Operator Theory Arising from Systems Engineering

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The research of this proposal is in functional analysis and operator theory related to engineering system theory. Many linear control problems can be phrased and studied in terms of matrix inequalities. These are formulated by a consideration of polynomials in noncommutative variables x. A basic result, due to the PI and obtained as part of his previous grant, is that every matrix positive noncommutative polynomial can be written as a sum of squares of noncommutative polynomials. In this proposal Helton will work on extending this theory, developing computer algorithms based on this theory, the analysis of such algorithms, and connections with other branches of mathematics. These are key optimization problems arising in designs of linear systems.The biggest advance in linear systems engineering during the 1990's is the realization that most linear control problems convert directly to matrix inequalities, abbreviated MIs. Many of these are badly behaved but a classical core of problems convert to linear matrix inequalities (LMIs) that are nicely behaved. Many different types of matrix inequalities have come up in the mathematics of the previous century, but the ones that dominate in engineering systems usually take the form of a polynomial or rational function of matrices being positive semidefinite. It is algebraic formulas like these that are programmed into modern computer packages in engineering. The goal of this project is to understand how one converts bad MIs to nice MIs. When is this possible?