Mathematical Sciences: Stability and Approximation for Distributed Parameter Systems

9527381法比亚诺这个鲁伊奖支持一个关于动态偏微分方程组和积分-微分方程组的稳定性和逼近的项目。所涉及的基本问题是系统在有限维近似下的稳定性保持问题。这是一个非常重要的问题，因为对于偏微分方程组或其他无限维系统的每一种计算格式都涉及这样或那样的有限维近似，并且众所周知，某些逼近格式会破坏稳定性或改变所谓的稳定裕度。我们将研究最近发展起来的一种近似方法，称为等价内积法，它可以保持原无穷维系统的稳定裕度。众所周知，这种方法对于有界算子(例如常微分方程组)的逼近是有用的。除了在所考虑的方程的背景下建立该方法的收敛性质之外，还将开发基于该方法的可靠的计算机算法。控制柔性弹性振动结构运动的数学方程被用来模拟各种物理系统，如机械臂、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。随着先进的执行器和材料等新技术的出现和性能要求的提高，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合理、可靠的数值算法来进行模型模拟和逼近。这个项目将集中在一种新的近似方法上，这种方法已经显示出很有希望精确地处理这些类型的问题。将开发该方法的基本收敛性质，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. ***