Mathematical Sciences: Commutant Lifting Methods in Operator Theory and Robust Control Theory

数学科学：算子理论和鲁棒控制理论中的交换提升方法

• 批准号: 9706838
• 负责人: Caixing Gu
• 金额: $ 6.02万
• 依托单位: California Polytechnic State University Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706838&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Commutant; Lifting; Methods; Mathematical; Sciences; Commutant; Lifting; Methods

Abstract Gu The proposed work is in two directions. The first direction lies at the interface of operator theory and system and control theory. The purpose of this part of the project is to extend to a more general class of operators the commutant lifting theory along with its connection with interpolation problems, Hankel and Toeplitz operators and robust control theory. The project will also include implementable algorithms for the computation of the norms of the type operators which appear in the solutions of interpolation problems and robust control problems by using commutant lifting methods. The qualitative and quantitative results for those operators are relevant to the design of controllers which stabilize the systems with parameter or modeling uncertainty while at the same time meet certain performance specifications. Specifically, the investigator will continue his study of a general operator extension problem which includes the causal commutant lifting approach for nonlinear systems. He will develop algorithms for the mixed sensitivity minimization problem of a class of multivariable distributed systems which is one of the fundamental problems in robust control theory. Another direction is related to the study of commutators of Toeplitz operators on certain important spaces of analytic functions including vector-valued Hardy spaces of the unit disk and Hardy spaces of the polydisk. These commutators have their origin in mathematical physics and non-commutative analysis. The research in this direction should be of interest to mathematical analysts as well as to scientists in many fields of applied sciences.

摘要gu拟议的工作是两个方向。 第一个方向在于操作者理论，系统与控制理论的界面。该项目的这一部分的目的是扩展到更一般的运营商类别的换议提升理论以及与插值问题，Hankel和Toeplitz运营商以及强大的控制理论的联系。 该项目还将包括可实施的算法，用于计算插值问题解决方案中出现的类型运算符的规范和通过使用换向的提升方法的鲁棒控制问题。这些操作员的定性和定量结果与控制器的设计有关，该控制器可以通过参数或建模不确定性来稳定系统，同时符合某些性能规格。具体而言，研究人员将继续他对普通操作员扩展问题的研究，其中包括非线性系统的因果接向提升方法。他将开发一类多变量分布式系统的混合敏感性最小化问题的算法，这是强大控制理论中的基本问题之一。另一个方向是与对分析函数的某些重要空间的换向器有关的研究，包括单位磁盘的矢量值Harty空间和Polydisk的Hardy空间。这些换向因素起源于数学物理学和非交通性分析。 数学分析师以及许多应用科学领域的科学家应该引起这个方向的研究。