Analysis and Control of Mathematical Models of Fluttering Plates

气动弹性领域的基本问题之一是气动弹性颤振不稳定性的预测和控制。当气动载荷激发柔性结构的固有振动模态时，柔性结构在气流中发生颤振；其结果是结构的位移和气体流场中的扰动之间潜在的剧烈相互作用。这种现象可能出现在许多应用中，包括：强风中的建筑物和桥梁，旗帜状结构，人体呼吸系统以及空中和陆地车辆上的面板和襟翼结构。在飞机的环境中，颤振是一个关键问题。如果颤振引起的结构位移量足够大，结构就会发生破坏。长时间持续的小振荡也可能给结构带来代价高昂和/或危险的疲劳。最近，利用颤振（自然发生的或诱发的）的想法已被建议通过压电“收获”提供一种替代能源。由于这些原因，人们对建立描述颤振现象的数学模型非常感兴趣，以便深入了解流-结构耦合的机制，并根据其物理参数预测系统的动力学。该项目包括控制偏微分方程（PDEs）分析与面板和襟翼颤振相关的主要模型。从PDE分析得出的结果是有价值的，原因有很多。它们：(i)指导和简化实验和数值颤振阈值的确定，（ii）可以提高实验的成本效益并缩短原型的设计时间，（iii）表明对于给定的流板配置，哪种类型和位置的阻尼是有效的。所提出的研究是基于最近在气动弹性方面的进展，该进展允许对过去50年来使用的经典流板模型进行扩展。由于界面处动力学不匹配规律和耦合中迹项定义不清或无界等问题，对流-结构系统的计算结果要求很高。双曲轨迹正则性理论、抽象耦合模型、带延迟的偏微分方程和几何约束阻尼的最新进展使流板模型的偏微分方程分析变得易于处理。本文主要研究一类非线性流板模型在反馈控制下的适定性和稳定性，其中包括部分自由板边界条件和板边缘附近的动态流板边界条件。考虑了结构面内、面外运动和非线性流体的全非线性模型。此外，最近的稳定性分析将扩展到中间的“跨音速”流动模式和活塞理论的高超音速模式。除了这些模型的适定性之外，动力学的时间收敛特性（即吸引子）将被考虑，以确定系统的非瞬态行为对板的边界条件和外部载荷的敏感性。目前的建议可以看作是对气动弹性模型的分析，通过将适定偏微分方程的定性性质与实验观察和/或数值近似行为进行比较。

One of the fundamental problems in the field of aeroelasticity is the prediction and control of the instability known as aeroelastic flutter. Flutter occurs in a flexible structure immersed in a gas flow when aerodynamic loading excites the natural oscillatory modes of the structure; the result is a potentially violent interaction between the displacements of the structure and perturbations in the gas flow field. This phenomenon may occur in a multitude of applications including: buildings and bridges in strong winds, flag-like structures, the human respiratory system, and panel and flap structures on air and land vehicles. In the context of aircraft, flutter is a key concern. If the magnitude of the structural displacements due to flutter is sufficiently large, structural failure can occur. Small oscillations sustained over long periods of time may also bring about costly and/or hazardous fatigue in the structure. Very recently, the idea of harnessing flutter (naturally occurring, or induced) has been suggested to provide an alternative source of energy via piezoelectric "harvesting". For these reasons there is great interest in producing mathematical models that describe the flutter phenomenon in order to gain insight into the mechanisms of flow-structure coupling and predict the dynamics of the system based on its physical parameters. This project comprises a control of partial differential equations (PDEs) analysis of the principal model associated to panel and flap flutter. Results derived from PDE analyses are valuable for a variety of reasons. They: (i) guide and streamline experimental and numerical flutter threshold determination, (ii) can improve cost-effectiveness of experimentation and cut-down on design time of prototypes, (iii) indicate what types and locations of damping will be effective for a given flow-plate configuration. The proposed investigations are based upon very recent progress in aeroelasticity that has permitted extensions of a classical flow-plate model used over the last 50 years. Obtaining results for flow-structure systems is demanding, as problems arise in the mismatching regularity of dynamics at the interface and ill defined or unbounded trace terms in the coupling. Recent advances in hyperbolic trace regularity theory, abstract coupled models, PDEs with delay, and geometrically constrained damping make modern PDE analysis of flow-plate models tractable. This proposal centers on well-posedness and stability in the presence of feedback controls for a class of nonlinear flow-plate models which include partially free plate boundary conditions and dynamic flow boundary conditions near plate edges. Fully nonlinear models accounting for both in-plane and out-of-plane motion in the structure, as well as nonlinear fluids, are considered. Moreover, recent stability analyses will be extended to the intermediary "transonic" flow regime and the piston-theoretic, hypersonic regime. Beyond well-posedness of these models, time convergence properties (i.e., attractors) of the dynamics will be considered to determine the sensitivity of non-transient behavior of the system to the plate's boundary conditions and external loading. The current proposal can be viewed as an analysis of models arising in aeroelasticity by providing a comparison between qualitative properties of well-posed PDEs to experimentally observed and/or numerically approximated behaviors.