一类二阶梯度系统的渐近行为

发布时间：2017-12-28 01:37

  本文关键词：一类二阶梯度系统的渐近行为　出处：《哈尔滨工业大学》2016年硕士论文　论文类型：学位论文

  更多相关文章： 梯度系统 Lyapunov函数 ?ojasiewicz不等式 Yosida逼近

【摘要】：梯度系统是一类十分重要的动力系统,在数学、物理学、工程科学等领域有广泛应用。对梯度系统的研究最初起源于物理上的最速下降问题。经过多年的研究,梯度系统模型越来越复杂,并且许多研究者在解的稳定性和渐近性方面取得了丰富的成果。二阶梯度系统是物理上标准的振动模型,其渐近行为是系统演变的最终状态,具有重要的研究价值。本文讨论一类带有不同阻尼的多个子系统耦合的二阶梯度系统解的存在唯一性及渐近行为。本文首先介绍梯度系统的研究背景、意义以及有关梯度微分方程、梯度微分包含的简要发展和研究进程,并简要介绍本文的研究模型和主要研究内容;然后,基于微分方程和非线性分析的理论知识,研究一类二阶梯度微分方程解全局存在唯一性,利用Lyapunov方法得到Lyapunov函数和方程解及其一阶导数、二阶导数的基本性质,利用?ojasiewicz不等式这个强有力的工具证明解析情形下解的收敛性并得到收敛速度估计,利用凸函数和其他函数ω-极限集、临界点集的性质,得到凸和一些特殊情形下解的收敛性;最后,基于凸函数Yosida逼近理论,研究一类二阶梯度微分包含解的全局存在性,并得到该类二阶微分包含和相应二阶微分方程的关系和该微分包含解的一阶导数的基本性质。
[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.
【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】：2016
【分类号】：O175
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本文编号：1344075