Operator Theory Arising from Systems Engineering

University Of California-San Diego, La Jolla CA

Investigators

Abstract

The research of this project deals with functional analysis and operator theory related to engineering systems theory. A very wide range of problems in engineering can be formulated and studied in terms of matrix inequalities. Furthermore, many linear system control problems have matrices as their natural variables and thus require an understanding of polynomials and rational functions in noncommutative variables. Of special practical importance is "convexity," for in its presence local minima (which can be found computationally) are in fact global minima (the ones that you really want to find). Yet more tractable are linear matrix inequalities. A surprising finding of the principal investigator and a collaborator that resulted from a previous grant is that any matrix inequality based on a convex noncommutative rational function is equivalent to a linear matrix inequality determined by the same function. In the current project the principal investigator will work on extending the classical (commuting) theory of polynomial inequalities (commonly called semialgebraic geometry) to noncommutative polynomials. He will explore aspects of this subject that pertain to convexity and curvature, develop computer algorithms based on this theory, analyze such algorithms, and look for connections with other branches of mathematics. The first line of mathematical models in engineering for nearly all things that move is a "system" ; namely, a box with inputs and outputs. Even if these are not linear, the first step in a design is often linearization, so the mathematics of linear systems plays a major role in areas such as control for planes, cars, satellites, structures in earthquakes, oil refineries, machine tools, flowing fluids, and much else. The biggest advance in linear systems engineering during the 1990s was the realization that most linear control problems convert directly to matrix inequalities. Many different types of matrix inequalities surfaced in twentieth-century mathematics, but the ones that dominate today in engineering systems usually require that a polynomial or rational function of matrices be "positive semidefinite," a technical property. Many engineering matrix inequalities are "ill-behaved," but there is a classical core of problems in which such inequalities convert to "well-behaved" linear matrix inequalities. The latter are computationally tractable and provably free from false optima (which can lead to unsafe designs). It is algebraic formulas like these that are programmed into modern computer packages in systems engineering. The goals of this project are the following: to test mathematically and computationally whether a given set of matrix inequalities is well behaved, to understand how to convert bad matrix inequalities into nice ones, and to develop corresponding computer algorithms. The principal investigator also devotes significant effort to promoting interactions between mathematicians and engineers. In addition to working with his own graduate students, he runs a modest computational lab in the summer that is staffed by students. Although many of these students go on to obtain Ph.D.'s in pure mathematics, the lab experience and exposure to engineering broadens them considerably.