*\(e(t,\phi )\)

*satisfies the condition*

*Here*\(p = \Vert P \Vert \).

### Proof

*P*is a positive symmetric matrix. It is easy to see that

### Remark 3

Although many studies on the synchronous control of complex network systems with time-varying delay have generated many and variable conclusions, the studies, especially those regarding the synchronization stability of systems, all used the same two requirements on time-varying delay. First, a time-varying delay should have boundaries; that is, \(0 \le \tau (t) \le \tau \), \(\forall t > 0\). Second, the time-varying delay should be derivable and smaller than 1; that is, \(0 \le \dot{\tau } (t) \le 1\), \(\forall t > 0\). For example [22, 25], the time-varying delay may not be derivable. In fact, the time-varying delay is not derivable when it changes quickly in a time period. In this study, only the first requirement of time-varying delay is needed. The second requirement can be solved by the Newton–Leibniz formula. The stability conditions of the exponential synchronization of the system are obtained through the Razumikhin theorem. Therefore, this theorem relieves the requirements to the system in contrast to the theorems used in other studies [20, 21, 22, 23, 24, 25, 26, 27].

### Remark 4

In [34, 35], the synchronization (or stability) of similar models, which all require a derivable time-varying delay, is investigated. Similarly, this requirement is unnecessary in Corollary 2.

According to the proof of Theorem 1, it is easy to get the following conclusion. We have the following corollary.

### Corollary 1

*The complex network system*(1)

*is exponentially synchronized if positive numbers*

*β*,

*δ*,

*λ*, \(L_{1}\), \(L_{2}\),

*and a positive matrix*

*P*

*exist such that the following inequality equation holds*:

*where*\(\gamma _{2} = [4\lambda ^{ - 1} + 4\lambda ^{ - 1}\tau + 2K]P + \delta I\).

*Moreover*,

*the error vector*\(e(t,\phi )\)

*satisfies the condition*

*Here*\(p = \Vert P \Vert \).

### 3.2 The synchronization control of complex dynamical network with time-varying coupling delay consisting of *N* identical nodes with linear couplings

*i*th leader, \(f:\mathbb{R}^{n} \times \mathbb{R}^{n} \to \mathbb{R}^{n}\) is a continuously differential nonlinear vector function, \(c_{1} > 0\) and \(c_{2} > 0\) are the non-delay and time-varying delay coupling strengths, \(\varGamma _{1} = \operatorname{diag}(\gamma _{1}^{1},\gamma _{1}^{2},\ldots,\gamma _{1}^{n})\) and \(\varGamma _{2} = \operatorname{diag}( \gamma _{2}^{1},\gamma _{2}^{2},\ldots,\gamma _{2}^{n})\) are positive definite diagonal matrices, which represent the inner connection matrices between each pair of nodes; \(A = (a_{ij}) \in \mathbb{R}^{N \times N}\), \(B = (b_{ij}) \in \mathbb{R}^{N \times N}\) are the non-delay and time-varying delay weight matrices. The entries are defined as follows: if is a link is present from the node

*i*to the node

*j*(\(i \ne j\)), then \(a_{ij} \ne 0\), \(b_{ij} \ne 0\); otherwise \(a_{ij} = 0\), \(b_{ij} = 0\), and the diagonal elements of matrices

*A*and

*B*are defined as \(a_{ii} = - \sum_{j = 1,j \ne i}^{N} a_{ij}\), \(b_{ii} = - \sum_{j = 1,j \ne i}^{N} b_{ij}\). Given that the networks in this paper are direct, \(a_{ij} \ne a_{ji}\) and \(b_{ij} \ne b_{ji}\) (\(i,j = 1,2,\ldots,N\)). The coupling time-varying delay \(\tau (t)\) is a bound function and a positive constant

*τ*satisfying \(0 \le \tau (t) \le \tau \), \(\forall t > 0\), \(u_{i} \in \mathbb{R}^{N \times N}\) is the control input of node

*i*. In this paper, we aim to design a suitable controller,

*i*th node.

