Analysis and Control Theory for Moving Boundary and Nonlinear Phenomena in Interactive Partial Differential Equations

交互偏微分方程中动边界和非线性现象的分析与控制理论

• 批准号: 1616425
• 负责人: George Avalos
• 金额: $ 32.89万
• 依托单位: University of Nebraska-Lincoln
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1616425&HistoricalAwards=false
• 关键词: Analysis; Control; Theory; Moving; Boundary

The theme of this research project is to conduct mathematical and computational studies on classes of partial differential equations (PDE) with moving boundaries. The first part of the project mathematically describes the phenomenon of acoustic sound waves within a chamber whose flexible walls are allowed to move freely as time evolves. For such acoustic wave dynamics, the goal is to determine which pre-assigned chamber geometries will give rise to an interior acoustic field that is as "quiet" as possible. That is to say, because of the analytically pre-determined chamber configuration, the acoustic pressure will be minimized and external noise optimally reduced. The second part of the project focuses on structure-fluid interactions as they occur in nature and engineering, for example the nucleus of a cell interacting with the cellular cytoplasm. In this example, the wall of the nucleus might be described by structural PDE, and the cytoplasm mathematically modeled by fluid flow PDE. The goal of the project is to develop and analyze more realistic models that comprise both fluid and structure PDE dynamics. The first part of the project will address the feasibility of stabilizing interior acoustic wave dynamics on a time-evolving bounded domain through the means of prescribing, in advance, the evolution of the geometry. That is, as time gets large, the prescribed moving boundary feedback law should cause the acoustic wave solutions to either decay or approach a certain set of finite energy states. Since the associated wave energy is majorized by wave boundary trace quantities that resemble velocity feedback mechanisms, this "geometry-control" problem might be viewed as a moving boundary analog to Neumann boundary stabilization of the wave equation on a fixed domain. The second part of the project will consider linear and nonlinear structure-fluid PDE models in which the structural components are mathematically modeled by physically relevant but quite complex Reissner-Mindlin-Timshenko (RMT) plate systems. By decoupling these structure-fluid dynamics, the associated pressure variables are eliminated in a nonstandard manner, intrinsic to the coupling mechanisms in place. Having resolved the well-posedness, the study will consider an associated optimal control problem that will reveal certain ocular dynamics. In addition, the question of global exact controllability for nonlinear von Karman plate dynamics will be investigated. The objective is to establish the exact steering property with respect to finite energy states of arbitrary size, not only target states within a small neighborhood of the origin. This control process will be simulated numerically and the dependence of the controllability time on the size of the initial and target states will be examined.

该研究项目的主题是对一类具有移动边界的偏微分方程（PDE）进行数学和计算研究。该项目的第一部分以数学方式描述了一个腔室中的声波现象，该腔室的柔性壁可以随着时间的推移自由移动。对于这种声波动力学，目标是确定哪些预先分配的腔室几何形状将产生尽可能“安静”的内部声场。也就是说，由于分析预定的腔室配置，声压将被最小化并且外部噪声被最佳地降低。该项目的第二部分侧重于自然界和工程中发生的结构-流体相互作用，例如细胞核与细胞质的相互作用。在这个例子中，细胞核的壁可以用结构PDE来描述，而细胞质可以用流体流动PDE来数学建模。 该项目的目标是开发和分析更现实的模型，包括流体和结构PDE动力学。该项目的第一部分将通过预先规定几何形状的演变来解决在时间演变的有界域上稳定内部声波动力学的可行性。也就是说，随着时间变大，规定的移动边界反馈定律应导致声波解衰减或接近某组有限能量状态。由于相关联的波能量是由类似于速度反馈机制的波边界迹量主导的，因此这个“几何控制”问题可以被视为固定域上的波动方程的诺依曼边界稳定化的移动边界模拟。该项目的第二部分将考虑线性和非线性结构-流体PDE模型，其中结构部件通过物理相关但相当复杂的Reissner-Mindlin-Timshenko（RMT）板系统进行数学建模。通过解耦这些结构-流体动力学，相关的压力变量以非标准的方式被消除，这是适当的耦合机制所固有的。在解决了适定性问题后，本研究将考虑一个相关的最优控制问题，该问题将揭示某些眼部动力学。此外，本文还研究了非线性冯卡门板动力学的全局精确能控性问题。我们的目标是建立精确的转向属性相对于有限的能量状态的任意大小，而不仅仅是目标状态在一个小的邻域内的起源。这个控制过程将进行数值模拟和可控性的时间上的初始和目标状态的大小的依赖关系将被检查。