Mathematical Sciences: Operator Theory and Systems Engineering

数学科学：算子理论与系统工程

• 批准号: 9501064
• 负责人: J. William Helton
• 金额: $ 13万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9501064&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Operator; Theory; Systems

9501064 Helton Most of the proposed research falls into three areas: 1. OPTIMIZATION OVER SPACES OF ANALYTIC FUNCTIONS Qualitative theory, computer algorithms based on this theory, analysis of such algorithms. Connections of this work with other branches of mathematics.The engineering motivation here is strong; these are the key optimization problems which arise il design of systems and amplifiers where there are competing constraints, time as well as frequency domain constraints, or uncertainty in the math model of the physical system one is trying to control. 2. NONLINEAR SYSTEMS The point here is to find canonical nonlinear operator theory which extends the classical linear theory of Nevanlinna-Pick and many followers. That theory is the core of linear control 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. Unfortunately, the mathematical theory is new and primitive. More general problems need to be solved and special cases need to be understood in greater detail. 3. COMPUTER OPERATOR ALGEBRA Linear engineering systems theory and operator theory are rife with calculations in a noncommuting algebra. Thus it is likely that someday there will be a branch of these subjects devoted to their symbolic computational aspects. Helton's group have, support, and are upgrading extensive software, called NCAlgebra, for performing noncommuting calculations in Mathematica. The software is currently at the level of a very powerful `yellow pad'. The main issue now is what `intelligence' should go in noncommuting packages. For example, calculations with energy conserving systems produce polynomials in matrices satisfying certain relations. Thus it is natural to find a list of rules for simplifying such polynomials tailored to this particular situation. The goal is to find and understand `complete' lists of simplifying rules. Helton and collaborat ors have successfully studied several situations of this type and more are in progress. ***

小行星9501064 大多数拟议的研究福尔斯分为三个方面：1。解析函数空间上的优化 定性理论，基于此理论的计算机算法， 分析这些算法。 这项工作与其他工作的联系 数学的分支。这里的工程动机很强;这些 是设计中出现的关键优化问题， 存在竞争约束的系统和放大器， 时域和频域约束，或试图控制的物理系统的数学模型中的不确定性。 2.非线性系统这里的重点是寻找正则非线性算子 理论，它扩展了经典的线性理论， 内万琳娜-皮克和许多追随者。这个理论是 线性控制理论的核心人们希望控制的许多系统 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。 不幸的是，数学理论是新的和原始的。 更多的问题需要解决， 特殊情况需要更详细地理解。 3.计算机算子代数线性工程系统理论和算子理论是 充满了非交换代数的计算 因此，很可能有一天会有一个分支， 专门研究符号计算方面的课题。 Helton的团队拥有、支持并正在进行广泛的升级 一个叫做NCAlgebra的软件，用于执行非交换 Mathematica中的计算 该软件目前 在一个非常强大的“黄色垫”的水平。 现在的主要问题是“情报”应该放进去什么 非通勤包裹例如，能量守恒系统的计算产生满足某些关系的矩阵多项式。 因此，找到一系列简化规则是很自然的 这种多项式适合于这种特定的情况。 目标是找到并理解“完整”的 简化规则。 Helton及其合作者 已经成功地研究了几种这种类型的情况， 更多的正在进行中。 ***