Operator Theory and Systems Engineering

项目摘要DMS-9732891，PI：Helton这是一项建议，继续支持与工程系统理论有关的算子理论和功能分析部分的几个项目。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1970年代中期的S和1980年代初的S，这一点被这位研究员和其他人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。这项提议的目的是将这一核心理论扩展到全新的方向：1.高度非线性的概括。目标是找到解析函数论和算子理论的相关部分的典型的“非线性推广”。人们希望控制的许多系统都是非线性的(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集、遗传算子和多项式是与Jim Agler合作的主题。3.解析函数空间上的最优化)(与Orlando Merino等人)。定性理论，以此理论为基础的计算机算法，分析此类算法。这项工作与其他数学分支的联系。这些都是在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.计算机算符代数。线性工程系统理论和算子论在非对易代数中充满了计算。例如，如果将上述经典核设置在工程上的“状态空间坐标”中，则主要定理可以主要由矩阵代数来推导。Helton的团队是数学计算软件(称为NCAlgebra)的主要供应商，该软件用于执行一般的非通勤计算。这项工作已经经历了几个阶段，在目前的研究中，研究人员现在有大量的软件来实现非对易的Groebner基算法，这是由于Mora在C++中的作用，以及链接的数学软件，用于以各种方式对输出进行排序和‘’收缩‘’。它被用于非常雄心勃勃的追求，试图找到计算机方法来帮助研究人员在算子理论和工程系统的分支中‘发现’(而不仅仅是验证)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.