无界域上波方程的能控性与反问题

项目名称： 无界域上波方程的能控性与反问题

项目编号： No.61473126

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 自动化技术、计算机技术

项目作者： 张志飞

作者单位： 华中科技大学

项目金额： 80万元

中文摘要： 波方程的控制问题和反问题是当前国内外十分活跃的研究课题。本项目拟研究无界域上波方程的边界能控性、系统物理能量的衰减以及方程系数识别等反问题。能控性研究是讨论系统状态能否在有限时间内到达目标状态；反问题研究是考察方程的系数对系统边界观测量依赖的唯一性和稳定性。对于无界域上的相关控制问题和反问题，许多有界域问题的处理方法，例如常用的吸收低阶项的紧性唯一性方法，还有研究能控性问题和反问题中经典的Carleman估计等都不再适用。本项目拟尝试通过几何分析的方法，将无界域上非线性波方程置于黎曼流形的框架下，利用几何上整体坐标下的计算技巧，建立适合于无界区域上波方程的改进Carleman型估计。基于新的Carleman型估计，我们将尝试结合对偶原理得到方程的能控性；结合波方程最优正则性结果，得到无界域上波方程反问题的唯一性和稳定性。本课题将在一定程度完善无界域上无穷维系统的控制理论以及反问题研究。

中文关键词： 波方程；能控性；反问题；无界域

英文摘要： Control theory of wave equations is an active topic in recent years. This project aims to investigate the controllability of wave equations on unbounded domains, as well as the energy decay rate of the system under boundary feedback control. Another main aspect of this project is to consider the inverse problems of wave equations on unbounded domains. We aim to establish the uniqueness and stability of the inverse problem, in which we will recover the damping coefficient and potential coefficient of the wave equation via just one boundary measurement. Here we only make some regularity assumptions on the initial data (in stead of assumptions on the solution of the equation as usual). For the related control problems on unbounded domains and inverse problems, many methods and estimates which hold true for bounded domains do not work, such as the compactness-uniqueness method for absorbing the low order terms. And the canonical Carleman estimates are no longer applicable. We attempt to use geometry analysis and the computing skills under the framework of Riemannian geometry to get the controllability, uniqueness and stability results for inverse problems of wave equations. In one word, this project will improve and enrich the control theory of nonlinear infinite dimensional system as well as the research of inverse problems.

英文关键词： Wave equations;Controllability;Inverse problems;Unbounded domains