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Mathematical Control Theory and Analysis of Partial Differential Equations Coupled Across a Boundary Interface

边界界面耦合偏微分方程的数学控制理论与分析

基本信息

批准号：
1907823
负责人：
George Avalos
金额：
$ 21.38万
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：
2019
资助国家：
美国
起止时间：
2019-07-15 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1907823&HistoricalAwards=false
关键词：
Mathematical Control Theory Analysis Partial

项目摘要

In one component of this project, the investigators will study equations which mathematically describe the flow of gas as it streams within a walled cavity and interacts with a vibrating portion of the cavity's boundary. Within this mathematical framework, the investigators will attempt to determine if the gas flow and cavity wall displacement can be favorably altered or "controlled" by means of the deliberate placement of forces which act on the cavity wall. The accomplishment of this program objective could lend insight into understanding and controlling the gas flows which are associated with high-speed aircraft and natural gas pipelines. Furthermore, a mathematical model which has recently been derived to describe a structure-fluid interaction in which an elastic body is wholly immersed within a given fluid will be studied. This new mathematical model differs from previous structure-fluid models, in that numerical simulations of solutions to the new model suggest that the associated structural displacements, as well as the fluid flow field, elicit a much more quiescent behavior than previously thought. The investigators intend to mathematically verify this observed "stability" for such structure-fluid dynamics. A successful conclusion to this work could potentially enhance the understanding of cellular dynamics: such structure-fluid interaction models have been invoked to describe a nuclear body, within a cell, as it interacts with the surrounding viscous cytoplasm. Moreover, a mathematical model which describes acoustic wave flow within a structural chamber, with one of the chamber walls being flexible, will be studied. In particular, it is intended to determine if the acoustic waves in this structural model will die out as time evolves. Such a conclusion of wave stability in this mathematical setting would have implications in real-world control engineering: With a view of promoting a quiescent acoustic field within the interior of an aircraft cabin, the stability results of the project could conceivably be taken into account in aircraft fuselage design. In the course of fulfilling the project objectives, graduate students, including first generation university students and women, will receive training in the necessary qualitative analysis and numerical computation. The gas flow-cavity model described above consists of the 3D compressible Stokes or Navier-Stokes flow equations coupled to an elastic plate equation which describes the displacements of the flexible portion of the cavity wall. This PDE system is subjected to cavity boundary control terms. The investigators will determine the optimal regularity of solutions to such controlled (gas) flow-structure PDE models under the influence of boundary control terms of prescribed smoothness. Subsequently, they intend to solve the associated null controllability problem with respect to the appropriate space of boundary control functions. Moreover, in the aforementioned structure-fluid interaction, the structure is composed of a "thick" and "thin" layer: the thick layer is described by a 3D wave equation, while the the thin layer is described by a 2D wave equation and constitutes the boundary interface between the thick wave equation and the 3D fluid component equation. The investigators intend to establish that classical solutions of this 3D wave-2D wave-3D heat PDE system manifest at least a polynomial rate of decay, which would be in line with what has been observed numerically. In addition, the rational decay problem for the structural acoustic interaction, in which an acoustic wave equation within a bounded domain interacts with a dissipative elastic equation across a boundary interface, will be considered. Moreover, an algorithm which will confirm that the polynomial rates obtained are optimal will be developed.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在该项目的一个组成部分中，研究人员将研究方程，这些方程以数学方式描述气体在有壁空腔内流动并与空腔边界的振动部分相互作用时的流动。在这个数学框架内，研究人员将尝试确定是否可以通过故意放置作用在腔壁上的力来有利地改变或“控制”气流和腔壁位移。该计划目标的实现可以帮助人们深入了解和控制与高速飞机和天然气管道相关的气流。此外，还将研究最近导出的用于描述结构-流体相互作用的数学模型，其中弹性体完全浸没在给定流体中。这种新的数学模型与以前的结构流体模型不同，因为新模型解的数值模拟表明，相关的结构位移以及流体流场引起了比以前想象的更加静态的行为。研究人员打算从数学上验证这种结构-流体动力学观察到的“稳定性”。这项工作的成功结束可能会增强对细胞动力学的理解：这种结构-流体相互作用模型已被用来描述细胞内的核体，因为它与周围的粘性细胞质相互作用。此外，还将研究一种描述结构室内声波流的数学模型，其中室壁之一是柔性的。特别是，其目的是确定该结构模型中的声波是否会随着时间的推移而消失。在这种数学背景下得出的波浪稳定性结论将对现实世界的控制工程产生影响：为了促进飞机机舱内部的静态声场，可以在飞机机身设计中考虑该项目的稳定性结果。在实现项目目标的过程中，研究生，包括第一代大学生和女性，将接受必要的定性分析和数值计算方面的培训。上述气体流动腔模型由 3D 可压缩斯托克斯或纳维-斯托克斯流动方程组成，耦合到描述腔壁柔性部分位移的弹性板方程。该偏微分方程系统受空腔边界控制项的约束。研究人员将在规定平滑度的边界控制项的影响下确定此类受控（气体）流结构 PDE 模型解的最佳规律性。随后，他们打算解决与边界控制函数的适当空间相关的零可控性问题。此外，在上述结构-流体相互作用中，结构由“厚”层和“薄”层组成：厚层由3D波动方程描述，而薄层由2D波动方程描述，并构成厚层波动方程与3D流体分量方程之间的边界界面。研究人员打算确定该 3D 波-2D 波-3D 热偏微分方程系统的经典解至少表现出多项式衰减率，这与数值观察到的结果一致。此外，还将考虑结构声相互作用的有理衰减问题，其中有界域内的声波方程与跨边界界面的耗散弹性方程相互作用。此外，还将开发一种算法来确认所获得的多项式率是最佳的。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。