Control at the interface of strongly coupled partial differential equations

本文主要研究强耦合偏微分方程系统的控制理论问题，其中（主动、被动）控制作用是在两介质界面处的传输条件下进行的。它们包括：稳定性和长期行为/吸引子理论，可控性，最优性，最小-最大博弈论，从单一系统组件的边界测量反演重建等。一个初步的问题是整个动态交互系统的适定性，通常由不同类型的组件组成，例如，抛物线与双曲线。重点讨论了数学和控制理论领域中尚未开发的问题，例如：气弹性中的气流-结构相互作用；弹性结构-流动与运动界面的相互作用；结构声相互作用与非浅壳作为主动弯曲壁和非线性声学Westervalt-Kuznetesov室（基于傅立叶热通量定律）或Jordan-Moore-Gibson-Thompson室（基于Maxwell-Cattaneo定律），两者都出现在高强度超声聚焦中。数学控制理论在微分方程的研究中注入了外部主体或操作者的主动观点，他们不满足于仅仅被动地观察或思考自然现象的演变；而是寻求干预，影响和修改其动态行为，以达到预定的结果。天体力学为被动的、非受控的观点提供了一个确定的例子：天文学家可以计算轨道，预测日食的时间和位置，但不能改变这些现象。相比之下，政府在社会或经济问题上的行动——如减少失业或刺激增长——说明了积极的、基于控制的战略。分析与综合。本项目应用合成的主动控制方法来研究复杂的多面现象或组合单元-在物理世界中无处不在-由强耦合偏微分方程系统组成，其中控制当局只能访问两种不同介质之间的界面。抑制湍流或颤振在流体-结构或空气/气体流动-结构相互作用。或者在声学室（如飞机内部）实现降噪。或在医疗应用中获得高强度超声聚焦，从碎石或热治疗到超声清洗和焊接。

The present proposal is focused on the study of control-theoretic issues for systems of strongly coupled PDEs, where the (active, passive) control action is exercised in the transmission conditions at the interface between two media. They include: stabilization and long-time behavior/theory of attractors, controllability, optimality, min-max game theory, inverse reconstruction from boundary measurements of a single system component, etc. A preliminary question is well-posedness of the overall dynamically interactive system, typically consisting of components of different type, e.g., parabolic versus hyperbolic. Emphasis is given to problems in mathematical and control-theoretic areas which are rather unexplored, such as: gas flow-structure interaction in aeroelasticity; elastic structure-flow interaction with moving interface; structural acoustic interaction with a non-shallow shell as active curved wall and a nonlinear acoustic Westervalt-Kuznetesov chamber (based on Fourier Law of Heat Flux) or Jordan-Moore-Gibson-Thompson chamber (based on Maxwell-Cattaneo Law), both arising in High Intensity Ultrasound focusing.Mathematical control theory injects in the study of differential equations the active viewpoint of an external agent or manipulator, who is not satisfied with just passively observing or contemplating the evolution of a natural phenomenon; but instead seeks to intervene as to influence and modify its dynamical behavior, in order to achieve a preassigned outcome. Celestial mechanics offers an established example of the passive, non-controlled viewpoint: The astronomer may calculate orbits, predict time and location of eclipses, but cannot alter these phenomena. In contrast, governmental action on social or economic issues---such as reducing unemployment or stimulating growth---illustrates the active, control-based strategy. Analysis versus synthesis. The present project applies the active control approach of synthesis to study complex multi-faceted phenomena or combined units---ubiquitous in the physical world---consisting of systems of strongly coupled partial differential equations, where the controlling authority has access only to the interface between two different media. To suppress turbulence or flutter in fluid-structure or air/gas flow-structure interactions. Or to achieve noise reduction in an acoustic chamber, such as the interior of an aircraft. Or to obtain high intensity ultrasound focusing in medical applications, ranging from lithotripsy or thermo-therapy to ultrasound cleaning and welding.