Mathematical Sciences: Classical Analysis: Complex Analysis,Computation, and Control

数学科学：经典分析：复分析、计算和控制

• 批准号: 9002852
• 负责人: Donald Marshall
• 金额: $ 4.8万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-05-01 至 1992-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9002852&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Classical; Analysis; Complex

Work on this project combines investigations into mathematical questions of classical complex analysis with applications to control problems and finitely generated algebras of functions of several complex variables. The optimal control problems, arising in engineering, are phrased in terms of vector-valued, bounded, functions defined on certain cylinder-like sets. One is interested in minimizing a certain functional over the class of functions. Sometimes there are no extermal functions, other times they turn out to be analytic. Sometimes there are unique solutions, but not always. The goal of this work is to describe the cylinders which give rise to bounded analytic optimal functions which are smooth on the boundary of the cylinder. A number of positive results in this area have focused the problem to one of taking analytic selections of functions on sets fibered over the unit circle and determining when such functions agree with a given function when restricted to the unit interval. Work on algebra generators evolves from an earlier result of Gleason which suggests that the classes of continuous and holomorphic functions in the two-dimensional domains which vanish at the origin are finitely generated algebras. This is easy to do for the bidisc, although the minimum norm of the generators is not yet known (it is probably unity). This particular question is to be addressed by computational methods first to verify the conjectured minimum. For the general Gleason problem, a new approach will be made by converting it to an equivalent problem in first order partial differential equations. Estimates for the norm of the generators should follow if this plan works out.

该项目的工作结合了调查， 经典复分析的数学问题 应用于控制问题和代数生成 多个复变量的函数。 工程中出现的最优控制问题是 用向量值、有界、定义在 某些圆柱状的集合。 一个是最小化 函数类上的某个函数。 有时 是没有外部功能的，其他时候它们被证明是 分析。 有时有独特的解决方案，但并不总是如此。 这项工作的目标是描述圆柱体， 上升到有界的解析最优函数是光滑的 圆柱体的边界。 取得了一些积极成果， 这一领域把问题集中到分析 在单位圆上纤维化的集合上选择函数， 确定这些函数何时与给定函数一致， 仅限于单位间隔。 关于代数生成器的工作是从 Gleason提出，连续和 二维域上为零的全纯函数 在原点处是n-生成代数。 这很容易 对于bidisc，尽管生成器的最小范数是 还不知道（可能是统一）。 这项问题 首先要通过计算方法来验证 最低限度。 对于一般的格里森问题，一个新的 将其转化为一个等价的问题 在一阶偏微分方程中。 估计数 如果这个计划成功，发电机的标准应该遵循。