- Research
- Open Access
Adaptive pinning impulsive synchronization of dynamical networks with time-varying delay
- Xiaoli Gong^{1} and
- Zhaoyan Wu^{1}Email author
https://doi.org/10.1186/s13662-015-0576-6
© Gong and Wu 2015
- Received: 4 May 2015
- Accepted: 14 July 2015
- Published: 6 August 2015
Abstract
In this paper, synchronization in dynamical networks with time-varying delay is investigated. Networks with time-varying coupling delay and node delay are both studied. By introducing adaptive strategy into pinning impulsive scheme, some effective and universal controllers are designed. In the proposed control schemes, for any given networks, the impulsive gains can adjust themselves to proper values when the impulsive intervals and some parameters are fixed. On the other hand, the impulsive instants can be estimated by solving a sequence of maximum value problems when the impulsive gains and some parameters are fixed. Based on the Lyapunov function method and mathematical analysis technique, several synchronization criteria are derived. Finally, numerical examples are performed to verify the effectiveness of the theoretical results.
Keywords
- synchronization
- dynamical network
- pinning impulsive control
- time-varying delay
1 Introduction
Dynamical networks consisting of nodes and edges are widely used to model the large-scale real systems coupled with interactive individuals [1–3]. The nodes denote the individuals and the edges denote the interactions between a pair of interactive individuals. Synchronization, as a typical collective dynamical behavior of dynamical network, has drawn more and more attention from different fields [4–22]. Due to the complexity of dynamical networks, especially those dynamical networks coupling with chaotic systems, they cannot achieve synchronization themselves without external control. Therefore, how to design effective and low-cost controllers for achieving synchronization becomes an important and challenging issue.
In the impulsive control scheme, the controllers are added onto nodes only at some discrete instants, i.e., it is low-cost and easier to implement. Thus, an impulsive control scheme has been widely used to design controllers for achieving synchronization [9–19]. In [9], Sun et al. studied the synchronization of impulsively coupled complex networks. In [10], Yang et al. considered the exponential synchronization of uncertain delayed complex networks with nonidentical nodes and stochastic perturbations via hybrid adaptive and impulsive control. In [12], Deng et al. investigated the cluster synchronization of community network via impulsive control. For any given dynamical network, one can choose proper impulsive gains and intervals such that the goal is realized.
As we know, many dynamical networks contain large number of nodes, which means that control of all nodes is high-cost and difficult to implement. Therefore, pinning control scheme, in which only a small fraction of nodes are controlled, has been widely adopted to design proper controllers combining with other control schemes [15–22]. Especially, in Refs. [17–19], stabilization and synchronization of dynamical networks are investigated by combining pinning and impulsive control, and some sufficient conditions are provided. From the sufficient conditions, for any given dynamical networks, one can easily estimate the impulsive gains and intervals for achieving the goals. However, different dynamical networks may have totally different system parameters and the number of controlled nodes may also be different, i.e., the pinning impulsive controllers with fixed impulsive gains and intervals are not universal. In Refs. [13–16], by introducing an adaptive strategy into (pinning) impulsive scheme, some adaptive (pinning) impulsive controllers are designed, which are universal to some extent. In Ref. [15], pinning impulsive synchronization of dynamical network without delay is investigated. However, time delays, including coupling delay and node delay, usually exist in many real networks. For example, the delays are usually time-varying in electronic implementation of analog networks due to the finite switching speed of amplifiers [17]. Therefore, pinning impulsive synchronization of dynamical network with delay, including node delay and coupling delay, deserves further studies.
Motivated by the above discussions, this paper investigates the synchronization of dynamical networks with time-varying delay via adaptive pinning impulsive control. Firstly, the dynamical network with time-varying coupling delay is considered and the corresponding controllers are designed. Secondly, the dynamical network with time-varying node delay is considered. According to the Lyapunov function method and mathematical analysis technique, the results are analytically proved. Compared with the obtained results in Refs. [15, 17–19], the main contributions of this paper are as follows: (1) effective and adaptive pinning impulsive controllers are designed for achieving synchronization of dynamical networks with time-varying delay, (2) the adaptive algorithms for not only the impulsive instants but also the impulsive gains are provided. That is, the obtained results extend those results obtained in Refs. [15, 17–19], to some extent.