Let \(\tilde{f}(x) = f(x) - f(s(t))\), \(K = \operatorname{diag}(k_{1},k_{2},\ldots,k_{N})\).

Therefore, if the error system (24) is exponentially stable, the complex network system (20) is exponentially synchronized.

### Theorem 2

*The complex network system*(18)

*is exponentially synchronized if positive numbers*

*β*,

*δ*,

*λ*, \(L_{1}\),

*and a positive matrix*

*P*

*exist such that the following inequality equation holds*:

*where*\(\gamma _{3} = L_{1}^{2}I + (1 + 4\tau )\lambda ^{ - 1}(P + \beta I) + \delta I + 2K(P + \beta I)\).

*Moreover*,

*the error vector*\(e(t,\phi )\)

*satisfies the condition*

*Here*\(p = \Vert P \Vert \).

### Proof

*P*is a positive symmetric matrix. It is easy to see that

### Remark 6

Many researchers who studied the synchronous control of complex network with time-varying delay hypothesized that dynamic behaviors include nonlinear function. Some of them did not consider time delay in nonlinear dynamic behaviors [36] and some did [37]. Notably, the time-varying delay of systems (time delay in dynamic behavior and node coupling) was considered derivable in previous studies. This condition is eliminated in our Theorem 2, which relieves the requirements on time delay and thereby reduces the conservation of conclusions.

According to Theorem 2, it is easy to get the following conclusion. We have the following corollary.

### Corollary 2

*The complex network system*(18)

*is exponentially synchronized when positive numbers*

*δ*,

*λ*, \(L_{1}\)

*and a positive matrix*

*P*

*are present and the following inequality equation holds*:

*where*\(\gamma _{4} = L_{1}^{2}I + \lambda ^{ - 1}(1 + 4\tau )P + \delta I + 2KP\).

*Moreover*,

*the error vector*\(e(t,\phi )\)

*satisfies the condition*

*Here*\(p = \Vert P \Vert \).

### 3.3 Numerical simulation

In this section, the exponential synchronization conditions obtained in this paper are illustrated with an example.

*K*and

*P*can be obtained, and the initial conditions of the system are given, \(t_{0} = 0\), and the initial state \(x(0) = [2.62342, - 5.74317,\ldots,2.62342, - 2.74317,1.89187]^{T}\) is given at random. Figures 1, 2, and 3 are for the node 1, 2, 3 status curves without controller and Figs. 4, 5, and 6 are for the node 1, 2, 3 status curves with controller.

As can be seen from the figure, after a period of time, the system state is synchronous, and the feasibility and effectiveness of the control method are demonstrated.

## 4 Conclusions and discussions

This work focuses on the problem of exponential synchronization in complex network systems with time-varying delay. In the light of the Razumikhin stability theorem combined with the Newton–Leibniz formula, we deduced an exponential synchronization condition, without considering the differentiability of the time-delay function. Finally, a numerical simulation is given to show its effectiveness. The following will be considered in our future work: how to achieve synchronization through a network that is less strict and sufficient, and how to extend existing methods to the problem of finite-time synchronization or fixed-time synchronization. This is the goal we will consider and study in the future.

## Notes

### Acknowledgements

The authors would like to thank X. Zou for her great help in many aspects.

### Availability of data and materials

The datasets used or analyzed during the current study are available from the corresponding author on request.

### Authors’ contributions

All authors contributed equally to the writing of this paper. Furthermore, all authors also read carefully and approved the final manuscript.

### Funding

This work was jointly supported by the National Natural Science Foundation of China (Grant No. 11372107), the Natural Science Foundation of Hunan Province (Grant No. 2017JJ5011) and the Natural Science Foundation of Hunan Province (Grant No. 2017JJ4004).

### Competing interests

The authors declare that they have no competing interests.

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