Operator Theory Arising from Systems Engineering

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This is a proposal for support of projects in parts of operator theory and functional analysis mostly related to Linear Matrix Inequalities (LMIs) and engineering system theory. Semidefinite programming, based on LMIs, is one of the main developments in optimization over the previous 15 years with applications ranging through most quantitative areas of science. A main thrust of this proposal is: what is the scope of semidefinite programming? Problems treatable with LMIs are convex, but conversely which convex problems are treatable with LMIs? With collaborators the principal investigator is currently making significant advances on the main aspects of this problem. Special emphasis goes to LMI problems motivated by systems engineering. While numerics for LMIs, is hotly pursued there is no systematic theory for simplifying or analyzing matrix inequalities algebraically. Major parts of this proposal are to develop such a theory. This produces a rich body of elegant problems in functional analysis including those described as follows. A symmetric noncommutative polynomial is called "free positive" provided that all of its values when evaluated on matrices is a positive semi-definite matrix. "Free convexity" is defined analogously. The set of all matrices solving a noncommutative polynomial inequality is called a "free semialgebraic set." Linear Matrix Inequalities concern conditions making a given linear pencil take positive semidefinite values. 1. Which sets are the set of all solutions to some LMI? 2. Can one change variables to achieve free convexity? 3. Find the convex hull of a semialgebraic set? These can be built as projections of certain convex sets, as was analysed by J. Nie and the investigator. Free convex hulls are very intriguing. 4. Noncommutative real algebraic geometry: In another direction, started by Hilbert's 17th problem, are algebraic certificates equivalent to statements like one polynomial is positive where another one is positive? The development of noncommutative analogs of this are going well and considerable work is in progress. 5. A cornerstone of classical real algebraic geometry is that projections of semialgebraic sets are semialgebraic. Recent theorems (by the investigator and collaborators) imply this is (overwhelmingly) false in the matricial world, producing a barrage of questions.The proposed work bears on, semidefinite programming, a type of convex optimization found in many branches of science and engineering. In particular what one sees in linear systems engineering and control are problems with matrix unknowns. Simplifying physical problems and converting them to convex ones is currently done (in thousands of papers) by ad hoc algebraic tricks. The goal in the proposal is to develop a theory (a noncommutative real algebraic geometry) which might be used to systematize this. In addition the investigator's group are the main providers of software (called NCAlgebra) for performing general noncommuting algebra calculations in the software program Mathematica. An emphasis now is on algorithms for treating noncommutative inequalities. Also the group does numerical optimization based on a floor of noncommutative algebra calculation. The project will engage graduate and some undergraduate students in summer research and computational projects. This lab experience with applications will broaden the training of students, many of whom get degrees in pure mathematics.