【摘要】：非線性時(shí)滯微分方程的穩(wěn)定性理論處于蓬勃發(fā)展之中,廣泛應(yīng)用于生命科學(xué)、物理科學(xué)、化學(xué)和經(jīng)濟(jì)學(xué)模型等各個(gè)領(lǐng)域。然而,受Lyapunov理論方法的限制,使得對(duì)非線性時(shí)滯微分方程的理論研究比較少,尤其是非線性中立型時(shí)滯微分方程的穩(wěn)定性的理論研究少之又少。近年來(lái),Ngoc等人通過(guò)Metzler矩陣譜的性質(zhì)以及比較原則研究了線性中立型時(shí)滯微分方程的穩(wěn)定性問(wèn)題,吸引了無(wú)數(shù)讀者。本文正是基于以上方法,研究了兩類系統(tǒng)的穩(wěn)定性問(wèn)題:一類是具有有界擾動(dòng)的非線性時(shí)滯微分系統(tǒng)的穩(wěn)定性問(wèn)題,另一類是非線性中立型時(shí)滯微分系統(tǒng)的穩(wěn)定性問(wèn)題。本文共分為四章,第一章主要包括課題背景以及研究意義、國(guó)內(nèi)外的研究現(xiàn)狀、主要工作及結(jié)構(gòu)安排;第二章介紹一些基本概念、引理、定理并給予證明,主要包括穩(wěn)定性的相關(guān)概念、Metzler矩陣的概念、Metzler矩陣的等價(jià)條件及譜的一些概念、Schur穩(wěn)定和Hurwitz穩(wěn)定的定理以及相應(yīng)的等價(jià)條件,為之后的研究?jī)?nèi)容做好基礎(chǔ);第三章,首先給出第一類問(wèn)題(具有有界擾動(dòng)的非線性時(shí)滯微分系統(tǒng))穩(wěn)定性的新判據(jù),其次用反證法證明此系統(tǒng)的穩(wěn)定性,然后給出數(shù)值算例并用Matlab軟件繪出相應(yīng)解的圖形,最后給出此類問(wèn)題的結(jié)論;第四章,首先針對(duì)第二類問(wèn)題(非線性中立型時(shí)滯微分系統(tǒng))穩(wěn)定性的新判據(jù),其次用反證法證明此系統(tǒng)的穩(wěn)定性,然后給出數(shù)值算例并用Matlab軟件繪出相應(yīng)解的圖形,最后給出此類問(wèn)題的結(jié)論。
[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.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

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本文編號(hào)：2411007