Mathematical Sciences: Operator Theory and Communications Engineering

赫尔顿教授的项目是应用于工程问题的数学研究。数学涉及平面区域（如圆盘）上的矩阵值函数。在这些函数中，有一类表现特别好的函数叫做解析函数。在反馈稳定控制系统的设计中，解析函数或多或少的任意函数的最佳逼近，以及在给定区域的指定点上求取指定值的解析函数是非常重要的问题。通过运用泛函分析和算子理论的方法，Helton寻求解决这类问题的有效方法。Helton的观点是，对于矩阵值函数，算子理论中的交换子提升几乎等价于插值理论。他将寻求发展一种非线性换向提升理论，并将其与现有的非线性控制系统理论联系起来。他将研究解析函数空间上最优化的定性和数值方面；在这一部分的项目中，计算实验和理论见解将相互支持。这项工作的另一个方面涉及矩阵补全，填充部分指定的矩阵的条目，以便使结果具有特定的性质，例如正性。在整个过程中，Helton将继续关注频域最优控制和最坏情况设计的应用。

Professor Helton's project is research in mathematics that has applications to problems in engineering. The mathematics involves matrix-valued functions on planar regions such as the disc. Among these there is a particularly well-behaved class of functions called analytic. Best-approximation of more or less arbitrary functions by analytic ones, and finding analytic functions that take on prescribed values at prescribed points in the region are problems that turn out to be important in the design of feedback stabilized control systems. By applying methods of functional analysis and operator theory, Helton seeks efficient ways of solving problems like these. Helton's point of view is that commutant lifting in operator theory is nearly equivalent to interpolation theory for matrix- valued functions. He will seek to develop a nonlinear commutant lifting theory, and to link it with the existing theory of nonlinear control systems. He will investigate qualitative and numerical aspects of optimization over spaces of analytic functions; computational experiment and theoretical insight will support one another mutually in this part of the project. A further aspect of the work involves matrix completion, filling in the entries of a partially specified matrix so as to make the result have a particular property such as positivity. Throughout, Helton will keep in view applications to optimal control and worst case design in the frequency domain.