噪声摄动下几类全局指数稳定的微分系统解的稳定性分析

发布时间：2018-04-13 15:23

本文选题：噪声 + 随机微分方程　；参考：《中国矿业大学》2017年硕士论文

【摘要】：稳定性理论是研究动态系统在遇到外界扰动时是否能继续保持稳定的理论[22].对于随机系统来说,稳定性不仅是非常重要的系统性能,而且在系统设计中起着相当重要的作用.随机系统的稳定性课题无论是在理论研究方面还是工业设计应用方面都具有十分重要的实践价值.本文在第二章和第三章中重点讨论了一定强度的噪声对非线性常微分系统以及非线性时滞微分系统稳定性的影响.本文直接利用解稳定性系数α和β来求解使随机系统继续保持稳定的噪声强度上界.通过不确定参数、放缩技巧、Holder不等式、Gronwall不等式及一些常用不等式,在Lipschitz条件下推导出摄动系统在一定条件下,可以保持全局指数稳定的结果;并得到了求解噪声强度的上界的超越方程,选择适当的不确定参数,可以估算最大允许噪声强度上界.特别的在第三章中针对非线性时滞系统,本文考虑了三种不同类型的噪声:随机项不含时滞的噪声、随机项含时滞的噪声和随机项含有时滞和不含时滞的混合噪声.本文主要得到两点结论:第一,不失稳定性的条件下,摄动系统能容许一定范围内的噪声强度,即当噪声的强度不超过给定的噪声上界,随机微分系统与随机时滞微分系统依旧可以保持全局指数稳定性质;第二,如果噪声的强度不超过导出的噪声上界,摄动系统的解具有更大的衰减速度,即噪声能进一步促进常微分系统与时滞微分系统解的衰减速度.
[Abstract]:Stability theory is a theory to study whether dynamic systems can continue to be stable in the presence of external disturbances [22].For stochastic systems, stability is not only a very important system performance, but also plays an important role in system design.The stability of stochastic systems is of great practical value in both theoretical research and industrial design applications.In the second and third chapters, the influence of noise intensity on the stability of nonlinear ordinary differential systems and nonlinear delay differential systems is discussed.In this paper, we directly use the stability coefficients 伪 and 尾 to solve the upper bound of noise intensity which can keep the stochastic system stable.By means of uncertain parameters, holder inequality and Gronwall inequality and some common inequalities, the results of global exponential stability of perturbed systems under certain conditions are derived under Lipschitz condition.The transcendental equation of the upper bound of noise intensity is obtained, and the maximum allowable noise intensity upper bound can be estimated by choosing appropriate uncertain parameters.In the third chapter, we consider three different types of noise: random term without delay, random term with delay and random term with and without delay.In this paper, two main conclusions are obtained: first, under the condition of no loss of stability, the perturbation system can tolerate the noise intensity in a certain range, that is, when the intensity of the noise does not exceed the upper bound of the given noise,Stochastic differential systems and stochastic delay differential systems can still maintain global exponential stability. Secondly, if the intensity of noise does not exceed the derived upper bound of noise, the solution of perturbed systems has a higher attenuation speed.That is, noise can further accelerate the decay rate of solutions of ordinary differential systems and delay differential systems.
【学位授予单位】：中国矿业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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