Cluster linear generalized outer synchronization in community networks via pinning control with two different switch periods
 Yuhong Liu^{1}Email author,
 Hui Li^{1},
 Qishui Zhong^{1} and
 Shouming Zhong^{2, 3}
https://doi.org/10.1186/s1366201711737
© The Author(s) 2017
Received: 20 December 2016
Accepted: 6 April 2017
Published: 20 April 2017
Abstract
This study investigates the problem of cluster generalized outer synchronization in community networks via pinning control with two different switch periods. Several pinning controllers have been designed to achieve linear generalized outer synchronization. Using Lyapunov stability theory, sufficient linear generalized outer synchronization criteria for community networks are derived. Finally, numerical examples are presented to demonstrate the effectiveness of the theoretical results.
Keywords
1 Introduction
Recently, complex networks have drawn increasing attention from researchers and engineers in virtue of its wide applications in many fields, such as World Wide Web, communication networks, social networks, neural networks, epidemic networks, traffic networks, etc. Lots of network models, such as weighted networks [1, 2], directed networks [3, 4], hierarchical networks [5], community networks [6–8] are introduced to explore the potential applications better. As is well known, the research on network synchronization is very important due to its potential applications in many fields including secure communication, laser transmission, image identification, information science, and so on [9–14]. In recent years, much literature reported the research results of network synchronization, and it has become a frontier issue [15–19]. As a result, different types of network synchronization have been put forward, for example, complete synchronization [20–22], phase synchronization [23, 24], projective synchronization [25, 26] and cluster synchronization [27, 28].
Furthermore, many real complex networks cannot synchronize themselves or synchronize with the desired orbits. Therefore, proper controllers should be designed to achieve the goals by adopting some control schemes, such as adaptive control [29], feedback control [30], observerbased control [31], impulsive control [32], intermittent control [33–35], pinning control [36, 37] and so on. As a matter of fact, there are many examples of relationships between different networks, which indicates that it is necessary and significant to investigate the dynamical systems between different networks. Recently, [38] investigated the synchronization between two unidirectionally coupled complex networks with identical topological structures. [39] discussed the synchronization between two complex dynamical networks with nonidentical topological structures via using adaptive control method. [40] discussed adaptive projective synchronization between two complex networks with timevarying coupling delay. In the above papers, it is assumed that each node in driveresponse networks has identical dynamics. Later, [41] studied the problem of generalized outer synchronization between two complex dynamical networks with different topologies and diverse node dynamics. Reference [42] discussed the linear generalized synchronization between two complex networks with the nondelay coupling and the same topological structure, each network has identical dynamics. However, detailed analysis of the linear generalized synchronization between two networks of different topological structures and timevarying coupling delay has not been attempted in [42].

By adding adaptive semiperiodically intermittent controllers to a small fraction of nodes in response network, several sufficient conditions are derived based on the Lyapunov stability theory and strict mathematical proofs.

