一类Euler-Bernoulli梁耦合格点系统解的存在唯一性及渐近性行为研究

发布时间：2018-06-02 16:51

本文选题：Euler-Bernoulli梁 + 格点系统　；参考：《太原理工大学》2016年硕士论文

【摘要】：作为非线性科学研究领域的一个重要课题,无穷维动力系统已成功地应用到许多重要方面,有着广泛的应用前景.尤其是,无穷维动力系统中的格点系统,受到了科学家的广泛关注,已被成功地应用在生物、力学、土木工程等诸多领域.近来,对格点系统解的研究,特别是解的存在唯一性及渐近性行为的研究,已经成为许多学者研究的热门课题.其中,渐近性行为上的指数吸引子是用来描述无穷维系统长时间动力学行为的重要概念之一,是在指数吸引轨道上的一个正不变集,并且包含整体吸引子.鉴于此,本文主要由两部分组成,第一部分研究了一类分数阶Euler-Bernoulli梁耦合格点系统解的存在唯一性.第二部分主要研究的是整数阶Euler-Bernoulli梁耦合格点系统指数吸引子的存在性.本文内容主要包括:第一章,介绍了Euler-Bernoulli梁耦合格点系统的相关研究背景和研究价值,给出了论文中所涉及到的基本定义、基本理论和常用不等式等基础知识.第二章,考虑的是一类分数阶Euler-Bernoulli梁耦合格点系统解的存在唯一性.首先对Euler-Bernoulli梁格点系统,考虑其热效应影响,并且将其推广到分数阶形式,从而研究了一类分数阶Euler-Bernoulli梁耦合格点系统解的初值问题.通过连续紧映射原理、Schauder’s不动点定理和巴拿赫压缩映射原理对该格点系统解的存在性和唯一性分别进行了讨论.第三章,考虑的是整数阶Euler-Bernoulli梁耦合格点系统解的长时间动力学行为中的指数吸引子.在希尔伯特空间中,解算子半群根据算子半群理论存在有界闭的吸收集,解算子满足一致Lipschitz连续.通过解的正交分解,解算子满足特定的不等式.应用压缩映射原理及Gronwall不等式证明了整数阶Euler-Bernoulli梁耦合格点系统指数吸引子的存在性.第四章,对本文进行了总结与展望.
[Abstract]:As an important subject in the field of nonlinear science, infinite dimensional dynamical system has been successfully applied to many important aspects and has a wide application prospect. In particular, lattice systems in infinite dimensional dynamical systems have been widely concerned by scientists and have been successfully applied in many fields, such as biology, mechanics, civil engineering and so on. Recently, the study of lattice system solutions, especially the existence, uniqueness and asymptotic behavior of solutions, has become a hot topic for many scholars. Among them, the exponential attractor in asymptotic behavior is one of the important concepts used to describe the long-time dynamical behavior of infinite dimensional systems. It is a positive invariant set on the exponential attraction orbit and contains the global attractor. In view of this, this paper mainly consists of two parts. In the first part, we study the existence and uniqueness of solutions for a class of fractional Euler-Bernoulli beam coupled lattice point systems. In the second part, the existence of exponential attractors for Euler-Bernoulli beam coupling lattice system of integer order is studied. The main contents of this paper are as follows: in the first chapter, the research background and research value of Euler-Bernoulli beam coupled lattice system are introduced, and the basic definitions, basic theories and commonly used inequalities are given. In chapter 2, we consider the existence and uniqueness of solutions for a class of fractional Euler-Bernoulli beam coupled lattice point systems. In this paper, we consider the thermal effect of Euler-Bernoulli lattice point system and extend it to fractional form, so as to study the initial value problem of the solution of a class of fractional Euler-Bernoulli beam coupled lattice point system. The existence and uniqueness of the solution of the lattice system are discussed by means of the Schauders fixed point theorem and Barnach contraction mapping principle. In chapter 3, we consider the exponential attractor in the long-time dynamical behavior of the coupled lattice system of Euler-Bernoulli beam with integer order. In Hilbert space, the solution operator semigroup has a bounded closed absorption set according to the operator semigroup theory, and the solution operator satisfies the uniform Lipschitz continuity. By orthogonal decomposition of the solution, the solution operator satisfies certain inequalities. By using the contraction mapping principle and Gronwall inequality, the existence of exponential attractors for Euler-Bernoulli beam coupled lattice point systems of integer order is proved. In the fourth chapter, this paper is summarized and prospected.
【学位授予单位】：太原理工大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175

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