Exact Controllability and Observation of Structural Acoustics and Thermoelastic Systems

0208121 Avalos这个项目是关于研究精确的边界可控性特性的耦合偏微分方程（PDE）的控制结构声室内的流动。精确和零边界可控性问题的二维系统的热弹性也将研究。在某种程度上，这项工作将需要研究的双重问题，即实现相关的观测不平等的解决方案齐次伴随方程。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，该项目的目的是寻找几何形状和规定控制的条件，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，将具有声流的精确可控性。预计这项工作的主要内容将包括以下内容：（i）在没有所谓的Lopatinski条件的情况下，波动方程的锐迹正则性（内在的波动方程下诺依曼边界条件）;（ii）微观局部分析估计，这将允许吸收的切向波的轨迹的时间导数的边界上;和（iii）最近的结果涉及Carleman的估计波动方程与控制Neumann部分的边界。此外，该项目将侧重于热弹性系统的线性和（全局）非线性精确可控性问题。特别地，将考虑具有适当的相关（非Lipschitz）非线性的热弹性PDE;例如，的vonKarman括号和拟线性出现在可伸展板的建模。这项工作将尝试使用，在一个重要的方式，现在已知的线性化热弹性模型的解析性和我们最近的不受控制的（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）的例子，如那些被调查，早已存在于文献中。然而，最近在智能材料技术的创新，以及这些创新的控制工程设计的背景下的潜在应用，大大增加了这些偏微分方程模型的兴趣。该项目旨在获得有关这些方程的某些定性性质的信息，这些信息反过来又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用于对飞行器机舱的内部声场与机舱的周围壁的相互作用进行建模。为了乘客的利益，希望消除或控制直接作用于内部声场的压力扰动。这些干扰通常从机舱环境外部发出;例如，由于飞机发动机和螺旋桨噪音引起的振动，或由于天气湍流引起的影响。在实践中，工程师试图通过将压电致动器/传感器放置在机舱壁的一部分上来控制这种外部噪声，这些设备以这样的方式起作用，以便消除或至少减轻有害的声压效应。然而，该技术的功效对机舱的形状以及机舱壁的放置致动器的特定区域非常敏感。该项目的目标包括：（i）精确的数学表征的机舱几何形状的主动控制设计的压电驱动确实是可能的;和（ii）当这样的控制设计是可行的，一个可靠的方法来规定的数量和区域的控制驱动，这将是必要的，以保持一个平静的声场机舱内的建设。

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.