Operator Theory Arising from Systems Engineering

源于系统工程的算子理论

• 批准号: 1201498
• 负责人: J. William Helton
• 金额: $ 28.66万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201498&HistoricalAwards=false
• 关键词: Operator; Theory; Arising; Systems; Engineering

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.

这是一个支持项目的建议，主要涉及线性矩阵不等式（LMI）和工程系统理论的算子理论和泛函分析部分。基于线性矩阵不等式的半定规划是过去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：什么是半定规划的范围？可以用LMI处理的问题是凸的，但反过来，哪些凸问题可以用LMI处理？与合作者的主要研究人员目前正在取得重大进展的主要方面，这一问题。特别强调LMI问题的系统工程的动机。虽然LMI的数值计算是热门的追求，但还没有系统的理论来简化或分析矩阵不等式的代数。本提案的主要部分是发展这样一种理论。这在函数分析中产生了丰富的优雅问题，包括下面描述的那些。一个对称的非交换多项式被称为“自由正”，只要它在矩阵上的所有值都是一个半正定矩阵。“自由凸性”的定义类似。解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线性束取半正定值的条件。1.哪些集合是某个线性矩阵不等式的所有解的集合？2.可以改变变量来实现自由凸性吗？3.求半代数集的凸船体？这些可以建立为某些凸集的投影，正如J. Nie和调查员所分析的那样。自由凸包是非常有趣的。4.非交换真实的代数几何：从希尔伯特第17个问题开始，从另一个方向来看，代数证书是否等价于一个多项式为正而另一个多项式为正的声明？非交换类似物的发展进展顺利，大量的工作正在进行中。5.经典真实的代数几何的一个基石是半代数集合的投影是半代数的。最近的定理（由研究者和合作者）暗示这是（压倒性的）错误的矩阵世界，产生了一连串的问题。拟议的工作承担，半定规划，一种类型的凸优化发现在许多科学和工程分支。特别是人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是通过特别的代数技巧来完成的（在数千篇论文中）。该提案的目标是发展一种理论（一种非交换的真实的代数几何），可以用来系统化这一点。此外，研究小组是软件（称为NCAlgebra）的主要提供商，用于在软件程序Mathematica中执行一般非交换代数计算。现在的重点是处理非交换不等式的算法。此外，该小组还根据非交换代数计算进行数值优化。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。这个实验室的应用经验将扩大学生的培训，其中许多人获得纯数学学位。