Analysis and Control of Layered Media

本文的研究重点是分析层状弹性介质偏微分方程组的能控性和镇定性质。我们所关注的特定层状弹性系统包括约束层阻尼模型、层合板、表面安装和/或嵌入执行器的板以及一般的多层板理论。对于给定的分层结构，通常有许多不同复杂性的PDE系统可用于对振动进行建模。通常，各种偏微分方程组通过可能是奇异的微扰连接在一起。中心问题将是确定稳定性和可控性对连接可能的模型的摄动参数的依赖性。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。我们的计划是将这种分析应用于实际的控制应用中，例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，出现了几个理论挑战。特别是，希望发展一种适用于分层系统边界控制的Carleman乘子理论。其目的是在边界控制方面取得最大可能的成果。随着材料科学、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。所考虑的模型是由涉及复合材料层状结构的实际应用所驱动的。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，利用复合结构来实现更优越的设计是可能的。由于层间耦合的复杂性，开发一种控制和反馈这些结构的理论是一个具有挑战性的问题。另一方面，为了使模型在实际工程应用中有用，需要进行各种粗略的近似以降低复杂性。因此，很大一部分目标是量化一个准确但复杂的数学模型和一个简化但更有用的工程模型之间的联系。需要解决的实际问题包括：使用合理布置的致动器对振动结构进行反馈控制，层状和层状复合材料中的最优减振设计和稳定性分析。

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.