【作者】 谢慧；

【导师】 郭振远；

【作者基本信息】 湖南大学 ， 数学， 2022， 硕士

【摘要】 目前,耦合神经网络的研究侧重于网络节点的非线性动力学性质,较少考虑网络结构的变化对于网络行为的影响.然而,网络的耦合结构在研究网络动态特性中有重要的作用.现实生活中的网络或系统经常会遭受一些突变因素的影响.比如周围环境的突然变化、随机发生的连接失败或者维修、子系统之间连接关系的改变等等,使得网络的耦合结构以及节点间的耦合强度发生随机跳变.这种大规模网络的跳变行为可以归类为随机切换神经网络的研究范畴.对这类系统的研究具有重要的理论价值和实际意义.本文主要研究一类具有随机扰动和时滞的耦合切换神经网络的有限时间同步和固定时间同步问题.此类切换神经网络中考虑两种类型的切换:(1)单个切换神经网络内部系统参数的状态依赖切换;(2)多个切换神经网络耦合结构间的Markov切换.这种具有随机扰动和时滞影响的耦合切换神经网络是一类特殊的随机系统.它具有两种动态形式(模态和状态).系统的模态由连续时间,离散状态的Markov链描述;系统的状态由每一模态下的状态方程描述.状态方程由一个随机微分方程表示,不同模态之间的转换受制于Markov切换规律.本文采用无领导式同步控制方案,将同步问题转化为误差系统的稳定性问题.因此,本文需要分析这类系统对应的Markov调制的随机微分方程的稳定性.首先,本文根据快速切换过程的It(?)-Doeblin公式和Markov链的定义,得到Markov切换过程的It(?)-Doeblin公式.在此基础上,本文利用Lyapunov稳定性理论和随机分析,分别得到Markov调制的随机时滞微分方程随机(依概率)渐近稳定的充分条件和随机(依概率)有限时间稳定的充分条件.其次,本文设计控制方案,得到状态依赖切换神经网络稳定化的充分条件.然后,本文在状态依赖切换神经网络中引入参数不确定性和随机扰动,得到受控的状态依赖切换神经网络的鲁棒稳定性.最后,本文采用无领导式同步方案,设计控制方案实现具有随机扰动和时滞影响的耦合切换神经网络的有限时间同步.本文使用固定时间同步控制技巧来改进控制方案,得到具有随机扰动和时滞的耦合切换神经网络固定时间同步的充分条件与同步时间估计.随之,本文分析控制方案中的参数对同步时间估计的影响,优化同步时间估计.进一步地,本文给出两个数值例子来说明理论结果的有效性.更多还原

【Abstract】 At present,the existing results on coupled neural networks focus on the nonlinear dynamics of nodes.The effects of the complexity of network structure are seldom considered.However,the coupling structure of a network plays an important role in the dynamic characteristics of the network.In applications,a network or system is often affected by some abrupt factors,such as stochastic disturbances in the environment,changes in the connection among subsystems,and so on.Thus,the coupling structure of the network and the coupling strength between nodes change randomly.The jump of this large-scale network can be classified into the research category of stochastic switched neural networks.The research on this kind of system has important theoretical value and practical significance.This thesis addresses finite-time and fixed-time synchronization of a general class of coupled switched neural networks(SNNs)with time delays subject to stochastic disturbances.Considering two types of switching rules in this class of coupled SNNs:(1)intraSNN state-dependent switching and(2)inter-SNN Markovian switching.This class of coupled SNNs with time delays subject to stochastic disturbances is a special stochastic system,which has two dynamic forms: mode and state.The mode of the system is described by a continuous-time and discrete-state Markov chain.The state of the system is described by a state equation in each mode.The state equation is represented by a stochastic differential equation.The switching among different modes is subject to the Markov chain.This thesis uses a leaderless synchronization control scheme and transforms the synchronization problem for the system into the stability problem for the error system.Therefore,it is necessary to analyze the stability of stochastic differential equations with Markovian switching.Firstly,according to the existing It(?)-Doeblin formula of the process with rapid switching and the definition of the Markov chain,the It(?)-Doeblin formula for Markov process is derived.By using Lyapunov stability theory and stochastic analysis,sufficient conditions for asymptotic stability in probability of stochastic differential equations with time delays and Markovian switching are obtained.Furthermore,the sufficient conditions for finitetime stability of stochastic differential equations with time delays and Markov switching are analyzed.Secondly,the control-law is designed to stabilize the state-dependent SNN.By introducing parameter uncertainty and stochastic disturbance into the state-dependent SNN,the robust stability of the controlled state-dependent SNN is obtained.Finally,according to the leaderless synchronization scheme,a contro-law is designed to achieve finite-time synchronization of coupled SNNs with stochastic disturbances and time delays.An improved control-law is designed for fixed-time synchronization.The sufficient conditions for fixed-time synchronization of coupled SNNs with stochastic disturbance and time delays are obtained.Several upper bounds of synchronization settling time are derived and their pros and cons are evaluated.Furthermore,two numerical examples are given to illustrate the viability of the theoretical results.更多还原