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 Avalos该项目需要对已知用于控制结构声学和热/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。在我们工作的第一部分中，我们将研究耦合双曲和抛物型偏微分方程的模型结构声学系统的作用下的无界逐点观测和控制，这里的重点是优化和精确边界可控性的问题。在特定的结构声学控制设计应用中，为了衰减声腔中的外部噪声的目的而实施逐点边界控制。此外，点边界控制的功效通过声压的点观测来测量。因此，支配偏微分方程模型涉及高度无界的控制和观测算子。在考虑控制和观测结构声学PDE的最优控制之前，必须首先建立这些点观测和控制影响下的动力学适定性。这一努力，以建立规律性将需要新的技术，伪微分分析，以建立微观局部偏微分方程估计，不能获得全球性的。此外，在PDE系统的优化可以进行之前，必须有一个状态空间的识别和表征，允许逐点观察的声压。这种正确表征状态空间的努力也将具有很重的微局部分量。随后，我们将考虑各种二次优化方案的结构声学PDE的下，这样的点控制和观察，着眼于发展的特征Riccati方程和有限元逼近的最优控制律。该项目的第二阶段涉及我们以前的研究的可控性和稳定性的一个现在经典的热弹性系统的延续。特别是，我们试图延长最近的结果的精确-近似可控性的情况下，热体积膨胀系数被允许随板材料的性能而变化。此外，我们将研究热弹性线性系统的零能控性的长期问题。这两个可控性调查将采用新的Carleman和pseudodifferential乘数和技术，而不是标准的差分乘数方法。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。这个项目的动机是从这样一个事实，即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的类。不严格地说，“智能”材料或结构是具有嵌入在其一个或多个组件中的小型化控制系统的材料或结构，以便诱导期望的结果或物理状态。例如，该控制系统可以以这样的方式起作用，以便消除不希望的外部影响，或者帮助材料获得最佳形状（相对于某些预定的设计规范是最佳的）。这样的智能材料的例子-这些包括传感器和致动器-可以在结构、结构声学、热/结构和流体/结构相互作用系统中找到。这些复合材料结构的数学建模往往最终出现上述耦合系统的PDE的。此外，由于控制机构的各种部件通常固定到结构的边界，或嵌入其介质内，和/或在结构的指定点处并置，描述特定相互作用的PDE的相关系统将具有存在于其中的控制输入项。本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理解和预见主动控制对复合材料的影响。此外，通过对这些偏微分方程模型进行严格的数值模拟，我们的目的是生成近似的控制律，可以实现“真实的--时间”的工程控制设计。