【作者】 谢东；

【导师】 蒋月评；

【作者基本信息】 湖南大学 ， 数学， 2017， 博士

【摘要】 复值神经网络是在复平面上处理信息的一类神经网络,其状态变量、连接权值与激励函数都是复值的.复值神经网络可视为实值神经网络的一种推广,但与实值神经网络有很多不同,具有更多复杂的性质.复值神经网络既可以处理现实生活中的复值问题,也可以处理实值问题,比实值神经网络功能更强大与更具优势.有些问题无法用实值神经网络解决,如XOR问题,而用复值神经网络则迎刃而解.因此,近年来,在神经网络的理论与应用研究中,复值神经网络受到广泛的关注.另外,复值神经网络在数学、物理学、工程学等领域有着广泛的应用,成功解决了许多实际问题,如:函数逼近、分类问题、图像处理、语音识别、信号处理、模式识别、安全通信以及人工神经信息处理等.众所周知,神经网络的众多应用都依赖于对其动力学行为的研究,因此神经网络的动力学行为分析显得尤为重要.由于复值神经网络有着巨大应用潜力与前景,近年来,复值神经网络吸引了国内外越来越多的研究者的浓厚兴趣,复值神经网络的动力学行为得到了深入的探讨,取得了较大进展,涌现了很多重要理论和一些非常优秀的成果.激励函数在神经网络的动力学行为分析中扮演着重要角色,而复值神经网络的激励函数的选取与实值神经网络非常不同,实值神经网络的激励函数通常可选取光滑、非常数的有界函数,但由Liouville定理可知,在复数域中,一个既解析又有界的函数就会退化为一个常数,因此,复值神经网络激励函数的选取是个重要的挑战.与实值神经网络类似,复值神经网络也有着非常丰富的动力学行为,如稳定性(包括渐近稳定、指数稳定、多稳定、-稳定、鲁棒稳定等)、周期性、分岔、同步性、耗散性与无源性等.但目前的研究还远远不够,继续深入探索复值神经网络的动力学行为以获得新的结果,无论在理论上还是实践上都非常重要,这也成了本文的主要目的与动机.本论文主要研究时滞复值神经网络的几种动力学行为,内容包括:复值神经网络周期解的存在性、唯一性和全局指数稳定性,复值神经网络的同步性、有限时间同步与有限时间滞后同步.主要研究工具和研究方法包括:重合度理论的延拓定理、矩阵测度理论、Halanay不等式理论、Lyapunov稳定性理论、有限时间稳定性理论等等.全文内容共分为五章.在第一章中,我们首先对复值神经网络的研究背景与意义进行概述,综述了研究现状与发展趋势及本文的写作动机.然后介绍了本文的主要工作与创新点.第二章介绍了一些预备知识,首先介绍了本文常用的一些数学符号,然后概述了复值神经网络的一些基本理论,包括几类复值神经网络模型、激励函数及基本定义与相关定理.第三章考虑了一类脉冲时滞复值神经网络的动力学行为.首先,利用重合度理论的Mawhin延拓定理,获得了复值神经网络的周期解的存在性的一些判别条件.其次,通过构造恰当的Lyapunov-Krasovskii泛函,得到了该复值神经网络周期解的全局指数稳定性的一些新的条件.第四章研究了一类具有时滞的复值神经网络的全局指数型同步性问题.基于Halanay不等式理论、Lyapunov稳定性理论和矩阵测度方法,通过分解复值神经网络的实部和虚部,给出了驱动响应复值神经网络的全局指数型同步的几个新准则.第五章考虑了两类时滞复值神经网络模型的有限时间同步性问题.首先,研究了时滞复值神经网络的有限时间滞后同步问题,利用Lyapunov函数方法和有限时间稳定性理论,通过设计反馈控制器,得到了一类时滞复值神经网络的有限时间滞后同步的两个新的充分条件.第二,在没有分解实部与虚部的情况下,把复向量作为一个整体,利用Lyapunov稳定性和有限时间相关理论,通过特定的控制器,得到了一类时滞复值神经网络的有限时间同步的新的充分条件.在本文的各章节中,通过一些数值例子利用数值软件来验证所得理论结果的正确性与有效性.通过对这些问题的讨论,一方面,在一定程度上加深和完善了复值神经网络网络的理论体系;另一方面,也为复值神经网络在科学技术上的实际应用提供理论支持.更多还原

