Operator Theory and Systems Engineering

项目摘要DMS-9732891，PI：Helton 这是一项继续支持以下几个项目的建议： 工程系统算子理论与泛函分析 理论几十年来，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。 事实上，大多数商业软件（至少在控制工程社区） 求解解析函数问题的方法就是基于函数和矩阵之间的这种相互作用。与应用密切相关的分析的一个分支是经典的Nevanlinna-Pick-Nehari理论，或等价的交换提升理论，该领域的一部分称为算子模型理论。 早期的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年代早期，这位研究人员和其他人证明，这对工程系统的设计至关重要， 系统的稳定性是关键约束。这激发了更多的数学发展，现在它是一个领域， 与控制最密切相关的功能分析 工程.多年来（自Norbert Wiener以来）， 优化的均方性能。上述理论最终导致 （商业上常见的）优化最坏情况的工具 频域性能这个建议的目的是将这个核心理论扩展到全新的方向：1.高度非线性的推广。 我们的目标是找到规范的"非线性推广"， 解析函数论和算子论的相关部分。 人们希望控制的许多系统都是非线性的（例如，喷气发动机）， 因此，对有效的非线性理论的需求很大。2.线性算子}矩阵区间的交，遗传算子和多项式 是吉姆·阿格勒的研究课题3.解析函数空间上的优化}（与奥兰多美利奴等人合著）。定性理论，计算机算法，基于此理论，分析此类算法。这项工作与其他数学分支的联系。这些是线性工程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。4.计算机算子代数。线性工程系统理论和算子理论中充斥着非交换代数中的计算。例如，如果上面的经典核心设置在工程中， “状态空间坐标”，主要定理大多可以由矩阵代数导出。赫尔顿的组织是主要的供应商 用于在Mathematica中执行一般非交换计算的软件（称为NCAlgebra）。这项工作已经经历了几个阶段，在目前的研究中，由于C++中的莫拉和链接的Mathematica软件，研究人员现在有大量的软件实现非交换Groebner基算法 用于以各种方式对输出进行排序和"收缩"。它被用于雄心勃勃的追求，试图找到计算机方法来帮助研究人员“发现”（而不仅仅是验证）算子理论和工程系统分支中的定理。

PROJECT ABSTRACT DMS-9732891, PI: Helton This is a proposal to continue support for several projects in the 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 by this investigator and others 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 aim of this proposal is to extend this core theory in radically new directions: 1.Highly nonlinear generalizations. The goal is to find canonical ''nonlinear generalizations" of analytic function theory and the related parts of operator theory. Many systems which people wish to control are nonlinear (e.g., jet engines), so there is a big demand for an effective nonlinear theory. 2.Linear Operators} Intersections of matrix intervals, and Hereditary operators and polynomials are the subject of work with Jim Agler. 3. Optimization over spaces of analytic functions} (with Orlando Merino and others). Qualitative theory, computer algorithms based on this theory, a nalysis of such algorithms. Connections of this work with other branches of mathematics. These are the key optimization problems which arise in designs of linear engineering systems where there are competing constraints, or uncertainty in the math model of the physical system one is trying to control. 4. Computer operator algebra.Linear engineering systems theory and operator theory are rife with calculations in a noncommuting algebra. For example, if the classical core above is set in engineering `statespace coordinates', the main theorems can be derived mostly by matrix algebra. Helton's group are the main providers of software (called NCAlgebra) for performing general noncommuting calculations in Mathematica. The work has gone thru several phases and in current research the investigators now have extensive software implementing noncommutative Groebner basis algorithms due to Mora in C++ and linked Mathematica software for sorting and ''shrinking" the output in various ways. It is used in the highly ambitious pursuit of trying to find computer methods for helping a researcher ''discover" (rather than just verify) theorems in branches of operator theory and engineering systems.