Mathematical Sciences: T+H*H Operators in Linear-Quadratic and H-Infinity Problems

本课题主要研究线性系统控制问题的数学理论中出现的一些问题。该项目的中心主题是T + H*H形式的算子，其中T代表Toeplitz， H代表Hankel。动机是双重的。首先，正如之前首席研究员所示，线性二次最优控制问题具有T + H*H算子，该算子揭示了许多传统方法无法识别的结构。其次，正如研究者所示，H -∞设计问题还涉及到一个T + H*H算子，其最大特征值提供了所谓的可实现反馈性能，这是一个关键参数。因此，这个算子揭示了线性二次问题和H -∞问题之间的奇怪联系。我们将对这种联系及其谱和计算方面进行研究。这项研究是由理论工程师和数学家在国内进行的一项更大的努力的一部分，目的是为具有鲁棒性要求的线性控制系统开发一种统一的设计方法。这项研究结合了系统工程和数学中的算子理论等领域。

This project focuses on some issues arising in the mathematical theory of control problems for linear systems. The central theme of this project is an operator of the form T + H*H where T is Toeplitz and H is Hankel. The motivation is two-fold. First, as previously shown by the principal investigator, the linear quadratic optimal control problem has a T + H*H operator that reveals a lot of structure not identifiable by conventional means. Secondly, as shown by the investigator, the H - infinity design problem alsdo involves a T + H*H operator, the largest eigenvalue of which provides the so called achievable feedback performance, a critical parameter. As such, this operator reveals a strange connection between the linear quadratic problem and H - infinity problems. This connection will be investigated along with its spectral and computational aspects. This research is part of a larger effort conducted in this country by theoretical engineers and mathematicians to develop a unified design methodology for linear control systems with robustness requirements on the design. This research combines fileds such as systems engineering and operator theory in mathematics.