Exponential and fixed-time synchronization of Cohen–Grossberg neural networks with time-varying delays and reaction-diffusion terms

Abstract

This paper is devoted to the global exponential and fixed-time synchronization of delayed reaction-diffusion Cohen–Grossberg neural networks. Adaptive controllers are designed such that the addressed system can realize global exponential synchronization goal under the framework of inequality techniques, Lyapunov method as well as some suitable assumptions. Furthermore, as corollaries, the corresponding conclusion is provided to ensure the delayed Cohen–Grossberg neural networks without reaction-diffusion term can reach fixed-time synchronization goal. In addition, the settling time of fixed-time synchronization can be adjusted to desired values regardless of initial conditions, which is more reasonable. Finally, two numerical examples and its simulations are given to show the effectiveness of the obtained results.

Introduction

As one of the most popular and typical neural networks, Cohen–Grossberg system was initially proposed in 1983 [1], which contains some well-known neural network models as its special cases, such as cellular NNs, Hopfield NNs, and bidirectional associative memory NNs. In the past few decades, this kinds of system have received increasing interesting due to its wide application in classification, pattern formation, association memories, parallel computation and solving optimization problems. Such applications heavily rely on its dynamical behaviors, thus, the qualitative analysis of its dynamical behaviors is a necessary step for the practical design and application of neural networks [2], [3], [4], [5], [6], [7], [8], [9].

In the course of studying neural networks, it is found that the diffusion phenomena could not be ignored in neural networks and electric circuits once electrons transport in a nonuniform eletromagnetic field. For example, the multilayer cellular neural networks (MCNNs) are arrays of nonlinear and simple computing elements characterized by local interactions between cells, therefore, the paradigm are well suited to describe locally interconnected simple dynamical systems showing a lattice-like structure [10]. In other words, the whole structure and dynamic behavior of MCNNs are seriously dependent on the evolution time of each variable and its position (space), as well as its interactions deriving from the space-distributed structure of whole networks. So, it is essential to consider the state variables varying with time and space variables. On the other hand, there are a large number of reaction-diffusion phenomena in nature and many discipline fields, especially in chemistry and biology fields. Thus, it is natural to consider the states of the neurons vary in space as well as in time [11], [12], [13], [14], [15], [16].

Synchronization, which means that two or more systems share a common dynamical behavior, and this common behavior can be induced by coupling or by external force. In the real word, synchronization phenomenon is very important in consideration of its potential applications in many different areas including secure communication, biology systems, optics, and information processing [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. By means of impulsive control technique, the corresponding lag synchronization criteria of chaotic delayed system was considered in [29], [17] pay attention to the non-fragile synchronization control issue of Markovian jumping model. Moreover, some interesting synchronization conclusions about the Cohen–Grossberg neural networks are proposed in [25], [26], [27]. However, in the above mentioned synchronization categories, the setting time is heavily depends on the initial values of the target model, which is unreasonable in some practical applications, i.e., the knowledge of the initial states may not be available in advance. To overcome this drawback, a new concept is put forward in [30], which is named as fixed-time stability, subsequently, some fixed time algorithms are also given in [31], [32], [33].

Fixed-time synchronization/stability implies that the system is globally finite-time stable and the upper bounds of the settling time can also be estimated for any initial values, which can effectively avoid the convergence time depending on the information of the initial states. However, there are very few existing papers concentrated on this topic [34], [35], [36], [37], [38], let alone the fixed-time synchronization of delayed Cohen-Crossberg neural networks, which contribute the motivation of this manuscript.

Motivated by the above discussion, the main objective of this paper is to design a new controller that provides a system with exponential synchronization and fixed-time synchronization property. Under the derived protocols, all the states of the system converge to a common bounded settling time, which is independent of any initial values. Moreover, it can be estimated a priori by the control parameters. When opposed to some related works, the main contribution of this present paper can be concluded in the following lines: (i) The reaction-diffusion term is considered in this paper, which is more tally with the actual; (ii) The well known Hardy–Poincarè inequality was carried out to evaluate the upper bound of the reaction-diffusion term, which makes the derived conditions contain much more factors that do affect the dynamic behaviors of the target system; (iii) Fixed-time synchronization criteria is introduced, which ensure the setting time is independent of any initial values.

The organization of this paper is as follows. In Section 2, some preliminaries are introduced. In Section 3, we shall derive several new sufficient conditions to check the synchronization problems of delayed Cohen–Grossberg neural networks with and without reaction-diffusion term. In Section 4, two numerical examples are presented to show the validity of the proposed results. Conclusions are drawn in Section 5.

Notation: $\mathbb{R}$ denotes the set of real numbers, ${\mathbb{R}}^n$ denotes the n-dimensional Euclidean space. $\Omega \subset {\mathbb{R}}^n$ is an open domain including the origin with smooth boundary ∂Ω and radially bounded by β, and mesΩ denotes the measure of Ω. Let $\mathcal{C}=\mathcal{C}([-\tau ,0]\times {\mathbb{R}}^m,{\mathbb{R}}^n)$ be the Banach space of continuous functions which maps $[-\tau ,0]\times {\mathbb{R}}^m$ into ${\mathbb{R}}^n$ with the topology of uniform converge. Let $\mathcal{C}=\mathcal{C}([-\tau ,0]\times {\mathbb{R}}^m,{\mathbb{R}}^n)$ be the Banach space of continuous functions which maps $[-\tau ,0]\times {\mathbb{R}}^m$ into ${\mathbb{R}}^n$ with the topology of uniform converge. Let ‖ · ‖ is either the Euclidean vector norm or the induced matrix norm. For any $e(t,x)={(e_1(t,x),e_2(t,x),\dots ,e_n(t,x))}^T\in {\mathbb{R}}^n,$ ‖e(t, x)‖₂ denotes ${\parallel e(t,x)\parallel }_2={\left({\int }_{\Omega }{\sum }_{i=1}^n{e}_i^2(t,x)dx\right)}^{1/2}$.