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

9501051霍恩这个项目专注于弹性系统的边界可控性和稳定化，有两个主要目标。首先讨论了一类Kirchhoff板模型的可控性问题，该模型的主部分为变系数的微分算子，控制作用为边界上的力矩。由于Kirchhoff板与含势的含时薛定谔方程之间的密切关系，作为第一步，我们利用解的局部光滑性的一般技巧来建立连续系数薛定谔方程的精确边界能控性。这些结果将被用来建立Kirchhoff板方程的精确边界能控性。第二个主要目标是解决三维线弹性方程的可控性问题，这是从二维板模型转移到一般三维弹性体的数学模型时自然产生的问题。即使当控制作用于整个边界时，目前可用的结果也假定域上有非常严格的几何条件。为了研究完全耦合系统的边界能控性问题，同时消除这些严格的几何限制，将使用一种基于能量估计和微局部分析技术的方法。求解线弹性力学系统的迹正则性估计将是这一过程中的关键步骤。将线性、常系数分布参数系统的边界能控性推广到变系数系统具有重要的数学和物理意义。许多重要物理系统的精确数学建模要求在模型中考虑空间不均匀，并且允许模型中的物理参数随时间变化。因此，开发一种基于这些更精确的物理世界模型的控制理论是重要的。特别是，寻求一种基于边界传感器和执行器的此类模型的控制理论，因为这种理论最重要的应用之一是控制大型柔性结构，其中测量和控制的自然位置都是结构的关节和外缘。由于弹性板模型和线弹性方程都可以用来模拟许多结构的部件，如翼型和太阳能电池板，因此了解如何控制这些部件对于开发复杂系统的综合控制策略是重要的一步。***

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. ***