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

这个项目的工作结合了对经典复分析的数学问题的研究，以及对控制问题和多个复变量函数的有限生成代数的应用。工程中出现的最优控制问题是用定义在某些类圆柱集上的矢量值、有界函数来表示的。人们感兴趣的是最小化函数类上的某个泛函。有时没有外函数，有时它们是解析的。有时会有独特的解决方案，但并不总是如此。这项工作的目的是描述产生有界解析最优函数的圆柱，这些函数在圆柱的边界上是光滑的。这一领域的一些积极结果将问题集中到对单位圆上的集合进行函数的解析选择，并确定当限制在单位区间内时，这些函数何时与给定函数一致。关于代数生成元的工作是从Gleason的早期结果发展而来的，该结果表明在原点消失的二维域中的连续函数和全纯函数类是有限生成的代数。对于双盘来说，这很容易做到，尽管生成器的最小范数还不知道(它可能是单位)。这个特殊的问题首先要用计算方法来解决，以验证猜想的最小值。对于一般的Gleason问题，通过将其转化为一阶偏微分方程组的等价问题，提出了一种新的方法。如果这项计划成功，对发电机标准的估计也应该遵循。

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.