Fixed-time synchronization control for a class of nonlinear coupled Cohen–Grossberg neural networks from synchronization dynamics viewpoint
Introduction
In recent years, close attention has been paid to CGNNs, which were proposed in 1983 by Cohen and Grossberg [1], because of their potential applications in parallel computation, image processing, classification, pattern recognition and so on [2], [3], [4], [5] Since such applications deeply rely on dynamical behaviours (including synchronization [6], [7], stability [8], [9], passivity [10], [11] and so on [12]), dynamic problems of CGNNs have been intensively studied by researchers. For example, in [6], by using inequality technology and Lyapunov stability method, Li et al. investigated global exponential and fixed-time synchronization of delayed CGNNs with reaction-diffusion terms under adaptive controllers. By using an adaptive control law, Hu et al. [7] made a class of complex-valued CGNNs with known and unknown parameters to realize adaptive exponential synchronization.
It should be noted that, as one of the most important dynamical behaviours, recently, synchronization problems of neural networks (NNs) have been a research hotspot [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]. For instance, in [13], Luo et al. derived some sufficient conditions based on Filippov’s sense to ensure globally exponential synchronization for a class of delayed memristor-based coupled NNs under aperiodically intermittent control. Lü et al. [14] considered an array of coupled delayed NNs with discontinuous or continuous activations and fixed-time synchronization. Linear feedback control and adaptive control strategies were respectively designed to realize asymptotical and adaptive synchronization for a class of CGNNs with proportional delays [23]. Wang et al. [24] investigated globally exponential and asymptotical synchronization for a class of mixed-delayed CGNNs with discontinuous activations under the framework of the Filippov solution and pinning control approach. It needs to be pointed out that until now, although many valuable and useful research achievements regarding the synchronization dynamics of neural networks have been obtained (e.g., single CG-NNs [5], [6], [7], [23], [24], [25], [26], [27], [28], coupled NNs [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]), only a few studies have considered synchronization dynamics for coupled CGNNs [29], [30], [31], [32]. In [29], by designing feedback controllers built on finite-time control theory, the finite-time synchronization of linear coupled CGNNs was studied. Based on fixed-time control theory and the Lyapunov functional method, Huang et al. [30] derived criteria for designing two types of fixed-time controllers so that the considered linear coupled CGNNs achieve fixed-time synchronization. In addition to these approaches, in some real physical systems, the nodes of the coupled network are entangled by nonlinear function schemes [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], such as the nonlinear interactions between different neuron elements in a brain dynamical network [35] and the nonlinear interactions between different electrical elements in an electrical gird dynamical network [36]. Because a linear coupling function is one special case of nonlinear coupling function [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], nonlinear couplings are more general than linear couplings. The above comments reveal that it is necessary and meaningful to further explore synchronization dynamic control problems related to NCCGNNs.
In fact, in many control systems (e.g., linear control systems [43], [44] and nonlinear control systems [45], [46], [47], [48]), it is not difficult to find that the control of a dynamical system is often required within a finite time interval [20], [25], [29], [44], [45]. Hence, recently, in the light of finite-time control theory, finite-time synchronization control problems of NNs have been quickly developed [49], [50], [51]. Through the existing works [20], [25], [29], [49], [50], [51], it can be easily observed that although in a finite time interval, finite-time control methods can make dynamical systems realize targets including synchronization [20], [25], [29], [49], [50], [51], [52], passivity [32], and so on [53], the derived finite-time estimation approaches are closely and heavily dependent on the initial conditions of dynamical systems. That is, if the initial conditions of dynamical systems cannot be obtained, finite-time control is essentially invalid. To overcome the limitation of finite-time control, in 2012, Polyakov proposed fixed-time control stabilization of dynamical systems [54]. Inspired by fixed-time control theory, over a few years, an increasing number of researchers have considered fixed-time synchronization control for NNs [6], [14], [22], [30], [55], [56], [57], [58]. Moreover, it is known that because of the finite transmission speed and the limited bandwidth, time delays can often occur in dynamical systems [14]. Therefore, some classes of delayed CGNNs were considered, and their synchronization issues were researched [5], [6], [7], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32]. Based on the above discussion, we conclude that it is valuable to investigate FTS of nonlinear coupled delayed CGNNs. To the best of our knowledge, until now, there have been no results regarding this research.
Motivated by the above analysis, in this paper, we address the FTS controller design problem of NCCGNNs with time-varying delay. A new fixed-time synchronization controller based on the derived criterion is designed to ensure FTS of the considered NCCGNNs with time-varying delay. Furthermore, by analysing the advantages and disadvantages of the obtained controller, which was built from a synchronization dynamics perspective, the design methods of several more feasible FTSCs are studied. In addition, the effectiveness of the theoretical results are further confirmed using two numerical examples.
