Analysis and Control of Layered Media

0205148 Hansen所提出的研究的重点是分析可控性和稳定性的偏微分方程（PDE）的分层弹性介质模型的系统。具体的层状弹性系统，我们关注的包括约束层阻尼模型，层合板，板表面安装和/或嵌入式致动器和一般的多层板理论。 对于一个给定的分层结构，通常有许多不同的PDE系统的不同的复杂性，可以用来模拟振动。通常，各种PDE通过扰动连接，该扰动可以是奇异的。中心问题是确定稳定性和可控性对连接可能模型的扰动参数的依赖性。通过这种方式，可以测试控制和反馈策略的模型依赖性、准确性和鲁棒性。该计划是将这种类型的分析应用到实际的控制应用，如约束层阻尼的优化设计和使用表面安装和嵌入式压电致动器的弹性结构的稳定。在试图开发对多层板有用的控制方法时，存在几个理论挑战。 特别是，它是希望发展的Carleman乘子理论，适用于分层系统的边界控制。目的是在边界管制方面取得尽可能明确的结果。由于材料科学、复合材料制造和加工速度的快速发展，理解层状弹性系统的建模、控制和稳定变得越来越重要。所考虑的模型的动机是实际的现实世界中的应用，涉及复合材料层结构。 这些包括船体和飞机机身的设计，汽车车身的隔音以及滑雪板和网球拍等体育用品的设计。通常，与单一材料相比，可以利用复合结构来实现更优越的上级设计。 由于层间耦合的复杂性质，开发这些结构的控制和反馈理论是一个具有挑战性的问题。 另一方面，一个模型是有用的，在实际工程应用中，各种粗略的近似需要作出，以减少复杂性。因此，目标的很大一部分是量化准确但复杂的数学模型和简化但更有用的工程模型之间的联系。 具体的实际利益，将解决的问题包括：振动结构的反馈控制，使用战略性地放置致动器，优化设计和稳定性分析的阻尼分层和层压复合材料。

0205148Hansen The focus of the proposed research is to analyze controllability and stabilization properties of systems of partial differential equations (PDE's) that model layered elastic media. Specific layered elastic systems we are concerned with include constrained-layer damping models, laminated plates, plates with surface-mounted and/or embedded actuators and general multilayer plate theory. For a given layered structure, generally there are many different PDE systems of varying complexity that could be used to model the vibrations. Often the various PDE's are connected through a perturbation, which may be singular. The central concern will be to determine the dependence of stability and controllability upon the perturbation parameters that connect the possible models. This way model dependence, accuracy, and robustness properties of control and feedback strategies can be tested. The plan is to apply this type of analysis to practical control applications such as optimal design for constrained-layer damping and the stabilization of elastic structures using surface-mounted and embedded piezoelectric actuators. Several theoretical challenges present themselves in attempting to develop control methods that are useful for multilayer plates. In particular, it is hoped to develop a theory of Carleman multipliers that is applicable to boundary control of layered systems. The objective is to obtain the sharpest possible results regarding boundary control. Understanding the modeling, control and stabilization of layered elastic systems is becoming increasingly important due to the rapid advances in material sciences, fabrication of composite materials, and speed of processing. The models considered are motivated by practical real-world applications involving composite layered structures. These include the design of ship hulls and airplane fuselage, sound-proofing in auto bodies and the design of sporting goods such as skis and tennis rackets. Often, it is possible to utilize a composite construction to achieve a far superior design compared to what is possible with a single material. Developing a theory for control and feedback of these structures is a challenging problem due the complicated nature of the coupling between the layers. On the other hand, for a model to be useful in practical engineering applications, various crude approximations need to be made to reduce the complexity. Thus, a large part of the goal is to quantify the connection between an accurate, but complex mathematical model and a simplified, but more useful engineering model. Specific problems of practical interest which will be addressed include: feedback control of vibrating structures using strategically placed actuators, optimal design and stability analysis for damping in layered and laminated composites.