Global exponential synchronization of interval neural networks with mixed delays via delayed impulsive control☆
Introduction
During the last twenty years, there has been a growing attention to the dynamics of neural networks including stability, synchronization and periodic attractors due to the widespread applications of signal processing, nonlinear optimization and biomedical engineering [1], [2], [3], [4], [5], [6], [7]. As a collective action of neural networks, synchronization has been widely investigated in various fields, such as image processing, safety communication, informatics and others [8], [9], [10], [11]. In these applications, due to network congestion or restricted change speed of amplifiers, there will inevitably appear some time delays, which may lead to undesirable dynamical behaviors and seriously affect the synchronization of systems [12], [13], [14], [15]. Consequently, the synchronization of neural networks with time delays has become an active and flouring research topic and some significative results have been reported in recent years [16], [17], [18], [19].
On the other hand, some key data of neural networks, such as neuron discharge rate, synaptic connection weight and image transmission delay, are usually measured and obtained via statistical estimation. Therefore, modeling errors and parameter deviations are inevitable in the implementation of the neural networks, which induces uncertainties in modeling neural networks, and even results in poor performance and asynchronization. Hence, it is very significant to study interval neural networks whose parameters values are unknown but bounded in given compact sets. Some results have been made on synchronization and stability of interval neural networks in recent years, see [20], [21], [22], [23]. However, because of the complex topology of the neural networks, appropriate external controllers are needed to realize synchronization, such as pinning control [24], state feedback control [25], impulsive control [26] and sampled control [27].
Among the above control methods, impulsive control has received much attention as it permits the system to possess control action just at some discrete cases, which can effectively reduce the amount of transmitted information and control cost. Such a simple controller structure makes it easier to implement in real applications [28], [29], [30], [31]. Actually, the processes of collecting and transferring impulse information are hard to achieve instantaneously due to the limited speed of signal transmission. It is necessary to introduce time delays into controllers. Hence, delayed impulsive control which depends on the current states of the system as well as its previous states, is supposed to be a better way of simulating many practical problems, such as financial systems, ecosystems and communication security systems [32], [33], [34], [35], [36], [37]. In recent years, it has attracted increasing attention. In [34], under impulsive control involving time delays, several sufficient criteria have been derived for the quasi-synchronization of neural networks with parameter mismatches. In [35], authors have obtained several synchronization criteria for nonlinear Lur’e systems with time delays under delay-dependent impulsive control. Zhang et al. [36] have studied exponential synchronization for complex-valued complex networks by delayed impulsive control. Leader-following synchronization of delayed coupled neural networks has been studied by a proper delayed impulsive controller in [37].
As we know, there normally have spatial properties on neural networks, which may rely on the state information of the entire past. Introducing unbounded distributed delays to model them is very necessary. Moreover, model error and parameter fluctuation are also inevitable in modeling of neural networks. However, most of the existing results, such as those in [34], [35], [36], [37], are only valid for neural networks with bounded time-varying delays and known parameters. There are few results concerning cases with mixed delays and uncertain parameters via delayed impulsive control, which motivates our research. Moreover, it should be noted that impulsive differential inequalities are important tools of studying impulsive control problems. Mixed delays including both time-varying and unbounded distributed delays are the most general case and more difficult to handle. When delayed impulses are involved, the existing impulsive differential inequalities cannot be applied to deal with such mixed delays. Therefore, it needs to explore and develop some new impulsive differential inequalities subjecting to delayed impulses, which is a very interesting and challenging work.
This paper aims to investigate the synchronization problem of interval neural networks with both time-varying and unbounded distributed delays by delayed impulsive control. The main contributions of this paper can be highlighted as follows: the model of neural networks under discussion is quite comprehensive, which takes uncertain parameters and mixed time-delays into simultaneous consideration; a novel impulsive differential inequality is developed to tackle the mathematical difficulty resulting from the presence of the mixed delays and delayed impulses, which plays a crucial role in the establishment of the main results; by designing a delayed impulsive controller, some new sufficient conditions are derived for neural networks to achieve the synchronization in the presence of mixed time-delays and uncertain parameters.
The remainder of this paper is organized as follows. Section 2 introduces several necessary preliminary knowledge and proves a new impulsive differential inequality. In Section 3, several synchronization criteria are proposed. A numerical result is given to demonstrate the validity of our methods in Section 4. Section 5 describes the conclusions of this paper.
Section snippets
Preliminaries
Notations. Let denote the set of real numbers. represents the n-dimensional real space which has the Euclidean norm , represents the set of positive integers and . () means that matrix A is a symmetric negative (positive) semi-definite matrix. () denotes the maximum (minimum) eigenvalue of A. The notation and denote the transpose and the inverse of A, respectively. For any set , interval is continuous everywhere
Main results
Theorem 1 Suppose that Assumptions 1–3 hold. If there exist some constants , an symmetric matrix diagonal matrices , four diagonal matrices , and matrices such that the following conditions hold:where
Example
In this section, a numerical example with simulation results is provided to show the validity of the conclusions.
Example. Consider the parameters of 2D interval network model (1) as follows:For simplicity, the initial condition is given as . Then the delay kernels , and
Conclusion
In this paper, the global exponential synchronization of interval neural networks with mixed delays has been investigated by delayed impulsive control. A new impulsive differential inequality considering mixed delays and delayed impulses has been presented. Based on the developed inequality, several synchronization criteria have been derived to ensure the synchronization of the systems (1) and (2). Finally, a numerical result has been given to demonstrate the validity of our methods. As we
CRediT authorship contribution statement
Yao Wang: Data curation, Writing - original draft. Yujuan Tian: Methodology, Writing - original draft, Writing - review & editing. Xiaodi Li: Methodology, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Yao Wang was born in Shandong Province, China, in 1996. She is currently a graduate student with control theory, School of Mathematics and Statistics, Shandong Normal University, Jinan, China. Her research interests include neural networks, stability and impulsive control theory.
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Yao Wang was born in Shandong Province, China, in 1996. She is currently a graduate student with control theory, School of Mathematics and Statistics, Shandong Normal University, Jinan, China. Her research interests include neural networks, stability and impulsive control theory.
Yujuan Tian received the B.S degree from Ludong University, Yantai, China, in 2004, and the Ph.D. degree from Dalian University of Technology, Dalian, China, in 2011, all in applied mathematics. She is currently an Associate Professor with the School of Mathematics and Statistics, Shandong Normal University. From May 2019 to Aug. 2019, she was a Visiting Research Fellow at Department of Mathematics and Statistics in University of North Florida, USA. She has authored or coauthored more than 10 research papers. Her research interests include impulsive control theory, time-delay systems, and switched systems.
Xiaodi Li received the B.S. and M.S. degrees from Shandong Normal University, Jinan, China, in 2005 and 2008, respectively, and the Ph.D. degree from Xiamen University, Xiamen, China, in 2011, all in applied mathematics. He is currently a Professor with the School of Mathematics and Statistics, Shandong Normal University. From Nov. 2014 to Dec. 2017, he was a Visiting Research Fellow at Laboratory for Industrial and Applied Mathematics in York University, Canada, and the University of Texas at Dallas, USA. In 2017, he was working as Visiting Research Fellow at the Department of Mathematics, City University of Hong Kong, Hong Kong. He has authored or coauthored more than 70 research papers. His current research interests include stability theory, delay systems, impulsive control theory, artificial neural networks, and applied mathematics.
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This work was supported by National Natural Science Foundation of China (11501333, 61673247), and the Research Fund for Distinguished Young Scholars and Excellent Young Scholars of Shandong Province (JQ201719). The paper has not been presented at any conference.