9304696 Ober This project is directed towards the study of infinite dimensional balanced realizations, both from the point of view of improving the basic understanding of these realizations and of providing further tools for model reductions of infinite dimensional systems. The project will focus on the specific questions of asymptotic and exponential stability of the balanced systems and the boundedness properties of the system operators. A further important point which will be addressed is the issue of whether general Trotter-Kato type convergence results can be proved for the balanced approximation scheme. Many engineering systems, such as highly flexible mechanical systems, inherently contain infinitely many degrees of freedom and therefore any accurate mathematical description of such a system must necessarily be infinite dimensional. This project is concerned with so-called balanced realizations for infinite dimensional linear systems. One of the important uses of balanced realizations of linear systems is as a tool in achieving model reductions of complex systems, which is a very important consideration at a practical and computational level. There already exists an extensive, elegant and important (from the point of view of applications) body of results concerning balanced realizations and their role in model reduction for finite dimensional systems. However, extensions of existing theory to infinite dimensional systems is extremely complex at both the conceptual and technical levels. The aim of this project is to develop the mathematical and computational tools that are of similar power to those available for systems of finite complexity. *** ??