Approximation Issues in Stability & Control of Distributed Parameter Systems

该项目通常涉及对复杂物理现象（如阻尼弹性结构、延迟过程、耦合结构（包括智能材料模型）等）的数学模型进行可靠和合适的近似方案的分析和开发。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，可以保留原始（无限维）系统的特性。但众所周知，一些近似系统不保留原系统的重要性质，如增长率、稳定性鲁棒性等（事实上，众所周知，一些具有边界阻尼的弹性结构模型具有标准“有限元”或“有限差分”近似系统所不能保留的能量衰减率）。这个项目的重点是如何以及为什么某些属性在近似下保留或丢失，并开发一种系统的方法来构建“属性保存”近似方案。关键是分析原始系统投影到近似系统上的几何形状（内积），其中一个要点是改变几何形状可以改善近似，从而更好地保留重要属性。这是最近的一个想法，PI已经通过一些初步的例子成功地实现了这个想法。这个项目将以重要的方式扩展方法和应用。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、机翼上的流体流动、振动建筑物等。在不断发展的“智能材料”和执行器技术的存在下，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。克服这个困难的一个标准策略是考虑一个更简单的模型（一个“近似方案”），它是原始复杂模型的一个相当好的近似，也可以在计算机上解决。对于任何给定的模型，通常都有许多近似方案，由于在近似中总会丢失一些东西，一些重要的问题是——“合理的近似”有多好？还有，哪些重要的属性会丢失或保留？此外，由于这些计算机模型用于更好地理解和控制这些复杂的系统，高保真的近似方案是必不可少的。本项目将重点关注这些问题，目标是为广泛的应用开发一种系统的方法，并基于该理论开发软件。

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.