Operator Theory Arising from Systems Engineering

本方案的研究方向是与工程系统论相关的泛函分析和算子理论。许多线性控制问题都可以用矩阵不等式来描述和研究。它们是通过考虑非交换变量x中的多项式来表示的。由于PI，并且作为他先前授权的一部分而获得的一个基本结果是，每个矩阵正非交换多项式都可以写成非交换多项式的平方和。在这份提案中，希尔顿将致力于扩展这一理论，开发基于这一理论的计算机算法，分析这种算法，并与数学的其他分支建立联系。这些都是线性系统设计中出现的关键优化问题。20世纪90年代S在线性系统工程方面的最大进步是实现了大多数线性控制问题直接转化为矩阵不等式，简称MI。这些问题中的许多都表现不佳，但经典的核心问题会转化为表现良好的线性矩阵不等式(LMI)。在上个世纪的数学中已经出现了许多不同类型的矩阵不等式，但在工程系统中占主导地位的那些通常采取矩阵的多项式或半正定的有理函数的形式。在工程上，就是这样的代数公式被编程到现代计算机程序包中。这个项目的目标是了解一个人如何将糟糕的管理信息转化为良好的管理信息。这在什么时候可能呢？

The research of this proposal is in functional analysis and operator theory related to engineering system theory. Many linear control problems can be phrased and studied in terms of matrix inequalities. These are formulated by a consideration of polynomials in noncommutative variables x. A basic result, due to the PI and obtained as part of his previous grant, is that every matrix positive noncommutative polynomial can be written as a sum of squares of noncommutative polynomials. In this proposal Helton will work on extending this theory, developing computer algorithms based on this theory, the analysis of such algorithms, and connections with other branches of mathematics. These are key optimization problems arising in designs of linear systems.The biggest advance in linear systems engineering during the 1990's is the realization that most linear control problems convert directly to matrix inequalities, abbreviated MIs. Many of these are badly behaved but a classical core of problems convert to linear matrix inequalities (LMIs) that are nicely behaved. Many different types of matrix inequalities have come up in the mathematics of the previous century, but the ones that dominate in engineering systems usually take the form of a polynomial or rational function of matrices being positive semidefinite. It is algebraic formulas like these that are programmed into modern computer packages in engineering. The goal of this project is to understand how one converts bad MIs to nice MIs. When is this possible?