Both community networks with identical nodes and nonidentical nodes are investigated. Therefore, our proposed control schemes are more applicable technically.
The rest of the current paper is organized as follows. Section 2 introduces the problem formulation and some necessary definitions, lemmas, and hypotheses. Some sufficient conditions for the linear generalized outer synchronization are obtained in Section 3. Section 4 gives some numerical examples to demonstrate the effectiveness of our main results. Finally, Section 5 draws the conclusion.
Notation
The superscripts T and \((1) \) stand for matrix transposition and matrix inverse, respectively; \(\mathbb{R}^{n}\) denotes the ndimensional Euclidean space; \(I_{l}\) means the ldimensional identity matrix. The notation \(X>Y\) (\(X\ge Y\)), where X, Y are symmetric matrices, means that \(XY\) is positive definite (positive semidefinite). ∗ denotes the term that is induced by symmetry. \(\Vert \xi \Vert \) indicates the 2norm of a vector ξ, i.e., \(\Vert \xi \Vert =\xi^{T}\xi\). \(\operatorname{col}\{x_{1}, x_{2}, \ldots, x_{n}\}\) means \([x_{1}^{T},x_{2}^{T}, \ldots,x_{n}^{T}]^{T}\) and \(\operatorname{Sym}\{X\}\) means \(X+X^{T}\). The shorthand notation \(\operatorname{diag}\{ M_{1}, M_{2}, \ldots, M_{n}\}\) denotes a block diagonal matrix with diagonal blocks being the matrices \(M_{1},M_{2}, \ldots, M_{n}\). \(\lambda_{\min}(\cdot)\) and \(\lambda_{\max}(\cdot)\) denote the smallest and largest eigenvalue of ⋅. The symbol ⊗ denotes the Kronecker product. Matrices, if their dimensions are not explicitly stated, are assumed to have appropriate dimensions for algebraic operations.
2 Problem formulation and preliminaries
Remark 2.1
The nonlinear vectorvalued functions \(f_{\varphi i}\) and \(\tilde{f}_{\varphi i}\) can be identical or nonidentical.
Remark 2.2
There are no limitations for the division of the clusters, the number of nodes in each cluster and the connections between nodes.
Remark 2.3
All nodes within a cluster have the same dynamics, and the dynamics of the nodes in different clusters can be different.
Remark 2.4
The proposed approach on the case with undirected topology is similar to the one that on the case with directed topology. So in this paper the underlying topology is assumed to be undirected.
Suppose that the networks (2) will be controlled onto some desired inhomogeneous state as \(\{y_{1}(t),\ldots,y_{m_{1}}(t)\}\rightarrow \phi_{1}(t)\), \(\{y_{m_{1}+1}(t),\ldots,y_{m_{2}}(t)\}\rightarrow \phi_{2}(t)\), … , \(\{y_{m_{s1}+1}(t),\ldots,y_{m_{s}}(t)\}\rightarrow \phi_{s}(t)\), i.e., \(\mathcal{M}=\{\{\phi_{1}(t),\ldots,\phi_{1}(t)\},\{\phi_{2}(t),\ldots,\phi_{2}(t)\}, \ldots,\{\phi_{s}(t),\ldots,\phi_{s}(t)\}\}\in \mathbb{R}^{n\times N}\) is desired cluster synchronization pattern under the pinning control.
Definition 2.1
Assumption 2.1
Lemma 2.1
For a diagonal matrix \(D=\operatorname{diag}\{\underbrace{d_{1},d_{2},\ldots,d_{l}}_{i= \{1,2,\ldots,l \} \subseteq \bar{V}_{\varphi i}}0,0,\ldots,0\}\) with \(d_{i}>0\), (\(i=1,2,\ldots,l\); \(1\leq l\leq N\)) and a symmetric matrix \(M\in \mathbb{R}^{N\times N}\), let \(MD= \bigl[ {\scriptsize\begin{matrix}{} E\bar{D}&S\cr \ast & M_{l} \end{matrix}} \bigr] \), where \(M_{l}\) is the minor matrix of M by removing its first l (\(1\leq l\leq N\)) rowcolumn pairs, E and S are matrices with appropriate dimensions, \(\bar{D}=\operatorname{diag}\{d_{1},d_{2},\ldots,d_{l}\}\). If \(d_{i}> \lambda_{\max}(ESM_{l}^{1}S^{T})\), then \(MD<0\) is equivalent to \(M_{l}<0\).
Proof
Let \(\bar{D}=\operatorname{diag}\{\underbrace{d_{1},d_{2},\ldots,d_{l}}_{i= \{1,2,\ldots,l \} \subseteq \bar{V}_{\varphi i}}\}\). Using matrix decomposition, \(M\in \mathbb{R}^{N\times N}\), let \(MD=\bigl[ {\scriptsize\begin{matrix}{} E\bar{D}&S\cr \ast&M_{l}\end{matrix}} \bigr]\), where \(M_{l}\) is the minor matrix of M by removing its first l (\(1\leq l\leq N\)) rowcolumn pairs, E and S are matrices with appropriate dimensions.
Using the Schur complement, it is easy to see that \(MD<0\) is equivalent to \(M_{l}<0\). We only need to prove that if \(M_{l}<0\), then \(MD<0\). When \(d_{i}>0\) (\(i=1,2,3,\ldots, l\)) are sufficiently large such that \(d_{i}> \lambda_{\max}(ESM_{l}^{1}S^{T})\) hold, it is easy to see that \(E\bar{D}SM_{l}^{1}S^{T}<0\). Then, using the Schur complement, we can conclude that \(MD<0\), so the proof is finished. □
Lemma 2.2
[37]
Assume that A, B are N by N Hermitian matrices. Let \(\alpha_{1}\geq \alpha_{2}\geq \cdots \geq \alpha_{N}\), \(\beta_{1}\geq \beta_{2}\geq \cdots \geq \beta_{N}\) and \(\gamma_{1}\geq \gamma_{2} \geq \cdots \gamma _{N}\) be eigenvalues of A, B and \(A+B\), respectively. Then one has \(\alpha_{i}+\beta_{N}\leq\gamma_{i}\leq \alpha_{i}+\beta_{1}\), \(i=1,2,\ldots, N\).
3 Main results
In this section, the CLGOS of the driveresponse community networks (1) and (2) will be investigated in three cases.
We denote \(\Xi_{1}^{m}=[mT,mT+\eta_{1}T_{1}]\) is the control width in period \(T_{1}\), \(\Xi_{2}^{m}=[mT+\eta_{1}T_{1},mT+T_{1}]\) is the nonfeedback control width in period \(T_{1}\), \(\Xi_{3}^{m}=[mT+T_{1},mT+T_{1}+\eta_{2}T_{2}]\) is the control width in period \(T_{2}\), \(\Xi_{4}^{m}=[mT+T_{1}+\eta_{2}T_{2}, (m+1)T]\) the nonfeedback control width in period \(T_{2}\), where \(m=0,1,2,\ldots\) .
Theorem 3.1
Proof

