Operator Theory and Systems Engineering

DMS-9732891， PI: Helton这是一份继续支持与工程系统理论相关的算子理论和功能分析部分的几个项目的提案。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，这个研究者和其他人证明了这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。本提案的目的是在全新的方向上扩展这一核心理论：高度非线性的推广。目标是找到解析函数理论和算子理论相关部分的典型“非线性推广”。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有很大的需求。2.线性算子}矩阵区间的交点，以及遗传算子和多项式是Jim Agler工作的主题。解析函数空间上的优化}（与Orlando Merino等人合作）。定性理论，基于此理论的计算机算法，分析这类算法。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，或者试图控制的物理系统的数学模型存在不确定性。4. 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。例如，如果上面的经典核心设置在工程“状态空间坐标”中，则主要定理主要可以通过矩阵代数推导出来。Helton的团队是在Mathematica中执行一般非交换计算的软件（称为NCAlgebra）的主要提供者。这项工作经历了几个阶段，在目前的研究中，研究人员现在有了广泛的软件来实现非交换格罗布纳基算法，这是由于c++中的Mora，并与Mathematica软件相关联，以各种方式对输出进行排序和“缩小”。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.