兰州大学机构库 >数学与统计学院
由抛物型偏微分方程描述的系统的控制问题
Alternative TitleThe Control Problems of Systems Governed by Parabolic Partial Differential Equations
董旺远
Thesis Advisor郑权
2006-05-04
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name博士
Keyword镇定 最优控制 有限维反馈控制 抛物型偏微分方程 Riccati代数方程 精确能控性
Abstract

本文主要研究由抛物型偏微分方程所描述的控制系统的镇定问题, 同时也考虑了这类控制系统的精确能控性问题. 通过对一类有应用背景的控制系统的能稳性的深入研究, 我们发现, 对于同一系统的不同平衡点可施加不同的控制使该平衡点成为局部渐近稳定的平衡点. 此外, 作为本文的一个相关论题, 我们还得到了所讨论系统的局部精确能控性的结果. 由于本文考虑的系统在不施加控制时都有爆破解, 施加控制使其局部能稳或精确能控就显得更有理论意义和应用价值. 在具体研究中, 本文采用的是最优控制理论研究中最近几年才出现的新方法和新技术. 这主要包括将状态空间分解技术和算子的Riccati代数方程相结合研究有限维反馈控制下系统的渐近性态的方法, 以及利用Carleman不等式研究控制系统的能控性的方法. 同时还结合了泛函分析和偏微分方程的一些重要工具和手段, 这其中包括算子半群、 算子的分数幂、 算子插值、 Sobolev空间理论、偏微分方程解的存在理论和上下解方法等. 与已有的工作相比较, 本文主要的特点是在处理系统的镇定问题时, 在同一系统的不同平衡点采用不同的控制, 其所得的结果不仅进一步完善和发展了分布参数系统的能稳性理论, 而且还有明显的应用价值.

Other Abstract

The main purpose of this dissertation is to study stabilization of systems governed by parabolic equations and exact controllability of this kind of systems. By studying stabilization of a class of parabolic equations with applied background, we find that for different steady-state solutions of this system one should add different controllers. In this dissertation, we mainly use some new methods and techniques of optimal control theory that recently appear. These mainly include the method to study asymptotic behavior of system under finite dimension feedback control by combining state space decomposition with operator Riccati equation, and the method to study exact controllability via Carleman estimate. Meanwhile, we also employ some tools and means from functional analysis and partial differential equation. Compared with the past-related works, the most important feature of the dissertation is to use different controllers at different steady-state solutions when dealing with stabilization. Our results not only further develop and perfect the theory of distributed parameter systems, but also possess obvious applied significance. to the system such that these steady-state solutions become locally asymptotic stable. Moreover, as a topic related to the dissertation, we obtain the locally exact controllability of this system.

URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/224384
Collection数学与统计学院
Recommended Citation
GB/T 7714
董旺远. 由抛物型偏微分方程描述的系统的控制问题[D]. 兰州. 兰州大学,2006.
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