Mathematical Sciences: Stability and Approximation for Distributed Parameter Systems

这个RUI奖支持一个研究动态偏微分方程和积分微分方程的稳定性和近似的项目。所涉及的基本问题是在有限维近似下保持系统的稳定性。这是一个非常重要的问题，因为偏微分方程系统或其他无限维系统的每一个计算方案都涉及到一种或另一种有限维近似，众所周知，某些近似方案会破坏稳定性或改变所谓的稳定裕度。一种最近发展的近似方法，称为等效内积法，已知可以保持原始无限维系统的稳定裕度，将被研究。这种方法对于有界算子（例如常微分方程组）的近似是有用的。除了在考虑的方程中建立该方法的收敛性外，还将开发基于该方法的可靠计算机算法。控制振动柔性弹性结构运动的数学方程被用于模拟各种物理系统，如机器人手臂、大型天线、建筑物（例如在地震中振动）、飞机机翼等。为了解决这些应用中的复杂问题，经常使用计算机算法来模拟这些方程。随着新技术的出现，如先进的致动器和材料，性能要求的提高，以及更好地理解和影响这些系统的运动的需要，需要更复杂的数学模型和理论上健全、可靠的数值算法来进行模型仿真和近似。本项目将重点研究一种新的近似方法，该方法对解决这类问题具有重要意义。将开发该方法的基本收敛特性，以及基于该方法的相关软件。＊＊＊

9527381 Fabiano This RUI award supports a project dealing with stability and approximation of dynamic partial differential equations and integro-differential equations. The basic issue involved is that of preservation of stability under finite dimensional approximation of the system. This is a very important question since every computational scheme for a system of partial differential equations or other infinite dimensional system involves a finite dimensional approximation of one sort or another, and it is well-known that certain approximation schemes will destroy stability or change the so-called stability margin. A recently developed method of approximation, called the method of equivalent inner products, that is known to preserve the stability margin of the original infinite dimensional system, will be studied. The method is known to be useful for approximation of bounded operators (systems of ordinary differential equations, for example). In additions to establishing convergence properties of the method in the context of the equations under consideration, reliable computer algorithms based on the method will be developed. The mathematical equations governing the motion of vibrating flexible elastic structures are used to model such diverse physical systems as robot arms, large antennae, buildings (vibrating in an earthquake, for example), airplane wings, and so forth. Computer algorithms are often used to simulate these equations in order to solve complex problems in these applications. With the advent of new technologies, such as advanced actuators and materials, and enhanced performance requirements, together with the need to better understand and influence the motion of these systems, there is a necessity for more sophisticated mathematical models and theoretically sound, reliable numerical algorithms for model simulation and approximation. This project will focus on a new approximation method which has shown significant promise for imp acting exactly these types of problems. Basic convergence properties of the method will be developed, as well as associated software based on the method. ***