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  本文關(guān)鍵詞：一類二階梯度系統(tǒng)的漸近行為　出處：《哈爾濱工業(yè)大學(xué)》2016年碩士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 梯度系統(tǒng) Lyapunov函數(shù) ?ojasiewicz不等式 Yosida逼近

【摘要】：梯度系統(tǒng)是一類十分重要的動力系統(tǒng),在數(shù)學(xué)、物理學(xué)、工程科學(xué)等領(lǐng)域有廣泛應(yīng)用。對梯度系統(tǒng)的研究最初起源于物理上的最速下降問題。經(jīng)過多年的研究,梯度系統(tǒng)模型越來越復(fù)雜,并且許多研究者在解的穩(wěn)定性和漸近性方面取得了豐富的成果。二階梯度系統(tǒng)是物理上標(biāo)準(zhǔn)的振動模型,其漸近行為是系統(tǒng)演變的最終狀態(tài),具有重要的研究價值。本文討論一類帶有不同阻尼的多個子系統(tǒng)耦合的二階梯度系統(tǒng)解的存在唯一性及漸近行為。本文首先介紹梯度系統(tǒng)的研究背景、意義以及有關(guān)梯度微分方程、梯度微分包含的簡要發(fā)展和研究進(jìn)程,并簡要介紹本文的研究模型和主要研究內(nèi)容;然后,基于微分方程和非線性分析的理論知識,研究一類二階梯度微分方程解全局存在唯一性,利用Lyapunov方法得到Lyapunov函數(shù)和方程解及其一階導(dǎo)數(shù)、二階導(dǎo)數(shù)的基本性質(zhì),利用?ojasiewicz不等式這個強(qiáng)有力的工具證明解析情形下解的收斂性并得到收斂速度估計(jì),利用凸函數(shù)和其他函數(shù)ω-極限集、臨界點(diǎn)集的性質(zhì),得到凸和一些特殊情形下解的收斂性;最后,基于凸函數(shù)Yosida逼近理論,研究一類二階梯度微分包含解的全局存在性,并得到該類二階微分包含和相應(yīng)二階微分方程的關(guān)系和該微分包含解的一階導(dǎo)數(shù)的基本性質(zhì)。
[Abstract]:The gradient system is a kind of very important power system, which is widely used in the fields of mathematics, physics, engineering science and so on. The study of the gradient system originally originated from the problem of the fastest descent in physics. After years of research, the gradient system model is becoming more and more complex, and many researchers have made great achievements in the stability and asymptotic behavior of the solution. The two step system is a physical model of vibration. The asymptotic behavior of the system is the final state of the evolution of the system. It has important research value. In this paper, the existence and uniqueness and asymptotic behavior of the solutions of a class of two - step systems coupled with different damping systems are discussed. This paper first introduces the gradient system research background, significance and related gradient differential equation, gradient differential inclusions briefly the research and development process, and briefly introduces the research model and main contents of this paper; then, analysis of differential equations and nonlinear theory of knowledge based on the study of a class of two degree ladder differential equations existence and uniqueness of global using Lyapunov method, and get the Lyapunov function and its derivative equations, two order derivatives of the basic properties of the ojasiewicz inequality? The powerful tool of analytic proof case solution convergence and convergence speed estimation, using the properties of convex function and other function Omega limit set, the critical point, be convex and some special cases of convergence; finally, convex function Yosida approximation theory based on the study of a class of two degree ladder of the global existence of solutions for differential inclusions, and The relation between the two order differential inclusions of the class and the corresponding two order differential equations and the basic properties of the first derivative of the solution are obtained.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2016
【分類號】：O175
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本文編號：1344075