Interface Control for Systems of Strongly Coupled Partial Differential Equations

交互式系统在技术应用中无处不在，对现代社会至关重要。一类重要的此类耦合系统具有不同动态特性的组件，其耦合发生在每个组件演化的不同介质的界面处。控制动作的作用是迫使系统以期望的、预先指定的方式运行：抑制颤振、抑制湍流、击中目标等。一个有力的例证是气动弹性中的颤振，当结构受到周围气体或流体流的影响时可能发生的现象。由于材料的疲劳，它会导致类似于岩石的不稳定性，这对结构的稳定性是致命的，例如1940年塔科马海峡大桥在强风中倒塌。在这样的流-结构相互作用中，颤振抑制的控制作用可以用来避免结构失效。控制的另一个例子是抑制流体中产生的湍流，作为流体-结构相互作用的一部分。例子包括在动脉壁或静脉内流动的体液和浸没在流体中的固体车辆的运动，无论是在空中飞行的飞机，还是在水中移动的船舶或潜艇等。另一个感兴趣的领域是非线性声学，特别是高强度超声，其在医疗和工业技术中有无数的用途：碎石术，热疗，超声波清洗，裂缝探测，隐藏武器探测，本项目旨在扩展和深化这些和其他重要应用的控制系统的数学基础。研究项目的重点是研究强耦合偏微分方程（PDE）的系统的控制理论问题，其中（主动，被动）的控制行动是行使在两种介质之间的界面处的传输条件。一个例子是双曲线/双曲线相互作用：位于水平面上的冯卡门板（飞机机翼的位移）浸入占据上半空间并在板上移动的3D气流中并与之相互作用。流动的数学模型由修改后的波动方程的流动潜力，允许各种类型的边界条件。通过另一个变量的迹线，在每个方程中发生耦合。通过流的标准化恒定速度决定了状态：亚音速、跨音速和超音速。该系统的数学研究包括：（i）各种模型在各种类型的边界条件下的适定性;（ii）结构的稳定性和吸引子，包括亚音速和超音速状态下颤振抑制的控制技术。另一个例子是抛物线-双曲相互作用，它将容器内的流体-气体方程与描述外壳的完整矢量卡门（壳）动力学方程耦合在一起。在这里，系统的数学研究包括（i）适定性和（ii）通过作用在板上的稳定控制来传递对不稳定流体的耗散的稳定性。在许多声学应用中，只有外部边界是可访问的，而不是内部。控制理论方法也可以应用于声学方程，一个三阶（在时间上）PDE与Dirichlet或Neumann边界控制。具体的研究课题包括：（一）从边界到内部的最优正则性及其轨迹;（二）精确边界可控性;（三）边界稳定性;（四）最优控制和最小最大博弈论与二次成本泛函。该研究涉及一套不同的工具，包括非线性泛函分析，微观局部分析和黎曼几何方法。

Interactive systems are ubiquitous in technological applications and critical for modern society. An important class of such coupled systems have components with differing dynamic properties, whose coupling occurs at the interface of the different media in which each component evolves. The role of a control action is to force the system to behave in a desired, pre-assigned way: suppressing flutter, suppressing turbulence, hitting a target, etc. One eloquent illustration is flutter in aero-elasticity, a phenomenon that may occur when a structure is subject to a surrounding gas or fluid flow. It results in a periodic-like instability that can be fatal for structural stability due to fatigue of the material, such as the collapse of the Tacoma Narrows Bridge in 1940 during strong winds. In such flow-structure interactions, the control action of flutter suppression can be used to avoid structural failure. Another illustration of control is the suppression of turbulence arising in a fluid as part of fluid-structure interaction. Examples include body fluids flowing within arterial walls or veins and motion of a solid vehicle immersed in a fluid, be it an aircraft flying in the air, a ship or a submarine moving in water, etc. Another area of interest is non-linear acoustics, in particular high intensity ultrasound, which has innumerable uses in medical and industrial technology: lithotripsy, thermotherapy, ultrasonic cleansing, detection of cracks, concealed weapon detection, etc. This project aims to extend and deepen the mathematical underpinnings of control systems for these and other important applications.The research project is focused on the study of control-theoretic issues for systems of strongly coupled partial differential equations (PDE), where the (active, passive) control action is exercised in the transmission conditions at the interface between two media. One example is a hyperbolic-like/hyperbolic interaction: a von Karman plate (displacement of an aircraft wing) sitting in the horizontal plane is immersed in, and interacts with, a 3D-gas flow which occupies the upper-half space and moves over the plate. The flow is mathematically modeled by a modified wave equation in terms of the flow potential, allowing various types of boundary conditions. Couplings occurs in each equation through the trace of the other variable. The normalized constant speed of the passing flow determines regimes: subsonic, transonic and supersonic. Mathematical studies of this system include: (i) well-posedness of various models under various types of boundary conditions; (ii) stability and attractors for the structure, to include control techniques for flutter suppression at both subsonic and supersonic regimes. Another example is the parabolic-hyperbolic interaction, which couples a fluid-gas equation inside the vessel with the full vectorial Karman (shell) dynamical equation, describing the external shell. Here again, mathematical studies of the system include (i) well-posedness and (ii) stabilization by means of stabilizing controls acting on the plate to transmit dissipation on the unstable fluid. In many acoustic applications, only the external boundary is accessible, not the interior. Control theoretic methods can also be applied to the acoustic equation, a third order (in time) PDE with either Dirichlet or Neumann boundary controls. Specific topics of study include: (i) optimal regularity from the boundary to the interior and its trace (ii) exact boundary controllability, (iii) boundary stabilization; (iv) optimal control and min-max game theory with quadratic cost functional. The research involves a diverse set of tools including non-linear functional analysis, micro-local analysis, and Riemannian geometry methods.