Mathematical Sciences: Stability and Approximation for Distributed Parameter Systems

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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. ***