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.

本研究项目的主题是对一类具有移动边界的偏微分方程组进行数学和计算研究。该项目的第一部分从数学上描述了一个房间内的声波现象，该房间的柔性墙壁可以随着时间的推移自由移动。对于这样的声波动力学，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”的室内声场。也就是说，由于分析上预先确定的腔体结构，声压将被最小化，外部噪声将被最佳地降低。该项目的第二部分侧重于在自然界和工程学中发生的结构-流体相互作用，例如细胞核与细胞质相互作用。在这个例子中，核壁可以用结构PDE描述，细胞质可以用流体PDE数学模型来描述。该项目的目标是开发和分析包括流体和结构PDE动力学的更现实的模型。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，随着时间的增长，规定的移动边界反馈律应该使声波解衰减或接近某一组有限能量状态。由于相关的波能量主要由类似于速度反馈机制的波边界迹量来控制，所以这个“几何控制”问题可以看作是一个移动边界，类似于固定区域上波动方程的Neumann边界镇定。该项目的第二部分将考虑线性和非线性结构-流体PDE模型，其中结构组件由物理上相关但相当复杂的Reissner-Mindlin-Timshenko(RMT)板块系统进行数学建模。通过分离这些结构-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。在解决了适定性之后，这项研究将考虑一个相关的最优控制问题，该问题将揭示特定的眼睛动力学。此外，还将研究非线性von Karman板动力学的全局精确能控性问题。目标是建立关于任意大小的有限能态的精确转向性质，而不仅仅是原点附近的目标态。这一控制过程将被数值模拟，并将考察可控性时间对初始和目标状态大小的依赖关系。