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

这个项目主要研究弹性系统的边界可控性和稳定性，有两个主要目标。第一个是解决Kirchhoff板模型的可控性问题，微分算子的主要部分是非常系数，控制作为边界上的力矩。由于Kirchhoff板与时变带势薛定谔方程关系密切，本文首先利用方程解的局部光滑性建立了连续系数薛定谔方程的精确边界可控性。这些结果将被用来建立基尔霍夫板方程的精确边界可控性。第二个主要目标是解决三维线弹性方程的可控性问题，这是一个从二维板模型转向一般三维弹性体数学模型时自然出现的问题。目前可用的结果假设域上有非常严格的几何条件，即使控制作用于整个边界。为了在消除这些严格的几何限制的同时检查全耦合系统的边界可控性问题，将使用基于能量估计和微局部分析技术的方法。线弹性系统解的轨迹正则性估计的推导将是这一过程中的关键步骤。将常系数线性分布参数系统的边界可控性结果推广到非常系数系统，在数学和物理上都具有重要意义。对许多重要的物理系统进行精确的数学建模，要求在模型中考虑到空间的非均质性，并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，这种基于边界传感器和执行器的模型的控制理论被寻求，因为这种理论最重要的应用之一是控制大型柔性结构的问题，其中测量和控制的自然位置是结构的关节和外缘。由于弹性板模型和线性弹性方程都可以用于模拟许多结构的组件，如机翼和太阳能电池板，因此了解如何控制这些元素是开发复杂系统综合控制策略的重要一步。＊＊＊

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. ***