The contributions of this paper are summarized as follows: (I) A new class of NCCGNNs with time-varying delay is considered. (II) Different from the existing synchronization controller design schemes such as [15], [21], [25], [29], [30], [31], [32], the proposed synchronization control methods of this paper can more effectively process the time-varying delay of the addressed system. (III) Compared with some fixed-time synchronization control approaches [6], [14], [22], [30], [55], [56], [57], [58], it should be noted that the design methods of the given fixed-time synchronization controllers in this paper are more reasonable. Comparisons of some of the above related works are shown in Tables 1–3 and remarks 1–4 and 6.
: Some necessary notations used throughout the article are introduced. The number N represents a positive integer. is the set of all n × m real matrices and denotes the n-dimensional Euclidean space. The superscript T denotes the matrix transposition. is an -dimensional identity matrix. If A is a matrix, λmax(A) and λmin(A) denote its maximal eigenvalue and its minimum eigenvalue, respectively. For matrix and vector ‖A‖2 and ‖e(t)‖1 are defined by and respectively. The Kronecker product of matrices and is which is a matrix. diag( · · · ) stands for a block-diagonal matrix, and big O notation is used to represent the computational complexity.
Section snippets
Model and preliminaries
First, we give a single CGNN with time-varying delay as follows:where ; n represents the number of neurons in the network; is the state variable of the ith neuron at time t; αi(xi(t)) > 0 and hi(xi(t)) are an amplification function and an appropriately behaved function, respectively; aij and bij correspond to the connection strengths of the jth neuron on the ith neuron; fj(xj(t)) and
Main results
In this section, the FTSCs of network (3) are designed to illustrate the reasonableness of the results in Remark 1.
A. Based on the fixed-time synchronization sufficient condition, the design of the fixed-time synchronization control rule for network (3). Theorem 1 Suppose that Assumptions 1–3 hold; then, ifnetwork (3) can achieve fixed-time synchronization under the following controllerwhere
Numerical examples
In this section, two examples are given to illustrate the main results. Let where e(t) stands for synchronization total error of network (3). In the next examples, we make . That is, the designed equilibrium state of network (3) is . According to network (2), we havewhere
Conclusions
This paper mainly emphasizes fixed-time synchronization control for a class of nonlinear coupled CGNNs with time-varying delay. By constructing a feasible Lyapunov–Krasovskii functional, we successfully derive a novel fixed-time synchronization criterion that can be used to design a fixed-time synchronization controller with time-varying delay. Moreover, based on synchronization dynamics, the considered network’s parameters and the obtained fixed-time synchronization criterion, several
Declaration of Competing Interests
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported in part by the National Natural Science Foundation of the People’s Republic of China under Grant No. 61603325 and in part by the Innovation Program of Shanghai Municipal Education Commission under Grant No. 13ZZ050.
Xin Wang received his M.S. degree in control theory and control engineering from Lanzhou University of Technology, Lanzhou, China, in 2006 and his Ph.D. degree in control theory and control engineering from Donghua University, Shanghai, China, in 2015. During October 2016 and December 2016, January 2017 and April 2017, he was the visiting scholar with the Xavier University of Louisiana, USA and the University of New Orleans, USA, respectively. Currently, he is an Associate Professor with the
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Xin Wang received his M.S. degree in control theory and control engineering from Lanzhou University of Technology, Lanzhou, China, in 2006 and his Ph.D. degree in control theory and control engineering from Donghua University, Shanghai, China, in 2015. During October 2016 and December 2016, January 2017 and April 2017, he was the visiting scholar with the Xavier University of Louisiana, USA and the University of New Orleans, USA, respectively. Currently, he is an Associate Professor with the School of Computer Science and Technology, Huaiyin Normal University, China. His current research interests include synchronization, control of neural networks and coupled system control.
Jian-an Fang received the B.S., M.S. and Ph.D. degrees in electrical engineering from Donghua University (China Textile University), Shanghai, China, in 1988, 1991 and 1994 respectively. Subsequently, he joined the College of Information Science and Technology, Donghua University, Shanghai, China, where he became a Dean and Professor in 2001. During February 1998 and May 1998, he was the visiting scholar in the University of Michigan at Ann Arbor. During May 1998 and February 1999, he was the visiting scholar in the University of Maryland at College Park. During May 2005 and August 2005, he was the senior visiting scholar in the University of Southern California. In 2005 and 2006, Prof. Fang was elected as a Council Member of Shanghai Automation Association and a Council Member of Shanghai Microcomputer Applications, respectively. His research interests are mainly in complex system modeling and control, intelligent control systems, chaotic system control and synchronization, and digitalized technique for textile and fashion.
Wuneng Zhou received a first class B.S. degree in Huazhong Normal University in 1982. He obtained his Ph.D. degree from Zhejiang University in 2005. Now he is a professor in Donghua University, Shanghai. His current research interests include stability, synchronization, control of neural networks and complex networks.