Synchronization analysis of singular dynamical networks with unbounded time-delays
- Song Liu^{1}Email author,
- Xiaoyan Li^{1},
- Xian-Feng Zhou^{1} and
- Wei Jiang^{1}
https://doi.org/10.1186/s13662-015-0529-0
© Liu et al. 2015
Received: 25 January 2015
Accepted: 5 June 2015
Published: 24 June 2015
Abstract
This paper deals with singular dynamical networks with non-delay coupling and unbounded time-delay coupling simultaneously, where the coupling configuration matrices are symmetric with zero row sums and nonnegative off-diagonal entries. A sufficient condition of synchronization is derived based on the Lyapunov-Krasovskii functional method and matrix analysis technique. A numerical example shows that our proposed method is simple and convenient in computation.
Keywords
MSC
1 Introduction
In general, complex networks consist of a large number of nodes, in which every node is a fundamental cell with specific activity. In the past two decades, complex networks have attracted scholars’ increasing attention. Several famous network models, such as scale-free model [1], small-world model [2, 3], which accurately characterize some important natural structures, have been presented. Synchronization is a universal phenomenon in various fields of science and society, and many significant works have been obtained in [4–8]; Wu et al. investigate synchronization of an array of linearly coupled identical systems in [9–11]; research in [12, 13] shows that the structure of networks must have an inevitable effect on the ability and speed of synchronization. It is well known that time-delays widely exist in a large number of concrete systems, and coupled dynamical networks are often associated with time-delays due to the finite speeds of transmission and spreading as well as traffic congestion. A lot of efforts have been made to study the synchronization of dynamical coupled systems with time-delays in [14–18].
At the same time, we notice that a large number of practical networks, such as economic networks, power networks and so on, are singular differential systems which are also named differential-algebraic systems or descriptor systems. Singular systems have some particular complex properties which need not be considered in normal systems. In singular systems, impulse behavior may appear (if the index is greater than one) and initial value problem may also be unsolvable or have more than one solution, regularity is closely related to the solution behavior of the corresponding singular systems [19]. In order to make a singular system solvable with no impulse behavior and unique solution, the system must be regular and the initial condition must be compatible, which can be acquired similarly to the method presented in [20]. Due to the effect of time-delays, coupled correlative terms will inevitably appear if the system is divided into two subsystems including a differential subsystem and an algebraic subsystem, which makes the problems become more complicated. Recently, Xiong et al. [21] have investigated synchronization of singular hybrid coupled networks without time-delays, the original system is divided into two subsystems including a differential subsystem and an algebraic subsystem, the authors present a sufficient condition of global synchronization, but the presented method cannot be applied to singular delayed networks. Koo et al. [22] and Li et al. [23] investigate synchronization of singular complex dynamical networks with time-varying bounded delays. Sufficient conditions for synchronization in terms of LMIs (linear matrix inequalities) are obtained, respectively. Motivated by this research, in this paper we study synchronization problem of singular dynamical networks with non-delay coupling and unbounded time-delay coupling simultaneously. Based on the Lyapunov-Krasovskii functional method, a simple sufficient condition of synchronization is derived by using matrix analysis and matrix inequality technique. Our presented method can also be applied to more general dynamical networks including the networks presented in [21–23]. Finally, a numerical example shows that our presented method is simple and convenient in computation.
Notation: The notation used throughout this paper is fairly standard. \(R^{n}\) denotes an n-dimensional Euclidean space, \(R^{n\times n}\) is the set of all \(n\times n\) real matrices, \(I_{n}\) stands for the identity matrix of order n, \(U^{T}\) means the transpose of a real matrix or vector U, \(\|x\|\) denotes the Euclidean norm of a real vector x. For a real matrix A, \(\lambda_{\mathrm{max}}(A)\) and \(\lambda_{\mathrm{min}}(A)\) denote the maximal and minimal eigenvalue respectively. \(\|A\|=\sqrt{\lambda_{\mathrm{max}}(A^{T}A)}\) denotes the spectral norm of matrix A, \(A>B\) (or \(A\ge B\)) means the symmetric matrix \(A-B\) is positive definite (or positive semi-definite) and \(A\otimes B\) denotes the Kronecker product between matrix A and B.
2 Preliminaries
To obtain our main results, the following lemmas will be used later.
Lemma 2.1
([20])
For any vectors \(x,y\in R^{n}\) and \(\varepsilon> 0\), the inequality \(2x^{T}y\leq\varepsilon x^{T}x+\frac{1}{\varepsilon} y^{T}y\) holds.
Lemma 2.2
([24])
3 Main results
In this section, the main results of this paper on asymptotical synchronization of singular delayed network (1) are derived. We first introduce the following definition.
