Mathematical Analysis and Control of Interactive Partial Differential Equations

交互偏微分方程的数学分析与控制

• 批准号: 0606776
• 负责人: George Avalos
• 金额: $ 12万
• 依托单位: University of Nebraska-Lincoln
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0606776&HistoricalAwards=false
• 关键词: Mathematical; Analysis; Control; Interactive; Partial

The project research will focus on the development of a robust control theory for those interactive partial differential equation (PDE) systems which model certain control engineering and physical phenomena. Such systems comprise a coupling of distinct PDE dynamics, each of these typically evolving on its own domain with boundary, and with the coupling typically being accomplished via the various boundary interfaces. The project investigator shall consider the problem of uniform stabilization for a fluid-structure interaction in three dimensions; this PDE constitutes a coupling of Stokes flow and a system of elasticity. In particular, the investigator will attempt to show that the asymptotic behavior of such fluid-structure PDEs can be uniformly controlled by boundary functions, enacted on the boundary of the structure (or possibly of the fluid, although boundary control via the structure has a direct pharmacological interpretation). Having gained some understanding of the boundary control process for this linearized dynamics, the research will in turn be directed to full-blown, nonlinear fluid-structure dynamics, this involving a coupling of the Navier-Stokes equations with systems of elasticity. In addition, research efforts will be made to understand controllability and optimal control theoretic properties of so-called composite "sandwich" plate models, three dimensional structural acoustic systems, and thermoelastic PDEs which come into contact with a moving obstacle and for which there will be a nonlinear generation of frictional heat.The principal intent of this research is to gain a large and precise understanding of those partial differential equations (PDE's) which fall within the class of so-called hybrid structures; such understanding will eventually have implications in control engineering and design. A hybrid structure--which typically describes some given physical phenomenon--is an equation which comprises two or more distinct PDE dynamics, usually by means of some sophisticated coupling mechanism which is intrinsic to the underlying physical modelling. These coupled systems have attracted, and will probably continue to attract, large interest within the mathematical community, inasmuch as such equations model physical interactions which are observed in the realms of science and engineering: specific examples include those under consideration in this project; namely, fluid-structure interactions, structural acoustic interactions, thin plates under the influence of thermal effects and frictional contact, and the "sandwich" plate models used to describe certain structures in material sciences. Qualitative information gleaned from a mathematical study of such hybrid PDEs can in fact have broader impacts upon other disciplines. For example, part of the project will deal with a three dimensional structural acoustic PDE, as it appears in the modelling of certain engineering design methodologies for the control of external engine noise entering the fuselage of an aircraft. The control design process involves the placement of piezoelectric devices on part of the aircraft wall; a voltage is subsequently applied through these devices to attenuate said noise. The principal investigator is aiming for results for three dimensional structural acoustic systems which will suggest the precise configuration the fuselage should take, so as to attain optimal noise attenuation.

该项目研究将侧重于为那些模拟某些控制工程和物理现象的交互式偏微分方程（PDE）系统开发鲁棒控制理论。此类系统包含不同 PDE 动力学的耦合，每个系统通常在其自己的边界域上演化，并且耦合通常通过各种边界接口完成。项目研究者应考虑三维流固耦合的均匀稳定问题；该偏微分方程构成了斯托克斯流和弹性系统的耦合。特别是，研究者将尝试证明这种流体结构偏微分方程的渐近行为可以通过在结构（或可能是流体，尽管通过结构的边界控制具有直接的药理学解释）边界上制定的边界函数来统一控制。在对这种线性动力学的边界控制过程有了一定的了解后，研究将转向全面的非线性流体结构动力学，这涉及纳维-斯托克斯方程与弹性系统的耦合。此外，还将开展研究工作，以了解所谓的复合“三明治”板模型、三维结构声学系统以及与移动障碍物接触并会产生非线性摩擦热的热弹性偏微分方程的可控性和最优控制理论特性。这项研究的主要目的是获得对属于以下类别的偏微分方程（PDE）的广泛而精确的理解： 所谓的混合结构；这种理解最终将对控制工程和设计产生影响。混合结构——通常描述一些给定的物理现象——是一个包含两个或多个不同的偏微分方程动力学的方程，通常通过底层物理建模所固有的一些复杂的耦合机制来实现。这些耦合系统已经吸引了并且可能会继续吸引数学界的巨大兴趣，因为这些方程模拟了在科学和工程领域中观察到的物理相互作用：具体的例子包括本项目中正在考虑的那些；即流体-结构相互作用、结构声学相互作用、热效应和摩擦接触影响下的薄板，以及用于描述材料科学中某些结构的“三明治”板模型。 从此类混合偏微分方程的数学研究中收集的定性信息实际上可以对其他学科产生更广泛的影响。例如，该项目的一部分将处理三维结构声学偏微分方程，它出现在某些工程设计方法的建模中，用于控制进入飞机机身的外部发动机噪声。控制设计过程涉及将压电器件放置在飞机壁的一部分上；随后通过这些装置施加电压以衰减所述噪声。首席研究员的目标是获得三维结构声学系统的结果，这将建议机身应采取的精确配置，以获得最佳的噪声衰减。