Collaborative Research: Control of Fluid-Elastic Structures and Related Topics

该研究旨在更好地理解流体-弹性耦合系统的能控性结构。具体地说，我们关注的是三维区域中流体的情况，该区域具有由板或膜方程描述的柔性边界。我们提出了几个问题，用来解决我们的基本问题：在耦合系统的一个或多个组件中需要多少控制才能获得(I)耦合系统的可控性或近似可控性，或(Ii)系统的一个或多个组件的部分可控性(近似或精确)？这项拟议的研究在几个方面扩展了最先进的技术。首先，对于我们正在考虑的基于物理的模型类型，流体弹性系统中的部分可控性问题尚未得到解决。其次，我们建议分析的许多模型甚至还没有被研究过，即使是适定性的。即使在某些模型的线性情况下(例如，在考虑自由边界条件时)，这也是不平凡的一步。然而，我们也希望解决与运动的柔性边界相关联的Stokes系统的自由边界问题相关的非线性问题的局部能控性。同样，我们希望在可压缩的Navier-Stokes流体与平板耦合的情况下取得一些进展。我们解决这些问题的主要分析工具将是Carleman估计与微局部分析相结合的方法。对于这些类型的耦合系统，Carleman估计的发展在更广泛的背景下可能被证明是非常有用的，因为在分析各种其他类型的耦合系统时，类似的方法可能是有用的。我们还建议通过适当定义与板-流体耦合系统的解相关的Dirichlet-to-Neumann映射来研究确定板系数的密切相关的反问题。我们要研究的流体弹性结构在许多实际应用中都会出现。例如，通过安装在飞机上和飞机内部的控制器将飞机机身一部分的引擎噪音降至最低的问题，与上文提到的部分可控性问题(Ii)密切相关。我们计划研究的一种特殊的流体弹性结构是耳蜗，它是内耳的听觉器官，将声音振动转换为传输到大脑的电脉冲。对耳蜗可控性结构的更好理解可能会导致人工耳蜗和助听器的改进设计。

The proposed research is aimed at better understanding the controllability structure of coupled fluid-elastic systems. Specifically, we focus on the situation of a fluid in a three-dimensional domain with a flexible boundary that is described by a plate or membrane equation. We suggest several problems which serve to address our fundamental question: How much control in one or more components of the coupled system is needed to obtain (i) controllability or approximate controllability of the coupled system or (ii) partial controllability (approximate or exact) of one or more components of the system? The proposed research extends the state-of-the-art in several ways. First, the partial controllability problem in the context of a fluid-elastic system has not been addressed for the types of physically-based models we are considering. Secondly, many of the models we propose to analyze have not been studied even for well-posedness. This is a nontrivial step even in the linear case for some of the models (e.g., when considering free boundary conditions). However we also hope to address local controllability for the nonlinear problems associated with the free boundary problem for the Stokes system associated with a moving flexible boundary. Likewise we hope to make some progress in the case of a compressible Navier-Stokes fluid coupled with a plate. Our main analytical tool to attack these problems will be the method of Carleman estimates together with microlocal analysis. Development of Carleman estimates for these types of coupled systems might turn out to be be very useful in a broader context, as it is likely that a similar approach may be useful in the analysis of a variety of other types of coupled systems. We also propose to investigate the closely related inverse problem of determination of plate coefficients through knowledge of the appropriately defined Dirichlet-to-Neumann maps associated with solutions of the coupled plate-fluid system. The fluid-elastic structures we aim to study arise in many practical applications. For example,the problem of minimizing the engine noise in a portion of an aircraft fuselage through controllers placed on and inside the aircraft is closely related to the partial controllability problem (ii) mentioned above. One specific fluid-elastic structure we plan to investigate is the cochlea, which is the auditory organ in the inner ear that translates sound vibrations to electrical impulses transmitted to the brain. A better understanding of the controllability structure of the cochlea could lead to an improved design of cochlear implants and hearing aids.