Exact Controllability and Observation of Structural Acoustics and Thermoelastic Systems

0208121本项目致力于研究控制腔内结构声学流动的耦合偏微分方程组(PDE‘s)的精确边界能控性。还将研究二维热弹性系统的精确边界能控性问题和零边界能控性问题。在某种程度上，这项工作将需要研究对偶问题；即，获得齐次伴随方程的解的相关可观测性不等式。根据预期的工程应用，重点将放在允许在尽可能小的(边界)控制区域上控制结构声学动力学的情况。此外，该项目的目的是找到几何条件和规定的控制，以便仅在声室的柔性部分实施控制，对于有限能量的任意初始数据，人们将具有对声流的精确可控性。预计这项工作的主要内容将包括：(I)波动方程在没有所谓洛帕廷斯基条件(Neumann边界条件下波动方程的固有条件)的情况下的锐迹正则性；(Ii)允许边界上的时间导数吸收切向波迹的微观局部分析估计；以及(Iii)最近涉及Carleman对边界具有受控Neumann部分的波动方程的估计的结果。此外，该项目还将重点研究热弹性系统的线性和(全局)非线性精确可控性问题。特别是，将考虑具有相关(非Lipschitz)非线性的热弹性偏微分方程组；例如，von Karman括号和在可伸缩板建模中出现的准线性。这项工作将试图以一种基本的方式使用目前已知的线性化热弹性模型的解析性，以及我们最近对无控制(但完全非线性)热弹性系统的稳定性工作。耦合偏微分方程组(PDE)的例子，例如那些要研究的例子，在文献中早已存在。然而，智能材料技术的最新创新，以及这些创新在控制工程设计中的潜在应用，极大地增加了人们对这些PDE模型的兴趣。该项目的目的是获得关于这些方程的某些定性性质的信息，这些信息反过来可以用来为这些方程所支配的结构/相互作用设计有效的控制律。例如，结构声学偏微分方程被用来模拟飞机机舱内部声场与机舱周围墙壁的相互作用。为了乘客的利益，希望消除或控制直接作用于室内声场的压力扰动。这些干扰通常来自机舱环境之外；例如，飞机发动机和螺旋桨噪音引起的振动，或天气湍流造成的影响。在实践中，工程师试图通过在舱壁的一部分放置压电致动器/传感器来控制这种外部噪声，这些设备的作用方式是消除或至少减少有害的声压影响。然而，这项技术的有效性对机舱的形状以及机舱壁上放置致动器的特定区域非常敏感。该项目的目标包括：(I)精确地描述那些确实可以通过压电驱动进行主动控制设计的舱室几何形状；以及(Ii)当这种控制设计可行时，构建一种可靠的方法来规定控制驱动的量和区域，这将是在舱内保持平静声场所必需的。

0208121AvalosThis project is concerned with studying exact boundary controllability properties of those systems of coupled partial differential equations (PDE's) which govern structural acoustic flow within a chamber. Exact and null boundary controllability problems for two-dimensional systems of thermoelasticity will also be studied. In part, the work will entail a study of the dual problem; namely, the attainment of related observability inequalities for solutions of homogeneous adjoint equations. In line with the intended engineering applications, the focus will be on situations which allow control of the structural acoustic dynamics on as small a (boundary) control region as possible. Moreover, this project is aimed at finding conditions on the geometry and prescribed controls so that, with control implemented on the flexible portion of the acoustic chamber only, one will have exact controllability of the acoustic flow, for arbitrary initial data of finite energy. It is anticipated that key ingredients in the work will include the following: (i) sharp trace regularity for the wave equation in the absence of the so-called Lopatinski condition (intrinsic to the wave equation under Neumann boundary conditions); (ii) microlocal analytical estimates which will allow the absorption of tangential wave traces by time derivatives on the boundary; and (iii) recent results involving Carleman's estimates for the wave equation with controlled Neumann part of the boundary. In addition, the project will focus on problems of linear and (globally) nonlinear exact controllability for thermoelastic systems. In particular, thermoelastic PDE's will be considered which have their associated (non-Lipschitz) nonlinearities in place; e.g., the von Karman bracket and the quasilinearities which appear in the modeling of extensible plates. This work will attempt to use, in an essential way, the now-known analyticity of linearized thermoelastic models and our recent stability work for uncontrolled (but fully nonlinear) thermoelastic systems. Examples of coupled partial differential equations (PDE's), such as those to be investigated, have long existed in the literature. However, recent innovations in smart material technology, and the potential applications of these innovations within the context of control engineering design, have greatly increased the interest in these PDE models. The project is aimed at obtaining information about certain qualitative properties of these equations, which in turn can be used to design effective control laws for the structures/interactions that these equations govern. For example, structural acoustic PDE's are used to model the interaction of an aircraft cabin's interior acoustic field with the surrounding walls of the cabin. For the benefit of the passengers, it is desirable to negate or control pressure disturbances that act directly on the interior acoustic field. These disturbances typically emanate from outside the cabin environment; e.g., vibrations due to aircraft engine and propeller noise, or effects due to weather turbulence. In practice, engineers attempt to control this external noise by placing piezoelectric actuators/sensors on a portion of the cabin wall, these devices to act in such a way so as to remove, or at least lessen, the harmful acoustic pressure effects. However, the efficacy of this technology is profoundly sensitive to the shape of the cabin, as well as to the particular region of the cabin walls where the actuators are placed. The goals of this project include: (i) the precise mathematical characterization of those cabin geometries for which active control design by piezoelectric actuation is indeed possible; and (ii) when such control design is practicable, the construction of a reliable method to prescribe the amount and region of control actuation which will be necessary to maintain a calm acoustic field within the cabin.