【Abstract】 Complex-valued neural networks(CVNNs)are the networks that deal with information in the complex plane,that is,their state vectors,connection weights and activation functions are complex-valued.Complex-valued neural networks are regarded as the extension of real-valued neural networks(RVNNs),but they are quite different from real-valued neural networks and have more complicated properties than RVNNs.Complex-valued neural networks have been used to solve complex-valued as well as real-valued problems in real life,also complex-valued neural networks have shown more powerful capability and advantage than realvalued neural networks.Complex-valued neural networks have the potential to solve some problems that cannot be solved with their real-valued counterparts,such as the exclusion XOR problem.So they have become increasingly popular and been paid more and more attention in the neural network community in recent years.Complex-valued neural networks have a diverse variety of applications in mathematics,physics,engineering,and other areas,a lot of problems have been solved successfully,such as function approximation,classification problems,image processing,speech recognition,signal processing,pattern recognition,secure communication,artificial neural information processing and so on.As we all know,the applications of neural networks depend on the study of its dynamical behaviors,so it is very important to analyze the dynamical behaviors of neural networks.Because of great potential and application prospect of complex-valued neural networks,recently,more and more researchers focus on their attentions and interests to CVNNs,dynamical behaviors of complex-valued neural networks are investigated deeply and have great progress,and there have been some important theories and good results.The activation function plays an important role in the dynamical behaviors of recurrent neural networks.The choice of activation function of CVNNs is quite different from real-valued neural networks.In RVNNs,their activation function is usually chosen to be a smooth,bounded and nonconstant function.However,in complex domain,according to the Liouville’s theorem,every bounded analytic function will reduce to a constant.Therefore,the choice of activation function of CVNNs is an important challenge.Be similar to real-valued neural networks,complex-valued neural networks have lots of dynamical behaviors,such as stability(including asymptotic stability,exponential stability,multistability,-stability,robust stability etc.),periodicity,bifurcation,synchronization,dissipation and passivity and so on.However,the present research is far from enough,thus it is very important and worthwhile to investigate CVNNs deeply in order to explore new capabilities and higher performance both theoretically and practically.This is also the main purpose and motivation of this thesis.The main aim of this thesis is to study several dynamical behaviors of delayed complex-valued neural networks,including existence,uniqueness and global exponential stability of periodic solutions of CVNNs,synchronization,finite-time synchronization and finite-time lag synchronization of CVNNs.The main research method include Mawhin’s continuation theorem of coincidence degree theory,matrix measure method,Halanay inequality,Lyapunov stability theory,finite-time stability theory and so on.The thesis is divided into five chapters.In the first chapter,we summarize the research background and significance of complex-valued neural networks,describe the current research status and development trend of complex-valued neural networks and the motivation of this thesis.And then we give a summary outlining of the main results and the innovations of the thesis.In the chapter 2,the preliminary knowledge about complex-valued neural networks are reviewed.Firstly,we introduce basic mathematical notations,which will be used throughout the thesis.Then,we introduce some basic theory of CVNNs,including several CVNNs models,the activation functions,which play an important role in CVNNs,and some basic definitions and related theorems of CVNNs,which will be used in the following chapters.In the chapter 3,a class of delayed complex-valued neural networks with impulses is investigated.By using Mawhin’s continuation theorem of coincidence degree theory,a series of useful criteria on existence of periodic solution are established for the complex-valued neural networks.By constructing appropriate Lyapunov-Krasovskii functional,some new sufficient conditions are derived for the global exponential stability of periodic solutions to the complex-valued neural networks.In the chapter 4,global exponential synchronization of a class of complexvalued neural networks with time delays is investigated.Based on Halanay inequality theory,Lyapunov theory and matrix measure method,by separating complexvalued neural networks into the real part and imaginary parts,several new criteria for the global exponentially synchronization of drive-response complex-valued neural networks are presented.In the chapter 5,the problems of finite-time synchronization of two classes of complex-valued neural networks are considered.In the section 1,the problem of finite-time lag synchronization of complex-valued neural networks with time delay is investigated.By means of the Lyapunov function method and finite-time stability theory,some new sufficient conditions are derived to guarantee the finite-time lag synchronization between two delayed complex-valued neural networks by designing a feedback controller.In the section 2,the issue of finite-time synchronization of complex-valued neural networks with time delay is investigated.We don’t use the method separating the real and imaginary parts,but treat the complex vector as a whole,based on the finite-time stability theory and by constructing appropriate Lyapunov function,some new sufficient conditions are derived to guarantee the finite-time synchronization between two delayed complex-valued neural networks by designing an appropriate controller.In each chapter of this thesis,some examples with numerical simulations are given to illustrate the correctness and effectiveness of our theoretical results via standard numerical software.These researches not only enrich and develop some basic theory of complex-valued neural networks,but also provide theoretical basis to solve many practical problems in science and technology.更多还原