- Research
- Open Access
Finite-time anti-synchronization of time-varying delayed neural networks via feedback control with intermittent adjustment
- Xin Sui^{1},
- Yongqing Yang^{2}Email author,
- Fei Wang^{1} and
- Lingzhong Zhang^{1}
https://doi.org/10.1186/s13662-017-1264-5
© The Author(s) 2017
Received: 18 November 2016
Accepted: 4 July 2017
Published: 7 August 2017
Abstract
This paper investigates the finite-time anti-synchronization of time-varying delayed neural networks. A simple intermittent adjustment feedback controller is designed to ensure the drive-response systems realize anti-synchronization in a finite time. By employing some differential inequalities and finite-time stability theory, some novel and effective finite-time anti-synchronization criteria are derived based on the Lyapunov functional method. This paper extends some traditional anti-synchronization criteria by using intermittent adjustment feedback control. Finally, two numerical examples are given to show the effectiveness of the proposed method.
Keywords
- finite-time anti-synchronization
- intermittent adjustment feedback control
- Lyapunov functional
1 Introduction
The nonlinear systems have been widely studied in the real world [1–7]. During the past 30 years, neural networks have become one of the most important nonlinear systems and have been intensively applied to solve various optimization problems [8]. On the other hand, time-delay may often occur in many practical systems such as neural networks [9–12]. At the same time, it can lead to many complex dynamic behaviors such as divergence, oscillation and instability of the neural networks [13]. It is very important that the stability analysis for delayed neural networks between theoretical and practical systems.
In the past decades, synchronization and anti-synchronization of neural networks have caused wide attention because of its potential application in various fields of science and humanity worldwide, such as image encryption [14], physical systems [15], secure communication [16] and laser technology [17]. Meanwhile, many control methods and techniques have been proposed to synchronize the behavior of neural networks. These control methods are divided into adaptive control [18], nonlinear control [19], active control [20], etc. Most of these methods are based on continuous control and discontinuous control strategies are rarely studied. The intermittent control [21] is a kind of discontinuous control methods and control of the system in a discontinuous time period. It is a more effective and economical approach than continuous control. So far, it is widely applied to solve the synchronization problem of neural networks [22–24].
We say that two systems achieve anti-synchronization when they have the same amplitude but opposite signs. Then the sum of two signals will converge to zero when the anti-synchronization phenomenon occurs. Therefore, the authors have studied the anti-synchronization phenomenon on neural networks and some relevant theoretical results have been established [25–27]. In the current research of synchronization theory, most authors studying the anti-synchronization of neural networks have based themselves on the convergence time being large enough. This indicates that the control can drive the slave system to anti-synchronize the master system after the infinite horizon. The control of infinite time is not difficult to realize in the actual system, but it takes a higher cost. We hope to control the cost as much as possible and to reach stability of the system as quickly as possible. In order to realize the stability and synchronization of the system quickly, one can use finite-time techniques to get a faster convergent speed. Finite-time synchronization of complex networks has been investigated in [28, 29]. Unfortunately, there are few papers considering the finite-time anti-synchronization of neural networks via feedback control with intermittent adjustment.
Motivated by the above discussion, this paper will investigate the finite-time anti-synchronization of time-varying delayed neural networks. Furthermore, by applying an inequality technique and the Lyapunov stability theorem, the finite-time anti-synchronization can be realized between the drive and response systems by designing an intermittent adjustment feedback controller. Lastly, two numerical examples are given to prove the correctness of the proposed method.
The remainder of this paper is organized as follows. In Section 2, the model formulation and some preliminaries are given. The main results will be obtained in Section 3. Two examples are given to show the effectiveness of our results in Section 4. Finally, conclusions are drawn in Section 5.
Notations: Throughout this paper, \(\mathbb{R}^{n}\) denotes n dimensional real numbers set, \(\mathbb{N}\) denotes natural numbers set. \(A_{m \times n}\) and \(I_{N}\) refer to \(m \times n\) matrix and \(N \times N\) identity matrix, respectively. The superscript T denotes vector transposition. \(\Vert \cdot \Vert \) is the Euclidean norm in \(\mathbb{R}^{n}\). If A is a matrix, \(\Vert A \Vert \) denotes its operator norm. If not explicitly stated, matrices are assumed to have compatible dimensions.
2 Model description and preliminaries
In this section, a neural network model and definition will be introduced. Furthermore, some useful lemmas will also be given, which will be used later.
Through this paper, in order to obtain the anti-synchronization results, we give the following assumption, definition and some useful lemmas.
Assumption 1
[30]
Definition 1
[23]
Lemma 1
[31]
Lemma 2
[32]
Lemma 3
[33]
3 Main results
Remark 1
When \(HT \le t < HT + \theta T\), \(H \in\mathbb{N}\). If \(0 < \alpha< 1\), the controller \(u(t)\) is a continuous function with respect t. If \(\alpha= 0\), \(u(t)\) turns to be discontinuous one, which is similar to the controller that have been considered in [34]. If \(\alpha= 1\), the controller will become typical feedback control issues, which only can realize an asymptotical anti-synchronization in an infinite time.
Theorem 1
Proof
Remark 2
The settling time of anti-synchronization can be estimated in a finite time. The sufficient conditions given in Theorem 1 can avoid the problem that the neural network only realizes anti-synchronization when time tends to infinity efficiently, and this has significant meanings in real engineering applications of network synchronization.
Remark 3
Corollary 1
4 Numerical simulations
In this section, two numerical examples are given to show the effectiveness of Theorem 1.
Example 1
Example 2
Remark 4
Remark 5
In this paper, our theorem conditions and the settling time can simply the calculation through Matlab to get the desired results. As regards references [19] and [30], our research is based on them, which can make the system stable in a shorter time. Therefore, it has higher research value in the actual system.
5 Conclusion
In this paper, we investigate the finite-time anti-synchronization by using Lyapunov functional method. A simple intermittent adjustment feedback controller is designed to control the states of two systems to achieve anti-synchronization within a finite time. Some sufficient conditions are put forward for the anti-synchronization of drive-response systems, it plays an important role in practical application. The main contribution of this paper is that system (1) and (3) can realize anti-synchronization in a finite time.
Furthermore, two numerical simulation examples are provided to verify the rightness of the proposed anti-synchronization criteria. It is important to note that the intermittent adjustment feedback control can reduce the control time and cost rather than the continuous control. Therefore, our method has very extensive application in transportation, communications and other areas. Finally, it is still a big challenge to investigate the finite-time anti-synchronization of neural networks with discontinuous dynamic behaviors; these problems will be considered in the next papers.
Declarations
Acknowledgements
This work was jointly supported by the Natural Science Foundation of Jiangsu Province of China under Grant No. BK20161126, the Graduate Innovation Project of Jiangsu Province under Grant No. KYLX16_0778, and the Fundamental Research Funds for the Central Universities JUSRP51317B.
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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