When \(t\in \Xi_{1}^{m}\), i.e., \(\frac{t\eta_{1}T_{1}}{T}< m\leq \frac{t}{T}\)$$\begin{aligned} V (t) \leq & V (m T)\operatorname{exp}\bigl(\beta (tmT) \bigr) \\ \leq & V (0)\operatorname{exp}\bigl(\beta mT +m\alpha \bigl( (1\eta_{1}) T_{1}+ (1\eta_{2})T_{2} \bigr) \bigr) \\ \leq & V (0) \operatorname{exp}\bigl( (\beta+\alpha\theta)t+\beta\eta_{1}T_{1} \bigr). \end{aligned}$$(25)

When \(t\in \Xi_{2}^{m}\), i.e., \(\frac{tT_{1}}{T}< m\leq \frac{t\delta_{1}T_{1}}{T}\),$$\begin{aligned} V (t) \leq &V (mT+T_{1}) \operatorname{exp}\bigl(t (mT+T_{1}) \bigr) \\ \leq&V (0)\operatorname{exp}\bigl( (\alpha\beta) (mT+T_{1})\alpha \bigl( (m+1)\eta_{1}T_{1}+m\eta_{2}T_{2} \bigr) \bigr) \\ \leq & V (0) \operatorname{exp}\biggl( (a_{\ast}\alpha\rho)t\alpha ( \eta_{1}\eta_{2}) \frac{T_{1}T_{2}}{T}+a_{\ast} (1 \eta_{1})T_{1} \biggr). \end{aligned}$$(26)

When \(t\in \Xi_{3}^{m}\), i.e., \(\frac{tT_{1}\eta_{2}T_{2}}{T}< m\leq \frac{tT_{1}}{T}\),$$\begin{aligned} V (t) \leq &V ( mT+T_{1}) \operatorname{exp}\bigl(\beta \bigl(t (mT+T_{1}) \bigr) \bigr) \\ \leq & V (0)\operatorname{exp}\bigl(\beta (mT+T_{1}) +\alpha \bigl( (m+1) (1 \eta_{1})T_{1} +m (1\eta_{2})T_{2} \bigr) \bigr) \\ \leq & V (0) \operatorname{exp}\biggl( (\beta+\alpha\theta)+\beta\eta_{2}T_{2} +\alpha\frac{\eta_{2}\eta_{1}}{T}T_{1}T_{2} \biggr). \end{aligned}$$(27)

When \(t\in \Xi_{4}^{m}\), i.e., \(\frac{t}{T}< m+1\leq \frac{t+TT_{1}\eta_{2}T_{2}}{T}\),$$\begin{aligned} V (t) \leq &V \bigl( (m+1)T \bigr) \operatorname{exp}\bigl(a_{\ast} \bigl(t (m+1)T \bigr) \bigr) \\ \leq &V (0)\operatorname{exp}\bigl(a_{\ast} (m+1)T  (m+1)\alpha ( \eta_{2}T_{2}+\eta_{1}T_{1}) \bigr) \\ \leq &V (0)\operatorname{exp}\bigl( (a_{\ast}\alpha\rho)t+a_{\ast} (1 \eta_{2})T_{2} \bigr). \end{aligned}$$(28)
Therefore, when \(t\in \Xi_{1}^{m}\cup \Xi_{3}^{m}\), if \(\beta\alpha\theta >0\) is satisfied, one has \(\lim_{t\rightarrow \infty}V(t)=0\); when \(t\in \Xi_{2}^{m}\cup \Xi_{4}^{m}\), if \(\alpha\rho a_{\ast}>0\) is satisfied, one has \(\lim_{t\rightarrow \infty}V(t)=0\). The conclusion of Theorem 3.1 holds. This completes the proof. □
Theorem 3.2
Proof
The proof is omitted here, as it is similar to that of Theorem 3.1. □
4 Numerical examples and simulation
In this section, two numerical examples will be provided to verify and demonstrate the effectiveness of the proposed method.
Example 1
Example 2
5 Conclusions
In this paper, we investigated the problems of CLGOS in community networks via pinning control with two different switch periods. Using Lyapunov stability theory, linear matrix inequality (LMI), sufficient CLGOS criteria for community networks are derived. Both community networks with identical nodes and nonidentical nodes are investigated. Therefore, our proposed control schemes are better applicable technically. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed method.
Declarations
Acknowledgements
The authors greatly appreciate the reviewers suggestions and the editors encouragement. The work is partially supported by the Sichuan Science and Technology Plan (2017GZ0165).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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