Operatory Theory and Systems Engineering

非线性算子理论。目标是找到解析函数理论和线性算子理论相关部分的典型“非线性推广”。许多经典的和许多新的解析函数定理都有一个纯线性算子的表述。令人惊讶的是，这些关于线性算子的定理实际上并不需要线性，而且很明显，它们将有广泛的非线性推广。这个广阔的开放领域接近于控制理论。计算机运算代数。线性工程系统理论和算子理论充斥着非交换代数的计算。Helton和M. Stankus的团队是Mathematica中用于执行一般非交换计算的软件（称为NCAlgebra）的主要供应商。软件的一个阶段是一个非常强大的“黄色记事本”，包含他们开发的许多算法。例如，下面将要讨论的“非交换凸性”算法。在另一个阶段，由于Mora和以各种方式对输出进行排序和“收缩”的算法（这在非交换情况下是至关重要的），有大量的软件实现了非交换的Groebner基算法。由于这些技术是新的，对传统问题进行实验是很重要的。非交换的不平等。最近，Helton和合作者在一个理论和结果算法上取得了进展，该理论和结果算法取非交换变量Z的有理函数F，并输出一系列不等式，这些不等式决定了Z的一个定义域G，其中F是“矩阵凸”。决定性的非平凡表明由算法确定的域G是f的矩阵凸性的“最大可能”域。这是工程师们现在用舒尔补技巧将设计问题转化为线性矩阵不等式的自动方法的第一次尝试。解析函数在空间上的优化。定性理论，基于此理论的计算机算法，分析这类算法。这些是线性系统设计中出现的关键优化问题，其中存在竞争约束，或者试图控制的物理系统的数学模型存在不确定性。本研究针对工程系统理论中算子理论和泛函分析部分的几个项目展开。几十年来，线性算子理论与解析函数理论和工程有着密切的相互作用。实际上，大多数用于解决解析函数问题的商业软件（至少在控制工程社区中）都是基于函数和矩阵之间的这种类型的相互作用。与应用密切相关的一个分析分支是经典的Nevanlinna-Pick-Nehari理论，或等效的交换提升理论，是算子模型理论的一部分。它的早期发展是出于最纯粹的数学原因，但在20世纪70年代中期和80年代初，它被证明对工程系统的设计至关重要，其中系统的稳定性是关键的约束。这激发了更多的数学发展，现在它是功能分析与控制工程最密切相关的领域之一。多年来（自Norbert Wiener以来），设计工具优化了均方性能。上述理论最终导致（商业上常见的）优化最坏情况频域性能的工具。许多拟议研究的目标是将这一理论扩展到几个全新的方向，我们列出了主要方向。高度非线性推广；目标是找到解析函数理论和线性算子理论相关部分的典型“非线性推广”。这一广阔的开放领域与控制理论密切相关。人们希望控制的许多系统都是非线性的（例如，喷气发动机）。非交换计算机代数；如果一个信号进入系统a，出来然后进入系统B，我们得到的是BA，而如果一个信号进入系统B，然后进入系统a，我们得到的是AB。AB很少等于BA，因此工程系统的设计需要（大量的）非交换计算。Helton的团队在开发计算机辅助此类计算的方法和理论方面做了广泛的努力。非可交换元素的不等式现在是工程学的一个主要课题，Helton的团队正在研究处理它们的系统方法。

ABSTRACTNONLINEAR 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 systemsrequires (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.