Approximation Issues in Stability & Control of Distributed Parameter Systems

该项目一般针对复杂物理现象的数学模型(如有阻尼的弹性结构、时滞过程、耦合结构(包括智能材料模型)等)分析和开发可靠且合适的近似方案。分析和解决此类系统问题的标准方法是考虑近似系统，理想情况下，近似系统既是有限维的，又足够丰富，足以保留原始(无限维)系统的性质。但众所周知，某些近似系统不能保持原系统的重要性质，如增长率、稳定性、稳健性等。(事实上，众所周知，某些具有边界阻尼的弹性结构模型的能量衰减率不是标准的有限元或有限差分近似系统所能保持的)。这个项目集中在如何以及为什么某些性质在近似下被保留或丢失，并开发一种系统的方法来构造“保性质”的近似方案。关键是分析原始系统在其下投影到近似系统的几何(内积)，其中一个要点是改变几何可以改进近似，从而更好地保留重要的性质。这是一个最近的想法，通过一些初步的例子，PI已经成功地实施了这一想法。这个项目将在很大程度上扩展方法和应用程序。在科学和工程中有许多涉及弹性结构的实际应用，如大型天线、翼型上的流体流动、建筑物振动等。随着不断发展的“智能材料”和致动器技术的存在，这些导致了复杂的数学模型，即使是最强大的计算机也无法完全解决。克服这一困难的标准策略是转而考虑一个更简单的模型(“近似方案”)，它是原始复杂模型的一个相当好的近似，并且也可以在计算机上求解。对于任何给定的模型，通常都有许多近似方案，由于在近似中总是会丢失一些东西，所以一些重要的问题是--一个“合理的好的近似”有多好？以及--哪些重要的财产被遗失或保存？此外，由于这些计算机模型被用来更好地理解和控制这些复杂系统，因此高保真近似方案是必不可少的。本项目将侧重于这些问题，目的是为广泛的应用程序开发一种系统的方法，并在此理论的基础上开发软件。

The project deals generally with the analysis and development of reliable and suitable approximation schemes for mathematical models of complex physical phenomena such as damped elastic structures, processes with delays, coupled structures (including models of smart materials), etc. A standard approach to analyzing and solving problems for such systems is to instead consider an approximate system, which ideally is both finite dimensional and rich enough to retain properties of the original (infinite dimensional) system. But it is well known that some approximate systems do not retain important properties of the original system, such as growth rate, stability robustness, etc. (In fact, it is well known that some models of elastic structures with boundary damping have an energy decay rate that is not preserved by standard "finite element" or "finite difference" approximate systems). This project focuses on how and why certain properties are retained or lost under approximation, and on developing a systematic method for constructing "property-preserving" approximation schemes. The key is analyzing the geometry (inner product) under which the original system is projected onto the approximating system, and one thrust is that changing geometries can improve approximations so that important properties are better preserved. This is a recent idea on which the PI has demonstrated successful implementation for some preliminary examples. This project will extend in significant ways the method and the applications.There are many practical applications in science and engineering that involve elastic structures, such as large antennae, fluid flow over an airfoil, vibrating buildings, etc. In the presence of evolving "smart materials" and actuator technologies, these lead to complex mathematical models that cannot be fully resolved on even the most powerful computers. A standard strategy for overcoming this difficulty is to instead consider a simpler model (an "approximation scheme") which is a reasonably good approximation of the original complex model, and which can also be solved on a computer. There are typically many approximation schemes for any given model, and since something is always lost in the approximation, some important questions are - how good is a 'reasonably good approximation'? and - which important properties are lost or preserved? Further, since these computer models are used to better understand and control these complex systems, high-fidelity approximation schemes are essential. This project will focus on these questions, with the goal of developing a systematic methodology for a broad range of applications, and of developing software based on the theory.