Mathematical Sciences: Control of Variable Coefficient Partial Differential Equations: Elastic Bodies and Schrodinger Equations

数学科学：变系数偏微分方程的控制：弹性体和薛定谔方程

• 批准号: 9501051
• 负责人: Mary Ann Horn
• 金额: $ 3.2万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-09-01 至 1998-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9501051&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Control; Variable; Coefficient

9501051 Horn This project focuses on boundary controllability and stabilization of elastic systems and has the two primary objectives. The first is to address the question of controllability of a Kirchhoff plate model with nonconstant coefficients in the principal part of the differential operator and with control acting as a moment on the boundary. Because of the close relationship between the Kirchhoff plate and a time-dependent Schroedinger equation with potential, as a first step, a general technique which relies on local smoothing properties of the solution is used to establish exact boundary controllability the Schroedinger equation with continuous coefficients. These results will then be utilized to establish exact boundary controllability of the Kirchhoff plate equation. The second major goal is to resolve the issue of controllability for the equations of three- dimensional linear elasticity, an issue which naturally arises when moving from two- dimensional plate models to mathematical models of general three-dimensional elastic bodies. Results currently available assume very strict geometric conditions on the domain even when the control is acting on the entire boundary. To examine the boundary controllability problem for the fully coupled system while eliminating these stringent geometric restrictions, an approach based on energy estimates and microlocal analysis techniques will be used. Derivation of trace regularity estimates for the solution of the system of linear elasticity will be a critical step in the process. Extensions of boundary controllability results from linear, constant coefficient distributed parameter systems to systems with nonconstant coefficients are mf great importance both mathematically and physically. Accurate mathematical modeling of many important physical systems requires that spatial inhomogeneities be admitted into the model and that the physical parameters in the model be permitted to vary with time. For this reason, it is important to develop a control theory based on these more accurate models of the physical world. In particular, a control theory for such models based on boundary sensors and actuators is sought, since one of the most important applications of such a theory is to the problem of controlling large flexible structures, where the natural locations for both measurements and controls are the joints and outer edges of the structure. Because both elastic plate models and equations of linear elasticity may be used to model components, such as airfoils and solar panels, of many structures, understanding how such elements can be controlled is a significant step to the development of a comprehensive control strategy for complex systems. ***

9501051喇叭这个项目的重点是边界可控性和稳定的弹性系统，有两个主要目标。 第一个是解决问题的可控性的Kirchhoff板模型的微分算子的主要部分中的非常数系数和控制作为一个时刻的边界上。 由于Kirchhoff板和含时势Schroedinger方程之间的密切关系，作为第一步，依赖于解的局部光滑性质的一般技术被用来建立具有连续系数的Schroedinger方程的精确边界能控性。 这些结果将被用来建立基尔霍夫板方程的精确边界能控性。 第二个主要目标是解决三维线性弹性方程的可控性问题，这个问题在从二维板模型转移到一般三维弹性体的数学模型时自然出现。 目前可用的结果假设非常严格的几何条件的域上，即使当控制作用于整个边界。 为了研究完全耦合系统的边界可控性问题，同时消除这些严格的几何限制，将使用基于能量估计和微局部分析技术的方法。 导出线弹性系统解的迹正则性估计将是这一过程中的关键步骤。 将边界能控性的结果从线性常系数分布参数系统推广到变系数系统，在数学和物理上都具有重要意义。 许多重要物理系统的精确数学建模要求模型中考虑空间不均匀性，并且允许模型中的物理参数随时间变化。因此，基于这些更精确的物理世界模型来发展控制理论是很重要的。 特别是，基于边界传感器和致动器的这种模型的控制理论的要求，因为这样的理论的最重要的应用之一是控制大型柔性结构的问题，其中的测量和控制的自然位置是关节和外边缘的结构。 由于弹性板模型和线性弹性方程都可以用来模拟许多结构的部件，如机翼和太阳能电池板，因此了解如何控制这些元件是开发复杂系统综合控制策略的重要一步。 ***