Exact Controllability and Observation of Structural Acoustics and Thermoelastic Systems

[20881 . 12]本项目旨在研究控制腔室内结构声流的耦合偏微分方程系统的精确边界可控性。还将研究二维热弹性系统的精确边界和零边界可控性问题。在某种程度上，这项工作将涉及对双重问题的研究；即齐次伴随方程解的相关可观察性不等式的获得。与预期的工程应用相一致，重点将放在允许在尽可能小的（边界）控制区上控制结构声学动力学的情况上。此外，本项目旨在寻找几何和规定控制的条件，以便仅对声室的柔性部分进行控制，对于有限能量的任意初始数据，人们将具有声流的精确可控性。预计这项工作的关键要素将包括以下内容：(i)在没有所谓Lopatinski条件（在诺伊曼边界条件下波动方程的固有条件）的情况下波动方程的明显迹规律性；（ii）允许边界上的时间导数吸收切向波迹的微局部分析估计；（iii）最近的结果，涉及对边界受控制的诺伊曼部分波动方程的Carleman估计。此外，该项目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。特别地，将考虑具有相关（非lipschitz）非线性的热弹性PDE；例如，von Karman支架和拟线性出现在可扩展板的建模中。这项工作将试图以一种基本的方式，利用现在已知的线性化热弹性模型的解析性和我们最近对非受控（但完全非线性）热弹性系统的稳定性工作。耦合偏微分方程（PDE's）的例子，如那些要研究的，早就存在于文献中。然而，最近智能材料技术的创新，以及这些创新在控制工程设计背景下的潜在应用，极大地增加了人们对这些PDE模型的兴趣。该项目旨在获得有关这些方程的某些定性性质的信息，这些信息反过来可用于为这些方程所控制的结构/相互作用设计有效的控制律。例如，结构声学PDE用于模拟飞机客舱内部声场与客舱周围墙壁的相互作用。为了乘客的利益，需要消除或控制直接作用于内部声场的压力干扰。这些干扰通常来自客舱外部环境；例如，由于飞机发动机和螺旋桨噪声引起的振动，或由于天气乱流引起的影响。在实践中，工程师们试图通过在机舱壁上放置压电致动器/传感器来控制这种外部噪音，这些装置以这种方式工作，从而消除或至少减少有害的声压影响。然而，这项技术的功效对机舱的形状以及放置执行器的舱壁的特定区域非常敏感。这个项目的目标包括：(i)精确的数学表征这些座舱几何形状，其中压电驱动的主动控制设计确实是可能的；（ii）当这种控制设计可行时，建立一种可靠的方法来规定控制驱动的数量和区域，这将是保持机舱内平静声场所必需的。

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.