Control of Fluid-Elastic Structures and Related Topics

提出的研究旨在更好地理解耦合流体-弹性系统的可控性结构。具体来说，我们关注的是具有柔性边界的流体在三维域中的情况，这种边界由板或膜方程描述。我们提出了几个问题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）？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-弹性系统背景下的部分可控性问题尚未得到解决。其次，我们提出要分析的许多模型甚至还没有经过适位性研究。即使在某些模型的线性情况下（例如，当考虑自由边界条件时），这也是一个重要的步骤。然而，我们也希望解决与运动柔性边界相关的Stokes系统的自由边界问题相关的非线性问题的局部可控性。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估计方法和微局部分析方法。对于这些类型的耦合系统的Carleman估计的发展可能在更广泛的环境中非常有用，因为类似的方法可能在分析各种其他类型的耦合系统中有用。我们还建议通过适当定义的Dirichlet-to-Neumann映射与耦合板-流体系统的解相关的知识来研究密切相关的板系数确定的反问题。我们要研究的流弹性结构有许多实际应用。例如，通过放置在飞机上和飞机内部的控制器来最小化飞机机身部分的发动机噪声的问题与上面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，它是内耳中的听觉器官，可以将声音振动转化为电脉冲，传输到大脑。更好地了解耳蜗的可控性结构可以导致改进设计的人工耳蜗和助听器。

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.