无穷维Hamilton系统下循环算子及反问题的探讨

发布时间：2018-04-25 11:54

本文选题：无穷维Hamilton正则系统 + 循环算子　；参考：《内蒙古工业大学》2017年硕士论文

【摘要】：无穷维Hamilton系统是一类具有特殊结构的偏微分方程(组),是解决物理、力学、控制等实际问题常用的基本形式.本文在Hamilton系统这个框架下,主要针对循环算子的获得和反问题进行了探讨,首先将一般微分方程系统的循环算子的某些理论平移到Hamilton体系中,得到一种获得循环算子的方法;然后对无穷维Hamilton系统的反问题的作了两点探讨.第一章,首先简述了研究对象及Hamilton系统的大致研究方向,并罗列了一些与无穷维Hamilton系统相关的定义;其次,阐述循环算子研究现状并对反问题方法作了综述;最后,介绍了全篇的主要研究思路及工作.第二章,对Hamilton系统循环算子的获得方法作出了探索,主要是将一般微分方程系统循环算子的一个结论移植到无穷维Hamilton系统,获得了常型Hamilton系统下循环算子的一般结论,并通过算例验证了该结论的可行性.第三章,在前人研究的基础上,对Hamilton系统的反问题作了两点探讨,第一,基于Taylor展开式的基础上改进了状态变量的计算方法,其简便性主要体现在分解因子Q的系数的计算上;第二,对文献已讨论过的一类Hamilton算子决定的高阶方程的正则化作了探索,并通过寻找变换,实现了一类新Hamilton算子决定的方程的正则形式化问题.最后,对所做的工作进行了一些简单地总结,分析出其中的不足之处,并对接下来可能开展的工作指明了方向.
[Abstract]:Infinite dimensional Hamilton system is a kind of partial differential equation with special structure, which is a basic form of solving practical problems such as physics, mechanics, control and so on. In this paper, under the framework of Hamilton system, we mainly discuss the problem of obtaining and inverse cyclic operators. Firstly, some theories of cyclic operators of general differential equation systems are translated into Hamilton system, and a method of obtaining cyclic operators is obtained. Then the inverse problem of infinite dimensional Hamilton system is discussed in two aspects. In the first chapter, the research object and the general research direction of Hamilton system are briefly introduced, and some definitions related to infinite dimensional Hamilton system are listed. Secondly, the research status of cyclic operator and the method of inverse problem are summarized. This paper introduces the main research ideas and work of the whole article. In the second chapter, the method of obtaining cyclic operators for Hamilton systems is explored. A conclusion of cyclic operators for ordinary differential equation systems is transplanted to infinite dimensional Hamilton systems, and the general results of cyclic operators in Hamilton systems of constant type are obtained. The feasibility of the conclusion is verified by an example. In chapter 3, the inverse problem of Hamilton system is discussed on the basis of previous studies. Firstly, the calculation method of state variables is improved on the basis of Taylor expansion, and its simplicity is mainly reflected in the calculation of the coefficient of decomposition factor Q; Secondly, the regularization of a class of higher order equations determined by Hamilton operators, which has been discussed in the literature, is explored. By searching for transformations, the regularization formalization of a class of equations determined by new Hamilton operators is realized. Finally, the work done is summarized briefly, the shortcomings are analyzed, and the direction of the possible work is pointed out.
【学位授予单位】：内蒙古工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.2

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