几类随机系统数值稳定性分析

发布时间：2018-02-04 11:39

本文关键词： 随机系统稳定性指数Euler方法 Poisson白噪声 Gauss白噪声　出处：《哈尔滨工业大学》2016年博士论文　论文类型：学位论文

【摘要】：近些年来,随着科技的进步,随机系统理论得到了不断地发展和完善。随机微分方程作为随机系统的数学描述,已经被广泛地应用到物理、生物、金融、电子工程和控制等各领域中。稳定性是随机微分方程理论中一个重要的性质,同时也是维持随机系统正常运转的必要条件。因此,稳定性的研究在理论意义和实际应用中具有非常重要的价值。本文以几类随机系统为研究对象,对数值方法的稳定性和系统的稳定性进行了分析。主要内容包含以下几个方面:研究了一类半线性随机比例微分方程的均方稳定性问题。构造了指数Euler方法,给出了关于解析解均方稳定的条件,并证明了在此条件下指数Euler方法对任意非零步长可以保持均方稳定性。最后,数值算例验证了所得结论。讨论了一类Poisson白噪声激励下随机延迟微分方程的稳定性。对于Poisson白噪声激励下线性的随机延迟微分方程,获得了解析解稳定的充分条件,当步长充分小时,指数Euler方法可以产生均方稳定性。进一步,对于Poisson白噪声激励下半线性的随机延迟微分方程,构造了补偿指数Euler方法,建立了解析解保持均方稳定的条件,并证实了该数值方法依任意步长保持原系统的均方稳定性。给出了相应的数值实验。考虑了一类Gauss白噪声激励下带有Mathieu-Duffing振子两质量相对转动系统的稳定性。在物理背景和实际意义下,建立了数学模型,利用Melnikov方法分析得知,系统出现了混沌动力学行为。在Gauss白噪声参激下,系统由原来不稳定状态转为稳定状态,从而实现系统的稳定化。数值仿真模拟进一步验证了结论。
[Abstract]:In recent years, with the progress of science and technology, stochastic system theory has been continuously developed and improved. As a mathematical description of stochastic systems, stochastic differential equations have been widely used in physics, biology and finance. In the fields of electronic engineering and control, stability is an important property in the theory of stochastic differential equations, and it is also a necessary condition to maintain the normal operation of stochastic systems. The study of stability is of great value in both theory and practice. In this paper, several kinds of stochastic systems are taken as research objects. The stability of the numerical method and the stability of the system are analyzed. The mean square stability of a class of semilinear stochastic proportional differential equations is studied and the exponential Euler method is constructed. The conditions for the mean square stability of analytic solutions are given and it is proved that the exponential Euler method can maintain the mean square stability for any nonzero step under this condition. The results are verified by numerical examples. The stability of a class of stochastic delay differential equations excited by Poisson white noise is discussed. The linear stochastic delay differential equations under Poisson white noise excitation are discussed. A sufficient condition for the stability of the analytical solution is obtained. When the step size is small enough, the exponential Euler method can produce the mean square stability. For the semilinear stochastic delay differential equation excited by Poisson white noise, a compensatory exponential Euler method is constructed, and the condition for the analytic solution to maintain the mean square stability is established. It is proved that the numerical method maintains the mean square stability of the original system according to arbitrary step size. The corresponding numerical experiments are given. A class of Gauss white noise excitation with Mathieu-Duffing is considered. The stability of the relative rotational system of the oscillator with two masses. In the physical background and in the practical sense. The mathematical model is established and the chaotic dynamics of the system is analyzed by using Melnikov method. Under the Gauss white noise parameter excitation, the system changes from the original unstable state to the stable state. Thus the stabilization of the system is realized, and the conclusion is further verified by numerical simulation.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O211.63

【相似文献】