Stabilization and Control in Nonlinear Structural-Acoustics, Magnetic Imaging, and Elasticity

该奖项由2009年《美国复苏和再投资法案》(公法111-5)资助。该项目集中在由非线性双曲型偏微分方程控制的动力系统：(I)通过限制在边界或区域内部的子集的非线性反馈来稳定电磁场；(Ii)结构与Reissner-Mindlin板模型所描述的弹性组件的声学相互作用的边界控制；(Iii)稳定在电磁场影响下结构振动的声噪声；(Iv)带记忆项和非线性阻尼的波动方程的稳定性和吸引子。主要目标之一是在这些系统的背景下，研究在某种意义上受到限制的控制和能量耗散机制：或者几何地限制到物理域的一部分，和/或在欠阻尼和过阻尼系统中的反馈的“强度”中。这项工作还将解决具有非耗散控制的方程(例如，Euler-Bernoulli梁和Kirchhoff板的剪力反馈)，在某些情况下，这些方程更适合实现，但其能量衰减效应并不明显，只能通过专门的技术来研究。该项目旨在建立操纵/稳定系统所需的控制的几何、初始数据和结构条件，或至少确保全局吸引子的某些性质。这项研究有望对分布参数系统(例如声学和机械振动、热效应、电磁场)控制的工程设计产生建设性的影响。麦克斯韦方程和电磁辐射稳定化是天线设计、非线性光学、半导体-超导体建模等领域的研究热点。结构-声学相互作用问题出现在从主动噪声控制到智能材料设计的各种领域。特别是，声磁弹性耦合的研究有助于理解如何最大限度地减少磁共振成像(MRI)设备中梯度线圈的噪声。人们希望设计出具有最小侵入性且便于实施的控制装置，从而促使对受限于空间及其强度的执行器和能量阻尼器进行研究。目标是创造条件，在这种条件下，这种机制对系统提供充分的控制，或者在无法实现充分有效性的情况下量化其不足之处。这项工作还将与研究和教育中的应用数值分析相联系：将开发本科生和研究生课程的补充项目，提供研究偏微分方程的数值方法的较低层次介绍；较长期的目标将是用数值算法加强上述研究的某些方面，以促进这些理论结果的实际应用。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project centers on control of dynamical systems governed by nonlinear hyperbolic partial differential equations: (i) stabilization of electromagnetic fields via nonlinear feedbacks restricted to a subset of the boundary or the interior of the domain; (ii) boundary control of structure-acoustic interactions with the elastic component described by the Reissner-Mindlin plate model; (iii) stabilization of acoustic noise from structures vibrating under influence of electromagnetic fields; (iv) stability and attractors for wave equations with memory terms and nonlinear damping. One of the primary goals is to investigate, in the context of these systems, control and energy dissipation mechanisms that are restricted in some sense: Either geometrically to a portion of the physical domain, and/or in the "strength" of the feedback as in under- and over-damped systems. The work will also address equations with non-dissipative controls (e.g. sheer force feedbacks for Euler-Bernoulli beams and Kirchhoff plates) which, in some cases, are more suitable for implementation, but whose energy-damping effects are not apparent and can only be studied via specialized techniques. The project is aimed at establishing the conditions on geometry, initial data, and structure of the controls necessary to steer/stabilize the system or, at least, ensure certain properties of the global attractors.This research is expected to constructively impact engineering design in control of distributed parameter systems (e.g. acoustic and mechanical vibrations, thermal effects, electro-magnetic fields). Maxwell equations and stabilization of electromagnetic radiation arise in antenna design, nonlinear optics, semiconductor-superconductor modeling. Structure-acoustic interaction problems show up in a variety of areas ranging from active noise control to design of smart materials. In particular, study of acoustic-magneto-elastic coupling helps understand how to minimize noise from the gradient coils in magnetic resonance imaging (MRI) devices. It is desirable to engineer controls that are minimally invasive and convenient for implementation, thus, prompting investigation of actuators and energy dampers that are restricted in space and their strength. The goal is to establish conditions under which such mechanisms provide sufficient control over the system, or to quantify their deficiencies when full effectiveness is unattainable. This work will also connect with applied numerical analysis both in research and in education: Supplementary projects for undergraduate and graduate course will be developed offering a lower-level introduction to numerical methods for studying partial differential equations; a longer-term objective will be to augment some aspects of the above research with numerical algorithms in order to facilitate practical applications of these theoretical results.