两类具有时滞和有界扰动的非线性系统的稳定性分析
发布时间：2019-01-18 18:51
【摘要】：非线性时滞微分方程的稳定性理论处于蓬勃发展之中,广泛应用于生命科学、物理科学、化学和经济学模型等各个领域。然而,受Lyapunov理论方法的限制,使得对非线性时滞微分方程的理论研究比较少,尤其是非线性中立型时滞微分方程的稳定性的理论研究少之又少。近年来,Ngoc等人通过Metzler矩阵谱的性质以及比较原则研究了线性中立型时滞微分方程的稳定性问题,吸引了无数读者。本文正是基于以上方法,研究了两类系统的稳定性问题:一类是具有有界扰动的非线性时滞微分系统的稳定性问题,另一类是非线性中立型时滞微分系统的稳定性问题。本文共分为四章,第一章主要包括课题背景以及研究意义、国内外的研究现状、主要工作及结构安排;第二章介绍一些基本概念、引理、定理并给予证明,主要包括稳定性的相关概念、Metzler矩阵的概念、Metzler矩阵的等价条件及谱的一些概念、Schur稳定和Hurwitz稳定的定理以及相应的等价条件,为之后的研究内容做好基础;第三章,首先给出第一类问题(具有有界扰动的非线性时滞微分系统)稳定性的新判据,其次用反证法证明此系统的稳定性,然后给出数值算例并用Matlab软件绘出相应解的图形,最后给出此类问题的结论;第四章,首先针对第二类问题(非线性中立型时滞微分系统)稳定性的新判据,其次用反证法证明此系统的稳定性,然后给出数值算例并用Matlab软件绘出相应解的图形,最后给出此类问题的结论。
[Abstract]:The stability theory of nonlinear delay differential equations is booming and is widely used in life sciences, physical sciences, chemical and economic models and other fields. However, due to the limitation of Lyapunov theory and method, there are few theoretical studies on nonlinear delay differential equations, especially on the stability of nonlinear neutral delay differential equations. In recent years, Ngoc et al studied the stability of linear neutral delay differential equations by using the properties of Metzler matrix spectrum and the principle of comparison. Based on the above methods, this paper studies the stability of two kinds of systems: one is the stability of nonlinear delay differential systems with bounded perturbations, the other is the stability of nonlinear neutral delay differential systems. This paper is divided into four chapters. The first chapter mainly includes the background of the subject and the significance of the research, the domestic and foreign research status, the main work and structure arrangement; The second chapter introduces some basic concepts, Lemma, theorems and proofs, including the related concepts of stability, the concept of Metzler matrix, the equivalent conditions of Metzler matrix and some concepts of spectrum. The theorems of Schur stability and Hurwitz stability and the corresponding equivalent conditions make a good foundation for the later research; In the third chapter, a new criterion for the stability of the first kind of problem (nonlinear delay differential system with bounded perturbation) is given, and then the stability of the system is proved by the counter-proof method. Then the numerical examples are given and the corresponding solutions are plotted by Matlab software. Finally, the conclusion of this kind of problem is given. In chapter 4, a new criterion for the stability of the second kind of problem (nonlinear neutral delay differential system) is presented, and then the stability of the system is proved by the counter-proof method. Then the numerical examples are given and the corresponding solutions are drawn by Matlab software. Finally, the conclusion of this kind of problem is given.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
本文编号：2411007
[Abstract]:The stability theory of nonlinear delay differential equations is booming and is widely used in life sciences, physical sciences, chemical and economic models and other fields. However, due to the limitation of Lyapunov theory and method, there are few theoretical studies on nonlinear delay differential equations, especially on the stability of nonlinear neutral delay differential equations. In recent years, Ngoc et al studied the stability of linear neutral delay differential equations by using the properties of Metzler matrix spectrum and the principle of comparison. Based on the above methods, this paper studies the stability of two kinds of systems: one is the stability of nonlinear delay differential systems with bounded perturbations, the other is the stability of nonlinear neutral delay differential systems. This paper is divided into four chapters. The first chapter mainly includes the background of the subject and the significance of the research, the domestic and foreign research status, the main work and structure arrangement; The second chapter introduces some basic concepts, Lemma, theorems and proofs, including the related concepts of stability, the concept of Metzler matrix, the equivalent conditions of Metzler matrix and some concepts of spectrum. The theorems of Schur stability and Hurwitz stability and the corresponding equivalent conditions make a good foundation for the later research; In the third chapter, a new criterion for the stability of the first kind of problem (nonlinear delay differential system with bounded perturbation) is given, and then the stability of the system is proved by the counter-proof method. Then the numerical examples are given and the corresponding solutions are plotted by Matlab software. Finally, the conclusion of this kind of problem is given. In chapter 4, a new criterion for the stability of the second kind of problem (nonlinear neutral delay differential system) is presented, and then the stability of the system is proved by the counter-proof method. Then the numerical examples are given and the corresponding solutions are drawn by Matlab software. Finally, the conclusion of this kind of problem is given.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
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