Nonsmooth exponential synchronization of coupled neural networks with delays: new switching design
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ORIGINAL ARTICLE
Nonsmooth exponential synchronization of coupled neural networks with delays: new switching design Chao Yang1 · Lihong Huang2 Received: 7 January 2017 / Accepted: 6 November 2017 © Springer-Verlag GmbH Germany, part of Springer Nature 2017
Abstract This paper considers the exponential synchronization for a class of coupled time-delayed neural networks with discontinuous activations. Based on differential inclusions theory, set-valued analysis, and by constructing suitable coupling function and Lyapunov function, designing a novel discontinuous controller, when the controller and activation functions are both discontinuous, the global exponential synchronization for the coupled neural networks can be achieved. Especially, we consider a new Lyapunov–Krasovskii functional which is time-dependent, and the results in this paper are applicable to the undirected weighted networks. Finally, to demonstrate the correctness of our results, a numerical example is provided to illustrate it. Our results extend previously known researches. Keywords Exponential synchronization · Discontinuous activations · Switched coupling · Pinning control · Neural networks
1 Introduction It is well known that neural networks with nonsmooth neuron activations have been widely used in a number of engineering tasks, such as switching in electronic circuits, system oscillating, automatic control and so on, see [1–5]. In addition, by studying the discontinuous neural networks, many useful and interesting traits of the dynamics system can be revealed. For example, the phenomenon of convergence time to the limit cycle or equilibrium point. Unfortunately, if one investigates the neural network dynamical system with discontinuous activations, it will arise many extra difficulties. Because many previous researches based on the classical theory of continuous differential equation have been indicated to be noneffective if the vector field is * Lihong Huang [email protected] Chao Yang [email protected] 1
Department of Mathematics and Computer Science, Changsha University, Changsha 410022, People’s Republic of China
School of Mathematical and Statistics, Changsha University of Science and Technology, Changsha 410114, Hunan, People’s Republic of China
2
discontinuous. In order to address this problem, the theory of differential inclusion was firstly introduced by Forti in [6], which rapidly become a powerful tool to investigate the dynamical behaviors systems with discontinuous activations. After that, discontinuous neural networks have attracted great attention and many results have sprung up in past decades [7–10]. However, many existing results focused on chaos, bifurcation, existence and convergence of periodic solution (or almost periodic solution). Synchronization, as far as we know, is a useful tool in nature which can help us to comprehend an unknown dynamical neural networks from the well-known dynamical neural networks. However, synchronization essentially mean that the dynamics of nodes share the same time-spatia
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