Definition 3.1
([21])
Remark 3.1
In [25], the authors presented a sufficient condition on the existence and uniqueness of solution of the system \(E\dot{x}(t)=Ax(t)+f(x(t),t)\).
Theorem 3.1
Proof
Obviously, the dynamical network (2) will achieve asymptotical synchronization if the error system (6) is asymptotically stable about the zero solution.
The proof is completed. □
In order to make singular system (1) or (4) solvable with no impulse, we suppose that the following assumption holds.
Assumption 1
Remark 3.2
The initial function space of differential systems with unbounded delays is not completed. Let \(\mathit{BC}:=\{\phi| \phi:(-\infty ,0] \rightarrow R^{n} , \phi \mbox{ is bounded and continuous}\}\), then \((\mathit{BC},\|\cdot\|)\) is a Banach space. We also may define a new complete initial function space.
Let \(x: [a,b]\rightarrow R^{n}\), with the norm \(\|x\|:=\sup_{s\in[a,b]}|x(s)|\). Denote \(C := \{\phi| \phi: (-\infty,0] \rightarrow R^{n} , \phi\mbox{ is continuous}\}\), then C and BC are both linear spaces.
Let \(h\in C\), \(h\geq0\), and \(0<\int^{0}_{-\infty}h(s)\, ds<\infty\). Denote \(C^{n}_{h}:= \{\phi| \int^{0}_{-\infty}h(s)\|\phi\|\, ds< \infty, \phi\in C \}\), the norm of \(\phi\in C^{n}_{h}\) is defined as \(\|\phi\|_{h}:= \int^{0}_{-\infty}h(s)\|\phi\|\, ds\), hence \(C^{n}_{h}\) is a linear subspace of C and \(\mathit{BC} \sqsubset C^{n}_{h}\).
Under condition (11), it follows from [27] that the pair \((E, A+J+c_{1}\lambda_{i} \Gamma)\) is regular and impulse free, hence the solution of Eq. (4) exists and is impulse free and unique on \([t_{0},\infty)\) for any admissible initial condition \(\phi \in C^{n}_{h} \).
Theorem 3.2
Suppose that matrix \(B\hat{B}\) is symmetric and Assumption 1 holds, then the singular networks with unbounded coupled delays (4) will asymptotically synchronize in the sense of (3).
Proof
From (14) and (15), we know \(\dot{V}_{i}(t)\) (\(i=1,2,\ldots,N\)) are negative definite. Therefore Eq. (4) is asymptotically stable about zero solution via the Lyapunov stability theory, then the singular delayed network (1) will achieve asymptotical synchronization. □
Remark 3.3
- (1)If \(b_{ij}=0\) (i.e., \(B=0\)) and \(\tau_{i}(t)=\tau (t)\) is bounded, Eq. (1) is reduced towhich is the network model presented in [22, 23].$$ E\dot{x}_{i}(t)=Ax_{i}(t)+f\bigl(x_{i}(t),t \bigr) +c_{2}\sum_{j=1}^{N} \hat{b}_{ij}\hat{\Gamma} x_{j}\bigl(t-\tau(t)\bigr),\quad i=1,2,\ldots ,N, $$(16)
- (2)
Hence, the model investigated in this paper may characterize many natural dynamical networks and our proposed method can also be applied to more general dynamical networks.
Remark 3.4
In reality, it is difficult to compute matrices \(P_{i}\) (\(i =1, 2, \ldots , N\)) and \(Q_{i}\) (\(i=2, 3,\ldots , N\)) for a general complex model with a large number N of nodes or with a large dimension from conditions (10) and (11). It should be pointed out that (10) and (11) cannot be solved directly by the LMI toolbox of Matlab. However, if matrix E is positive semi-definite and matrix A is negative definite, one can easily choose positive definite matrices \(P_{i}\) and \(Q_{i}\) satisfying conditions (10)-(12) (see the following numerical example). Comparing with [21–23], our proposed method is simple and convenient in computation.
4 An illustrative example
5 Conclusions
This paper investigates singular complex networks with non-delay coupling and unbounded time-delay coupling simultaneously. Based on the Lyapunov stability theory and matrix inequalities and singular system theory, a simple sufficient condition of synchronization is derived, which can be easily realized and is simple and convenient in computation. The proposed method also can be applied to more general complex networks comparing with [21–23]. Finally, a simple example is given to illustrate the effectiveness of our theoretical results.
Declarations
Acknowledgements
This research is supported by the National Natural Science Foundation of China (11371027, 11326115, 11471015), Research Fund for the Doctoral Program of Higher Education of China (20133401120013), Program of College Natural Science of Anhui Province (KJ2013A032, KJ2011A020), Doctoral Starting Fund of Anhui University (023033190181), Young Scientist Fund of Anhui University (023033050055), Young Outstanding Teacher Fund of Anhui University (023033010264).
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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