随机时滞系统数值解的稳定性分析

发布时间：2018-01-01 23:23

本文关键词：随机时滞系统数值解的稳定性分析　出处：《南京师范大学》2016年硕士论文　论文类型：学位论文

【摘要】：众所周知,随机时滞系统在理论上有着丰富的研究成果,并在经济、金融、医学以及其他科学领域都有广泛的应用。稳定性研究是这个领域中最重要课题,因为一切系统能够正常运行的前提是其必须保持稳定.然而一些随机时滞系统并没有显式解.因此,我们去研究随机时滞系统的数值解就显得尤为重要.本文主要研究了两类比较有代表性的随机时滞系统,即中立型随机时滞系统与随机时滞Hopfield神经网络系统.对于上述系统,我们主要研究了四种数值方法,包括随机线性θ法、分步θ法、Euler法和向后Euler法.具体内容如下：一、中立型随机时滞微分方程的两类θ方法的均方稳定性分析本部分首先介绍和分析了关于中立型随机时滞微分方程的随机线性θ方法.我们给出了一些关于中立项、漂移系数、扩散系数的条件,这些条件允许扩散系数可以高度的非线性,同时不满足线性增长条件以及全局Lipschitz条件.我们证明了,对于所有的正的步长,当θ∈[1/2,1]时,随机线性θ法是均方渐近稳定的.当θ∈[0,1/2)时,在一个强的假设条件下,当△∈(0,△t0)时,随机线性θ法是均方渐近稳定的.其次,我们考虑分步θ方法并得到了相似但更好的结果.我们证明了,对于所有的正的步长,当θ∈[1/21]时,分步θ法是均方稳定的.当θ∈[0,1/2)时,在一个强的假设条件下,当△∈(0,△t0)时,分步θ法是均方稳定的.二、随机时滞Hopfield神经网络系统的Euler法和向后Euler法的几乎必然指数稳定性分析本部分运用Euler法和向后Euler法来研究随机时滞Hopfield神经网络系统的数值解的几乎必然指数稳定性.在简单、合理条件下,我们证明了Euler法和向后Euler法是几乎必然指数稳定的.值得一提的是,不同于以往文献利用Chebyshev不等式和Borel-Cantelli引理,我们的工具是非负半鞅收敛定理.
[Abstract]:As everyone knows, the random time-delay systems have abundant research achievements in theory, and in the economic, financial, medical and other fields are widely used. The stability is the most important research topic in this field, because all system can normal operation of the premise is must remain stable. However, some stochastic time-delay systems was not significantly the solution. Therefore, it is particularly important for us to study the numerical stochastic time-delay system solution. This paper mainly studies two kinds of comparison with stochastic time-delay systems representative, i.e. neutral stochastic time-delay systems with random time delay Hopfield neural network system. For this system, we mainly study four kinds of numerical methods, including random linear theta theta method, step method, Euler method and the backward Euler method. The main contents are as follows: first, the mean square stability of two kinds of theta method of neutral stochastic delay differential equations of the The first part introduces and analyzes the method of stochastic linear theta neutral stochastic delay differential equations. We give some about the project, drift coefficient, diffusion coefficient conditions, these conditions allow the nonlinear diffusion coefficient can be high, but does not satisfy the linear growth condition and global Lipschitz conditions. We prove that, for all positive step, when an angle e [1/2,1], random linear 0 method is mean square asymptotically stable. When in [0,1/2, theta) in a strong assumption, when delta, (0, t0), random linear theta method is asymptotically stable in the mean square. Secondly, we consider the step by step. Methods and obtained similar but better results. We show that, for all the positive step, when an angle e [1/21], step by step method is mean square stable theta theta epsilon [0,1/2). When, in a strong assumption, when delta, (0, t0), step by step Wilson is The mean square stability. Two, almost sure exponential stability numerical analysis of almost sure exponential stability of Euler method and Hopfield neural networks and stochastic delay backward Euler method to this part of the application of Euler method and the backward Euler method to study Hopfield neural network system of random solutions of delay. In simple and reasonable conditions, we prove that the Euler method and the backward Euler method is almost surely exponential stability. It is worth mentioning that, different from the previous literature by using the Chebyshev inequality and Borel-Cantelli lemma, our tool is nonnegative semi martingale convergence theorem.

【学位授予单位】：南京师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175

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