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

在该项目的一个组成部分中，研究人员将研究数学上描述气体在有壁腔内流动并与腔边界振动部分相互作用的方程。在这个数学框架内，研究人员将试图确定气体流动和腔壁位移是否可以通过故意施加作用于腔壁的力来有利地改变或“控制”。这一项目目标的实现将有助于理解和控制与高速飞机和天然气管道相关的气体流动。此外，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，其中弹性体完全浸入给定流体中。这个新的数学模型不同于以前的结构-流体模型，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发了比以前认为的更安静的行为。研究人员打算用数学方法验证这种结构流体动力学中观察到的“稳定性”。这项工作的成功结论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，一个数学模型，描述声波流动的结构室，其中一个室壁是柔性的，将进行研究。特别是，它的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。在这种数学环境下，波稳定性的结论将对现实世界的控制工程产生影响：从促进飞机机舱内部的静态声场的角度来看，该项目的稳定性结果可以想象在飞机机身设计中被考虑在内。在实现项目目标的过程中，研究生，包括第一代大学生和妇女，将接受必要的定性分析和数值计算方面的培训。上述气体流腔模型由三维可压缩Stokes或Navier-Stokes流动方程与描述腔壁柔性部分位移的弹性板方程耦合组成。该PDE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这种受控（气体）流结构PDE模型解的最优规则性。随后，他们打算在边界控制函数的适当空间中解决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：厚层由三维波动方程描述，薄层由二维波动方程描述，构成了厚波方程和三维流体分量方程之间的边界界面。研究人员打算建立这种三维波-二维波-三维热PDE系统的经典解至少表现出多项式的衰减速率，这将与数值观察到的结果一致。此外，将考虑结构声相互作用的理性衰减问题，其中有界域内的声波方程与边界界面上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得到的多项式速率是最优的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.