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

热/结构和结构声相互作用的偏微分方程的数学控制理论

• 批准号: 9972349
• 负责人: George Avalos
• 金额: $ 8.09万
• 依托单位: Texas Tech University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-15 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9972349&HistoricalAwards=false
• 关键词: Mathematical; Control; Theory; Partial; Differential

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.

9972349价值这个项目需要对那些控制结构、声学和热/结构相互作用的耦合偏微分方程组(PDE)的控制问题进行数学分析。在我们的第一部分工作中，我们将研究在无界逐点观测和控制作用下的耦合双曲和抛物型偏微分方程组，它模拟结构声学系统，这里的重点是最优化和精确边界可控性问题。在一种特殊的结构声学控制设计应用中，实现逐点边界控制的目的是为了衰减声室中的外部噪声。此外，通过对声压的点观测来衡量点边界控制的有效性。因此，控制PDE模型涉及高度无界的控制和观测算子。在考虑受控和观测的结构声学偏振器的最优控制之前，必须首先确定在这些点观测和控制影响下的动力学适定性。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在全球范围内获得的微局部PDE估计。此外，在进行PDE系统的优化之前，必须识别和表征允许逐点观测声压的状态空间。这种适当地刻画状态空间的努力也将具有沉重的微局部成分。随后，我们将考虑在这样的点控制和观测下结构声学偏微分方程的各种二次优化方案，以期建立一个特征化的Riccati方程和最优控制律的有限元逼近。该项目的第二阶段包括我们先前关于现在经典的热弹性系统的可控性和稳定性性质的研究的继续。特别地，我们尝试将最近的精确-近似可控性结果推广到允许热体积膨胀系数随板材料的性质变化的情况。此外，我们还将研究线性热弹性系统的零能控性问题。这两个可控性研究都将使用新的Carleman和伪微分乘法器和技术，而不是标准的微分乘法器方法。本研究的动机来自于在智能材料技术的发展和应用中经常出现的耦合偏微分方程类。宽泛地说，“智能”材料或结构是指在其一个或多个组件中嵌入一个微型控制系统，以诱导期望的结果或物理状态的材料或结构。例如，该控制系统可能以这样一种方式起作用，以消除不想要的外部影响，或帮助材料获得最佳形状(相对于某些预定的设计规范而言是最佳的)。这种智能材料的例子-包括传感器和致动器-可以在结构、结构声学、热/结构和流体/结构交互系统中找到。这些复合结构的数学建模通常将最终出现前述的PDE的耦合系统。此外，由于控制机构的各种组件通常被固定在结构的边界上，或嵌入在其介质中，和/或在结构的指定点的配置中，描述特定相互作用的关联的PDE系统将具有其中存在的控制输入项。因此，这个项目的主要目的是通过分析其相应的控制方程的行为来更好地了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。