Mathematical Sciences: Operator Theory and Systems Engineering

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9501064 Helton Most of the proposed research falls into three areas: 1. OPTIMIZATION OVER SPACES OF ANALYTIC FUNCTIONS Qualitative theory, computer algorithms based on this theory, analysis of such algorithms. Connections of this work with other branches of mathematics.The engineering motivation here is strong; these are the key optimization problems which arise il design of systems and amplifiers where there are competing constraints, time as well as frequency domain constraints, or uncertainty in the math model of the physical system one is trying to control. 2. NONLINEAR SYSTEMS The point here is to find canonical nonlinear operator theory which extends the classical linear theory of Nevanlinna-Pick and many followers. That theory is the core of linear control theory. Many systems which people wish to control are nonlinear (e.g., jet engines), so there is a big demand for an effective nonlinear theory. Unfortunately, the mathematical theory is new and primitive. More general problems need to be solved and special cases need to be understood in greater detail. 3. COMPUTER OPERATOR ALGEBRA Linear engineering systems theory and operator theory are rife with calculations in a noncommuting algebra. Thus it is likely that someday there will be a branch of these subjects devoted to their symbolic computational aspects. Helton's group have, support, and are upgrading extensive software, called NCAlgebra, for performing noncommuting calculations in Mathematica. The software is currently at the level of a very powerful `yellow pad'. The main issue now is what `intelligence' should go in noncommuting packages. For example, calculations with energy conserving systems produce polynomials in matrices satisfying certain relations. Thus it is natural to find a list of rules for simplifying such polynomials tailored to this particular situation. The goal is to find and understand `complete' lists of simplifying rules. Helton and collaborat ors have successfully studied several situations of this type and more are in progress. ***