Operatory Theory and Systems Engineering

抽象非线性算子理论。目标是找到解析函数论和线性算子理论的相关部分的典型的“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粹的线性算子的表述。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，它们将被广泛的非线性推广。这一广阔的领域接近于控制理论。计算机算子代数。线性工程系统理论和算子理论充斥着非对易代数中的计算。Helton的团队和M.Stankus是在数学中执行一般非对易计算的软件(称为NCAlgebra)的主要提供商。该软件的一个阶段是一个非常强大的“黄色记事本”，包含了他们开发的许多算法。例如，下面讨论的“非对易凸性”算法将进入。在另一个阶段，有大量的软件实现了非对易的Groebner基算法，这是由于Mora和以各种方式对输出进行排序和“收缩”的算法(这在非对易的情况下是至关重要的)。由于这些技术是新技术，因此在传统问题上进行实验是重要的。非对易不等式。最近，Helton和他的合作者在一个理论和结果算法上取得了进展，该算法取非对易变量Z的有理函数F，并输出一族确定Z的一个区域G的不等式，在该区域上，F是“矩阵凸的”。绝对不平凡的是，表明由算法确定的域G是F的矩阵凸性的“最大可能”域。这是工程师现在用Schur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。解析函数空间上的最优化。定性理论，以此理论为基础的计算机算法，分析此类算法。这些都是在线性系统设计中出现的关键优化问题，其中存在竞争约束，或试图控制的物理系统的数学模型中的不确定性。研究针对与工程系统理论相关的算子理论和函数分析的部分项目。几十年来，线性算子理论与解析函数理论和工程学有着很强的相互作用。事实上，大多数用于解决解析函数问题的商业软件(至少在控制工程领域中)都是基于函数和矩阵之间的这种相互作用。与应用密切相关的一个分析分支是经典的Nevanlinna-Pick-nehari理论，或等价的交换提升理论，该领域的一部分被称为算子模型理论。它的早期发展是出于最纯粹的数学原因，但在20世纪70年代中期的S和80年代初的S，这被证明是工程系统设计的关键，其中系统的稳定性是关键的制约因素。这推动了更多的数学发展，现在它是与控制工程联系最密切的泛函分析领域之一。多年来(自诺伯特·维纳以来)，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)用于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，我们列出了主要的几个方向。HIGHLY非线性推广；目标是找到解析函数论和线性算子理论的相关部分的典型的“非线性推广”。这一广阔的领域与控制理论密切相关。人们想要控制的许多系统都是非线性的(例如，喷气发动机)。非共通计算机代数；如果一个信号进入系统A，出来，然后进入B，我们得到的是BA，而如果一个信号进入B，然后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(繁重的)非对易计算。希尔顿的团队在开发计算机辅助这类计算的方法和理论方面有着广泛的基础。带有非对易元素的不等式现在是工程学中的一个主要话题，Helton的团队正在研究系统的方法来处理它们。

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.