Operatory Theory and Systems Engineering

University Of California-San Diego, La Jolla CA

Investigators

Abstract

ABSTRACT NONLINEAR OPERATOR THEORY. The goal is to find canonical "nonlinear generalizations" of analytic function theory and the related parts of linear operator theory. Many classical and many new analytic function theorems have a statement purely in terms of linear operators. The surprising thing is that these theorems about linear operators do not actually require linearity, and it has become clear that there will be extensive nonlinear generalizations of them. This wide open area is close to control theory. COMPUTER OPERATOR ALGEBRA. Linear engineering systems theory and operator theory are rife with calculations in a noncommutative algebra. Helton's group with M. Stankus are major providers of software (called NCAlgebra) for performing general noncommutative calculations in Mathematica. One phase of the software is at the level of a very powerful `yellow pad', and contains numerous algorithms they developed. For example, the "noncommutative convexity" algorithm discussed below will go in. In another phase there is extensive software implementing noncommutative Groebner basis algorithms due to Mora and algorithms for sorting and "shrinking" the output in various ways ( this is crucial in the noncommutative case). Since the techniques are new, experimentation on traditional problems is important. NONCOMMUTATIVE INEQUALITIES. Recently, Helton and collaborators made progress on a theory and resulting algorithm which takes a rational function F of noncommutative variables Z and outputs a family of inequalities which determine a domain G of Z on which F is "matrix convex". Decidedly non-trivial is showing that the domain G determined by the algorithm is "the largest possible" domain of matrix convexity for F. This is a first attempt at an automatic method for what engineers now do with Schur complement tricks to convert a design problem to Linear Matrix Inequalities. OPTIMIZATION OVER SPACES OF ANALYTIC FUNCTIONS. Qualitative theory, computer algorithms based on this theory, analysis of such algorithms. These are the key optimization problems which arise in designs of linear systems where there are competing constraints, or uncertainty in the math model of the physical system one is trying to control. The research is directed at several projects in parts of operator theory and functional analysis related to engineering system theory. Linear operator theory has had a strong interplay with analytic function theory and engineering for many decades. Indeed most commercial software (at least in the control engineering community) for solving analytic function problems is based on this type of interplay between functions and matrices. One branch of analysis closely related to applications is classical Nevanlinna-Pick- Nehari theory, or equivalently commutant lifting theory, a part of the area called operator model theory. The early development of this was done for the purest of mathematical reasons, but in the mid 1970's and early 1980's this was shown to be critical to the design of engineering systems where stability of the system is the key constraint. This motivated much more mathematical development and now it is one of the areas of functional analysis most closely associated with control engineering. For many years (since Norbert Wiener) design tools optimized mean square performance. The theory above ultimately lead to (commercially commonplace) tools for optimizing worst case frequency domain performance. The goal of much of the proposed research is to extend this theory in several radically new directions and we list the main ones. HIGHLY NONLINEAR GENERALIZATIONS; the goal is to find canonical "nonlinear generalizations" of analytic function theory and the related parts of linear operator theory. This wide open area is closely related to control theory. Many systems which people wish to control are nonlinear (e.g., jet engines). NONCOMMUTATIVE COMPUTER ALGEBRA; if a signal goes into a system A, comes out and then goes into B what we get is BA, while if a signal goes into B and then A what we get is AB. Seldom does AB equal BA, thus the design of engineering systems requires (heavy) noncommutative calculations. Helton's group has a broad based effort to develop methods and theory for computer assistance of such calculations. Inequalities with noncommuting elements is now a major topic in engineering and Helton's group is working out systematic methods for treating them.