一维障碍波方程有限元半离散化一致指数稳定性

发布时间：2018-03-24 23:33

本文选题：一维障碍波方程　切入点：有限元半离散化　出处：《渤海大学》2017年硕士论文

【摘要】：一维障碍波方程是分布参数控制系统研究的主要对象之一,其指数稳定性、精确可观性和精确可控性以及它们之间的关系得到广泛而深入的研究。为了便于计算和工程上的实现,从上世纪末开始,一些学者们从数值分析的角度对此系统的可控性、可观性和指数稳定性进行了研究,也就是半离散化数值逼近系统能否一致保持连续系统的可控性、可观性和稳定性问题。在这样一种背景下,本文对一维障碍波方程进行有限元空间半离散化,研究离散系统解的能量的一致指数衰减性。首先,通过对离散系统及其共轭系统的分析,指出原离散系统的指数稳定性等价于共轭系统的精确可观性。其次,通过对共轭系统的特征值分析和对相应特征向量的边界观测,得到离散系统不具有一致指数稳定性;最后,在离散系统中添加数值粘性项,通过乘子法进一步证明了此时系统具有一致指数稳定性。本文结构如下:第一章是绪论部分,简单介绍本文的研究背景和研究意义。第二章是预备知识,介绍一维障碍波方程指数稳定性与精确可观性的等价关系,并对一维波方程有限差分和有限元空间半离散化进行了介绍。第三章,根据有限元半离散化建立一维障碍波方程系统的数学模型,进而研究系统的一致指数稳定性和非一致指数稳定性的条件。最后,对本文进行总结以及发展期望。
[Abstract]:The one-dimensional obstacle wave equation is one of the main objects in the study of distributed parameter control system, and its exponential stability. The accuracy of observability and precision controllability and the relationship between them have been extensively and deeply studied. In order to facilitate the calculation and engineering realization, since the end of the last century, some scholars have controlled the system from the point of view of numerical analysis. In this paper, observability and exponential stability are studied, that is, whether the semi-discretization numerical approximation system can uniformly maintain the controllability, observability and stability of continuous systems. In this paper, the finite element semi-discretization of the one-dimensional barrier wave equation is carried out, and the uniform exponential decay of the energy of the solution of the discrete system is studied. Firstly, the discrete system and its conjugate system are analyzed. It is pointed out that the exponential stability of the original discrete system is equivalent to the exact observability of the conjugate system. Secondly, by analyzing the eigenvalue of the conjugate system and observing the boundary of the corresponding eigenvector, it is found that the discrete system does not have uniform exponential stability. The numerical viscosity term is added to the discrete system and the exponential stability of the system is further proved by the multiplier method. The structure of this paper is as follows: the first chapter is the introduction part. In chapter two, the preparatory knowledge is introduced, and the equivalent relation between the exponential stability of one-dimensional obstacle wave equation and the exact observability is introduced. The finite difference of one-dimensional wave equation and the semi-discretization of finite element space are introduced. In the third chapter, the mathematical model of one-dimensional obstacle wave equation system is established according to the finite element semi-discretization. Then we study the conditions of uniform exponential stability and non-uniform exponential stability of the system. Finally, we summarize and develop the expectation of this paper.
【学位授予单位】：渤海大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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