A Mathematical Control Theory for the Partial Differential Equations of Thermal/Structure and Structural Acoustic Interactions

9972349 avalos这个项目需要对那些已知控制结构声学和热/结构相互作用的耦合偏微分方程（PDE）系统的控制问题进行数学分析。在我们的工作的第一部分，我们将研究耦合的双曲和抛物线-类PDE模型在无界点观测和控制作用下的结构声系统，重点是优化和精确边界可控性问题。在一个特殊的结构声学控制设计应用中，点向边界控制是为了衰减声室中的外部噪声而实现的。此外，还通过对声压的点观测来衡量点边界控制的效果。因此，控制PDE模型包含高度无界的控制和观测算子。在考虑对被控和观测的结构声学PDE进行最优控制之前，必须首先建立这些点观测和控制影响下的动力学的适定性。这种建立规律性的努力将需要新的伪微分分析技术来建立无法获得全局的微局部PDE估计。此外，在对PDE系统进行优化之前，必须对状态空间进行识别和表征，以便对声压进行逐点观测。这种适当地描述状态空间的努力也将包含大量的微局部组件。随后，我们将考虑在这种点控制和观测下的结构声学PDE的各种二次优化方案，以期建立表征Riccati方程和最优控制律的有限元逼近。该项目的第二阶段涉及我们先前对现在经典热弹性系统的可控性和稳定性特性的研究的延续。特别是，我们试图将最近的精确-近似可控性的结果扩展到允许热体膨胀系数随板材料的性质而变化的情况。此外，我们将研究长期存在的热弹性线性系统的零可控性问题。这两种可控性研究都将采用新的Carleman和伪微分乘法器和技术，而不是标准的微分乘法器方法。在没有转动惯性的情况下，热弹性系统的非线性稳定问题也将被考虑。这个项目的动机来自于这样一个事实，即在智能材料技术的开发和应用中经常出现耦合PDE的类别。粗略地说，“智能”材料或结构是指在其一个或多个组件中嵌入了小型化的控制系统，从而产生期望的结果或物理状态。例如，这种控制系统可能以这样一种方式工作，以消除不必要的外部影响，或帮助材料获得最佳形状（相对于某些预定的设计规范的最佳形状）。这种智能材料的例子——包括传感器和执行器——可以在结构、结构声学、热/结构和流体/结构相互作用系统中找到。这些复合结构的数学建模通常会在前面提到的PDE耦合系统的出现中达到高潮。此外，由于控制机制的各种组件通常附着在结构的边界上，或嵌入其介质中，和/或在结构的指定点上搭配，描述特定相互作用的PDE的相关系统将在其中存在控制输入项。该项目的主要目的是通过分析复合材料的控制方程的行为，更好地理解和预见主动控制对复合材料的影响。此外，通过对这些PDE模型进行严格的数值模拟，我们的目的是生成可用于“实时”工程控制设计的近似控制律。

9972349AvalosThis project entails the mathematical analysis of control problems for those systems of coupled partial differential equations (PDE's) known to govern structural acoustic and thermal/structure interactions. In the first part of our work, we will study the coupled hyperbolic and parabolic--like PDE's which model structural acoustic systems under the action of unbounded pointwise observation and control, with a focus here upon issues of optimization and exact boundary controllability. In a particular structural acoustic, control design application, pointwise boundary control is implemented for the purpose of attenuating external noise in an acoustic chamber. Moreover, the efficacy of the point boundary control is measured by point observations of the acoustic pressure. Accordingly, the governing PDE model involves highly unbounded control and observation operators. Before one can consider the optimal control of the controlled and observed structural acoustic PDE, wellposedness of the dynamics under the influence of these point observations and control must first be established. This effort to establish regularity will require new techniques of pseudodifferential analysis to establish microlocal PDE estimates that cannot be obtained globally. Moreover, before optimization of the PDE system can proceed, there must be an identification and characterization of a state space which allows pointwise observations of the acoustic pressure. This effort to properly characterize the state space will also have a heavy microlocal component. Subsequently, we will consider various quadratic optimization schemes for the structural acoustic PDE's under such point control and observation, with a view toward developing a characterizing Riccati Equation and finite element approximation of the optimal control laws. The second phase of the project involves a continuation of our previous studies on controllability and stability properties of a now classical system of thermoelasticity. In particular, we attempt to extend a recent result of exact--approximate controllability to the case where the coefficient of thermal volume expansion is allowed to vary with the properties of the plate material. In addition, we will study the longstanding problem of null controllability for the linear system of thermoelasticity. Both these controllability investigations will employ novel Carleman and pseudodifferential multipliers and techniques, as opposed to standard differential multiplier methods. Nonlinear stabilization problems for systems of thermoelasticity in the absence of rotational inertial will also be considered.The motivation for this project is drawn from the fact that the classes of coupled PDE's to be considered frequently arise in the development and application of smart materials technology. Loosely speaking, a "smart" material or structure is that which has a miniaturized control system embedded within one or more of its components, so as to induce a desired result or physical state. For instance, this control system might function in such a way so as to negate unwanted external influences, or to assist the material in attaining an optimal shape (optimal with respect to some predetermined design specification). Examples of such smart materials---these comprising both sensor and actuator---may be found in structural, structural acoustic, thermal/structure and fluid/structure interaction systems. The mathematical modeling of these composite structures will often culminate in the appearance of the aforementioned coupled systems of PDE's. Moreover, inasmuch as the various components of the control mechanism are typically affixed to the structure's boundary, or embedded within its medium, and/or in collocation at specified points of the structure, the associated system of PDE's which describes the particular interaction will have a control input term present therein. The principal intent of this project then is to obtain a better understanding and foresight concerning the effect of active control on composite materials by analyzing the behavior of their corresponding governing equations. In addition, by performing rigorous numerical simulations on these PDE models, it is our intent to generate approximate control laws that can be implemented for "real--time" engineering control design.