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

该奖项是根据2009年美国复苏和再投资法案资助的（公法111-5）.该项目的中心是由非线性双曲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）用Reissner-Mindlin板模型描述的弹性元件对结构-声相互作用进行边界控制;（iii）在电磁场影响下结构振动的噪声稳定;（iv）具有记忆项和非线性阻尼的波动方程的稳定性和吸引子。主要目标之一是调查，在这些系统的背景下，控制和能量耗散机制，在某种意义上是有限的：无论是几何的一部分物理域，和/或在“强度”的反馈，在欠阻尼和过阻尼系统。 这项工作还将解决方程与非耗散控制（例如剪切力反馈的欧拉-伯努利梁和基尔霍夫板），在某些情况下，更适合实施，但其能量阻尼效果并不明显，只能通过专门的技术进行研究。该项目的目的是建立的几何条件，初始数据，和必要的控制转向/稳定系统的结构，或至少，确保某些属性的全球attractor.This研究预计将建设性地影响工程设计中的分布参数系统（如声学和机械振动，热效应，电磁场）的控制。 麦克斯韦方程组和电磁辐射的稳定性出现在天线设计、非线性光学、超导体-超导体模型中。结构-声学相互作用问题出现在从有源噪声控制到智能材料设计的各个领域。特别是，声磁弹性耦合的研究有助于了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。这是可取的工程控制，是微创和方便的实施，因此，促进调查的致动器和能量阻尼器，在空间和它们的强度受到限制。目标是建立条件，使这些机制能够对系统进行充分控制，或在无法实现充分有效性时量化其不足之处。这项工作也将与应用数值分析在研究和教育：补充项目的本科和研究生课程将开发提供一个较低层次的介绍数值方法研究偏微分方程;一个长期的目标将是增加上述研究的某些方面与数值算法，以促进这些理论成果的实际应用。

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.