Operator Theory Arising from Systems Engineering

这是一份支持算子理论和泛函分析部分主要与线性矩阵不等式（LMIs）和工程系统理论相关的项目的建议。基于lmi的半确定规划是过去15年来优化领域的主要发展之一，其应用范围遍及大多数定量科学领域。这个建议的主旨是：半确定规划的范围是什么？可以用lmi处理的问题是凸问题，但反过来，哪些凸问题可以用lmi处理？在合作者的帮助下，首席研究员目前在这个问题的主要方面取得了重大进展。特别强调由系统工程引起的LMI问题。虽然lmi的数值问题一直是研究的热点，但目前还没有系统的理论来简化或分析矩阵不等式的代数问题。这个建议的主要部分是发展这样一个理论。这在功能分析中产生了大量优雅的问题，包括以下描述的问题。如果对称非交换多项式在矩阵上的所有值都是正的半定矩阵，则称其为“自由正”。“自由凸性”的定义类似。求解一个非交换多项式不等式的所有矩阵的集合称为“自由半代数集”。线性矩阵不等式涉及使给定的线性铅笔取正半定值的条件。1. 哪些集合是某个LMI的所有解的集合？2. 可以通过改变变量来实现自由凸性吗？3. 找到半代数集的凸包？正如J. Nie和研究者所分析的那样，这些可以作为某些凸集的投影来构建。免费凸包非常有趣。4. 非交换实代数几何：在另一个方向上，从希尔伯特的第17个问题开始，代数证明是否等同于一个多项式为正而另一个多项式为正的陈述？它的非交换类似物的开发进展顺利，大量的工作正在进行中。5. 经典实代数几何的一个基础是半代数集的投影是半代数的。最近的定理（研究者和合作者）暗示这在物质世界中（绝对）是错误的，产生了一连串的问题。所提出的工作与半确定规划有关，半确定规划是在科学和工程的许多分支中发现的一种凸优化。特别是我们在线性系统工程和控制中看到的是矩阵未知数的问题。简化物理问题并将其转化为凸问题，目前（在数千篇论文中）是通过特殊的代数技巧完成的。在建议的目标是发展一个理论（一个非交换的实代数几何），可能用于系统化这一点。此外，研究小组是软件（称为NCAlgebra）的主要提供者，用于在软件程序Mathematica中执行一般的非交换代数计算。现在的重点是处理非交换不等式的算法。此外，该小组还在非交换代数计算的基础上进行了数值优化。该项目将吸引研究生和一些本科生参加夏季研究和计算项目。这种实验室应用经验将拓宽学生的训练，其中许多人获得了纯数学学位。

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.