Mathematical Analysis and Control of Interactive Partial Differential Equations

该项目的研究将集中于发展一种对某些控制工程和物理现象进行建模的交互偏微分方程(PDE)系统的鲁棒控制理论。这样的系统包括不同的PDE动力学的耦合，其中每一个通常在其自己的域上随着边界而演变，并且该耦合通常通过各种边界接口来实现。项目调查员应考虑三维流固相互作用的均匀稳定问题；该PDE由Stokes流和弹性系统耦合而成。特别是，研究人员将试图证明，这种流体结构PDE的渐近行为可以由结构(或可能是流体)边界上的边界函数统一控制，尽管通过结构进行的边界控制具有直接的药理学解释。在对这种线性化动力学的边界控制过程有了一些了解后，研究将转向全面的、非线性的流体-结构动力学，这涉及到Navier-Stokes方程和弹性系统的耦合。此外，还将努力了解所谓的“夹层”板模型、三维结构声学系统和热弹性偏微分方程组的可控性和最优控制理论性质，这些模型与移动的障碍物接触，并将产生非线性摩擦热。本研究的主要目的是对属于所谓混合结构的偏微分方程组有一个广泛而精确的理解，这种理解最终将对控制工程和设计产生影响。混合结构--它通常描述一些给定的物理现象--是一个由两个或更多不同的PDE动力学组成的方程，通常借助于一些复杂的耦合机制，这是基本物理模型所固有的。这些耦合系统已经并可能继续引起数学界的极大兴趣，因为这样的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。从对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，该项目的一部分将处理三维结构声学PDE，就像它在控制进入飞机机身的外部发动机噪声的某些工程设计方法的建模中所显示的那样。控制设计过程包括在飞机机壁的一部分放置压电装置；随后通过这些装置施加电压以减弱所述噪音。主要研究人员的目标是三维结构声学系统的结果，该结果将建议机身应该采用的精确外形，以便获得最佳的噪声衰减。

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.