Mathematical Analysis and Control of Interactive Partial Differential Equations

该项目研究的重点是为那些对某些控制工程和物理现象进行建模的交互式偏微分方程（PDE）系统开发鲁棒控制理论。这样的系统包括不同PDE动力学的耦合，这些动力学中的每一个通常在其自身的具有边界的域上演化，并且耦合通常经由各种边界界面来实现。项目研究人员应考虑三维流体-结构相互作用的均匀稳定问题;该PDE构成斯托克斯流和弹性系统的耦合。特别是，研究人员将试图表明，这种流体结构偏微分方程的渐近行为可以统一控制的边界功能，制定的边界上的结构（或可能的流体，虽然边界控制通过结构有一个直接的药理学解释）。在对这种线性化动力学的边界控制过程有了一定的了解之后，研究将转向成熟的非线性流体-结构动力学，这涉及到Navier-Stokes方程与弹性系统的耦合。此外，还将努力研究所谓的复合材料“三明治”板模型、三维结构声学系统、和热弹性偏微分方程，接触到一个移动的障碍物，并将有一个非线性产生的摩擦热。这项研究的主要目的是获得一个大的和精确的理解，这些偏微分方程（偏微分方程的），属于所谓的混合结构类，这种理解将最终在控制工程和设计的影响。混合结构-通常描述某些给定的物理现象-是一个方程，它包括两个或多个不同的PDE动力学，通常通过一些复杂的耦合机制，这是内在的基础物理建模。这些耦合系统已经吸引并可能继续吸引数学界的极大兴趣，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。 从这种混合偏微分方程的数学研究中收集到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，该项目的一部分将涉及三维结构声学PDE，因为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。控制设计过程涉及在飞机壁的一部分上放置压电装置;随后通过这些装置施加电压以衰减所述噪声。主要研究人员的目标是三维结构声学系统的结果，这些结果将建议机身应采取的精确配置，以达到最佳的噪声衰减。

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.