The rest of this paper is organized as follows. In Section 2, the network models are introduced and some preliminaries are given. In Section 3, the adaptive pinning impulsive controllers for achieving synchronization are designed and the sufficient conditions are provided. In Section 4, several numerical simulations are performed to verify the results. In Section 5, the conclusion for this paper is given.
2 Model description and preliminaries
Networks (1) and (2) are said to achieve synchronization if \(\lim_{t\rightarrow\infty}\|x_{i}(t)-s_{1}(t)\|=0\) and \(\lim_{t\rightarrow\infty}\|x_{i}(t)-s_{2}(t)\|=0\), where \(s_{1}(t)\) and \(s_{2}(t)\) satisfy \(\dot{s}_{1}(t)=f(s_{1}(t))\) and \(\dot{s}_{2}(t)=g(s_{2}(t-\tau(t)))\), respectively.
Assumption 1
Assumption 2
Assumption 3
Suppose that the time-varying coupling delay \(\tau(t)\) is differentiable and there exists a constant \(\mu<1\) such that \(\dot{\tau}(t)\leq\mu\).
3 Main results
Let \(e^{(m)}(t)= ((e_{1}^{(m)}(t))^{T}, (e_{2}^{(m)}(t))^{T},\ldots, (e_{N}^{(m)}(t))^{T} )^{T}\) (\(m=1,2\)), \(\tau_{k}=t_{k}-t_{k-1}\) be the impulsive intervals, I be an identity matrix with appropriate dimension, \(\lambda_{1}\) and \(\lambda_{2}\) be the largest eigenvalues of \(2L_{1}I+c(A\otimes H)^{T}(A\otimes H)+cI/(1-\mu)\) and \(2L_{2}I+2c(A\otimes H)+L_{3}I/(1-\mu)\), \(\beta(t_{k})=(1+b(t_{k}))^{2}\), \(\rho (t_{k})=1-p(1-\beta(t_{k}))/N\) for \(t=t_{k}\) and \(\rho(t)=1\) for \(t\neq t_{k}\).
Theorem 1
Proof
Theorem 2
Proof
Remark 1
Remark 2
From conditions (7) and (8), if p, N, \(b(t_{k})\) and α are fixed, one can estimate the control instants \(t_{k}\) through finding the maximum value of \(t_{k}\) subject to \(t_{k}< t_{k-1}- (\ln\rho(t_{k})+\alpha )\widehat {L}^{-1}(t_{k})/2\) with \(t_{0}=0\), \(k=1,2,\ldots \) .
Remark 3
Compared with the results in Ref. [15], this paper considers not only the dynamical network with time-varying coupling delay but also the one with time-varying node delay. Besides the adaptive algorithm for solving the impulsive instants, the algorithm for determining the impulsive gains is also given. Compared with the results in Refs. [17–19], this paper designs effective and adaptive controllers through introducing a proper adaptive strategy. From the derived conditions and proofs in Theorems 1 and 2, the largest eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) need not be calculated. That is, the obtained results in Refs. [15, 17–19] are extended to some extent.
4 Numerical illustrations
Example 1
Example 2
5 Conclusion
In this paper, the synchronization of dynamical networks with time-varying delay is well studied via adaptive pinning impulsive control. Dynamical networks with both time-varying coupling delay and node delay are considered. Based on the Lyapunov function method and mathematical analysis technique, several sufficient conditions for achieving synchronization are derived. According to the discussions in Remarks 1 and 2, the impulsive gains or instants can adjust themselves to proper values or be estimated by solving a sequence of maximum value problems. Noticeably, some constants with respect to the node dynamics and topology of network need not be calculated beforehand. Finally, the obtained results are verified to be correct and effective by performing several numerical simulations.
Declarations
Acknowledgements
This work is jointly supported by the National Natural Science Foundation of China under Grant No. 61463022 and the Natural Science Foundation of Jiangxi Educational Committee under Grant No. GJJ14273.
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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