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

本文提出的工作有两个方向。第一个方向是算子理论与系统控制理论的结合。本课题的目的是将换向提升理论及其与插值问题、Hankel和Toeplitz算子以及鲁棒控制理论的联系扩展到更一般的算子类。该项目还将包括可实现的算法，用于计算类型运算符的规范，这些运算符出现在使用换向提升方法解决插值问题和鲁棒控制问题中。这些操作者的定性和定量结果与控制器的设计有关，该控制器既能稳定具有参数或建模不确定性的系统，又能满足一定的性能要求。具体地说，研究者将继续他的一般算子扩展问题的研究，其中包括非线性系统的因果交换提升方法。他将开发一类多变量分布式系统的混合灵敏度最小化问题的算法，这是鲁棒控制理论的基本问题之一。另一个方向是研究一些重要解析函数空间上Toeplitz算子的换向子，包括单位盘的向量值Hardy空间和多盘的Hardy空间。这些换向子来源于数学物理和非交换分析。这个方向的研究应该引起数学分析人员和许多应用科学领域的科学家的兴趣。

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.