Operator Theory Arising from Systems Engineering

本课题的研究涉及工程系统理论中的泛函分析和算子理论。工程中的许多问题都可以用矩阵不等式来表述和研究。此外，许多线性系统控制问题以矩阵作为其自然变量，因此需要理解多项式和非交换变量中的有理函数。具有特殊实际重要性的是“凸性”，因为在它的存在下，局部极小值（可以通过计算得到）实际上就是全局极小值（你真正想要找到的）。然而更容易处理的是线性矩阵不等式。一个令人惊讶的发现是，任何基于凸非交换有理函数的矩阵不等式都等价于由同一函数确定的线性矩阵不等式。在当前的项目中，首席研究员将致力于将多项式不等式（通常称为半代数几何）的经典（交换）理论扩展到非交换多项式。他将探索这门学科中与凸性和曲率有关的方面，根据这一理论开发计算机算法，分析这些算法，并寻找与其他数学分支的联系。在工程中，几乎所有运动的物体的第一行数学模型都是一个“系统”；也就是说，一个有输入和输出的盒子。即使这些不是线性的，设计的第一步通常是线性化，所以线性系统的数学在飞机、汽车、卫星、地震结构、炼油厂、机床、流动流体等领域的控制中起着重要作用。20世纪90年代线性系统工程的最大进步是认识到大多数线性控制问题直接转化为矩阵不等式。在20世纪的数学中出现了许多不同类型的矩阵不等式，但今天在工程系统中占主导地位的矩阵不等式通常要求矩阵的多项式或有理函数是“正半定的”，这是一种技术性质。许多工程矩阵不等式是“表现不佳的”，但有一个经典的核心问题，其中这些不等式转化为“表现良好的”线性矩阵不等式。后者在计算上是可处理的，并且可以证明不存在虚假最优（这可能导致不安全的设计）。在系统工程中，像这样的代数公式被编入现代计算机程序包。这个项目的目标是：在数学和计算上测试一组给定的矩阵不等式是否表现良好，了解如何将糟糕的矩阵不等式转化为良好的矩阵不等式，并开发相应的计算机算法。首席研究员还致力于促进数学家和工程师之间的互动。除了和自己的研究生一起工作外，他在夏天还经营着一个由学生组成的小型计算实验室。虽然这些学生中有许多人继续获得纯数学博士学位，但实验室经验和对工程的接触大大拓宽了他们的视野。

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.