- Research
- Open Access
Impulsive synchronization of drive-response chaotic delayed neural networks
- Zhaoyan Wu^{1}Email author and
- Hui Leng^{1}
https://doi.org/10.1186/s13662-016-0928-x
© Wu and Leng 2016
- Received: 26 April 2016
- Accepted: 28 July 2016
- Published: 11 August 2016
Abstract
In this paper, we investigate synchronization of drive-response neural networks with time-varying delays via impulsive control. Based on impulsive stability theory, we design proper impulsive controllers and derive some sufficient conditions for achieving synchronization. Noticeably, we adopt adaptive strategy to design unified controllers for different neural networks and relax the restrictions on the impulsive interval. All the obtained results are verified by several numerical examples.
Keywords
- synchronization
- delayed neural network
- impulsive control
- chaotic
1 Introduction
Neural networks, including Hopfield neural networks and cellular neural networks, have been widely investigated in past decades [1–23]. Synchronization, as a typical collective dynamical behavior of neural networks, has attracted more and more attention in various fields. For achieving the synchronization of neural networks, especially of chaotic neural networks, many control methods and techniques have been adopted to design proper and effective controllers, such as feedback control, intermittent control, adaptive control, impulsive control, and so on.
In real world, because of switching phenomenon or sudden noise, many real systems have been found to be subject to instantaneous perturbations and abrupt changes at certain instants. That is, these systems cannot be controlled by continuous control and endure continuous disturbance. Therefore, impulsive control, as a typical discontinuous control scheme, has been widely adopted to design proper controllers for achieving synchronization or stability [1–9]. Based on the Lyapunov function method, the Razumikhin technique, or the comparison principle, many valuable results have been obtained, and synchronization criteria have been derived. For a given neural network, we can estimate the largest impulsive interval from the derived criteria by fixing impulsive gains and calculating some system constants, for example, Lipschitz or Lipschitz-like constants with respect to neuron activation functions, and vice versa. As we know, different neural networks usually have totally different system parameters and activation functions, which means that the impulsive controllers with fixed impulsive gains and intervals are not unified. In other words, the system parameters have more restrictions on the choice of impulsive gains and intervals. For relaxing the restrictions, adaptive strategy is introduced to design adaptive impulsive controllers. The Lipschitz (or Lipschitz-like) and other constants with respect to system parameters and activation functions need not be known beforehand and can be calculated according to the proposed adaptive strategy [24–26].
On the other hand, due to the transmission speed of signals or information between neurons is finite, neural networks with coupling delay should be considered. Motivated by the above discussions, in this paper, we investigate the impulsive synchronization of drive-response chaotic delayed neural networks. Firstly, we give some sufficient conditions for achieving synchronization, from which we can easily estimate the largest impulsive intervals for given neural networks and impulsive gains. Secondly, we adopt adaptive strategy to design adaptive impulsive controllers for relaxing the restrictions. Noticeably, the designed controllers are universal for different neural networks. Finally, we perform some numerical examples to verify the obtained results.
The rest of this paper is organized as follows. In Section 2, we introduce the model and some preliminaries. In Section 3, we study the impulsive synchronization of drive-response chaotic delayed neural networks. In Section 4, we provide several numerical simulations to verify the effectiveness of the theoretical results. Section 5 concludes this paper.
2 Model and preliminaries
The drive-response networks (2) and (3) are said to achieve synchronization if \(\lim_{t\to\infty}\|y(t)-x(t)\|=0\).
Assumption 1
Assumption 2
The time-varying delay \(\tau(t)\) is differentiable and satisfies \(\dot{\tau}(t)\leq\mu<1\).
Lemma 1
[27]
3 Main result
In what follows, let \(d_{k}=t_{k}-t_{k-1}\), λ be the largest eigenvalue of matrix \(-C+(1-\mu)^{-1}I_{n}+AA^{T}+L_{f}^{2}I_{n}+L_{g}^{2}BB^{T}\), \(\beta(t_{k})=(1+b(t_{k}))^{2}\), and \(\beta(t)=1\) for \(t\neq t_{k}\).
Theorem 1
Proof
Remark 1
For any given neural network (2), the positive constants \(L_{f}\) and \(L_{g}\) in Assumption 1 and the largest eigenvalue λ can be estimated by simple calculations. Thus, if the constant α and the impulsive gain \(b(t_{k})\) are fixed, then from conditions (6) the impulsive intervals \(d_{k}\) can be estimated. However, the neuron activation functions and coefficient matrices are usually nonidentical for different neural networks, that is, the proposed impulsive controllers with fixed impulsive intervals are not universal. In the following, adaptive strategy is adopted to design universal impulsive controllers.
Theorem 2
Proof
Remark 2
From conditions (9) in Theorem 2 it is clear that some constants with respect to the neuron activation functions and coefficient matrices need not be known beforehand. For any given neural network, the constants can be estimated by \(\hat{L}(t)\) with proper adaptive gain δ. When impulsive instants or intervals are fixed, we can give the updating law of impulsive gain with time from conditions (9). When impulsive gains are fixed, we can give a method for estimating the impulsive instants as well. Detailed methods are provided in the following remarks. That is, the proposed adaptive impulsive control scheme is universal for those neural networks, provided that their activation functions satisfy Assumption 1.
Remark 3
Remark 4
By conditions (9), for any given \(b(t_{k})\) and α, we can estimate the control instants \(t_{k}\) through finding the maximum value of \(t_{k}\) subject to \(t_{k}< t_{k-1}-(\ln\beta(t_{k})+\alpha)\hat{L}^{-1}(t_{k})\) with \(t_{0}=0\), \(k=1,2,\ldots\) .
4 Numerical simulations
Example 1
Example 2
Consider the synchronization of the same neural network via the adaptive impulsive control scheme proposed in Theorem 2. In numerical simulations, choose \(\delta=1\), the initial value of \(\hat{L}(t)\) as \(\hat{L}(0)=1\), and the other parameters as in the previous example.
Example 3
5 Conclusions
In this paper, the synchronization problem of drive-response chaotic delayed neural networks has been investigated via impulsive control scheme. Firstly, some sufficient conditions for achieving synchronization were provided according to the Lyapunov function method and impulsive stability theory. For given neural networks, the largest impulsive interval can be estimated by fixing impulsive gains, and vice versa. Secondly, an adaptive strategy, combined with impulsive control scheme, was used to design universal controllers for different neural networks and relax the restrictions on impulsive intervals and gains. Finally, some numerical examples were performed to verify the correctness and effectiveness of the obtained results.
Declarations
Acknowledgements
This work is jointly supported by the NSFC under Grant No. 61463022 and the NSF 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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