Mathematical Sciences: Research Problems in Stochastic Control and Numerical Approximation

本文将研究以下几个主题：(1)一些奇异随机控制问题：两个问题：一个是分析一类多维的奇异控制问题，另一个是一维奇异随机控制的改编版本。(2)脉冲控制问题：将研究脉冲控制理论在最优维修和顺序调度问题中的两种新应用。(3)一类具有长期平均准则的跳跃扩散随机控制的遍历控制问题。重点是对相关不变测度的研究。(4)随机控制问题的数值分析：求解一类多维奇异控制问题的数值算法。一些Hamilton-Jacobi-Bellman方程的数值逼近。(5)随机分布参数系统中的若干问题：研究非线性随机分布系统的最优控制引起的无穷维h - j - b方程的解。本文将研究具有可能简并的系统的稳定性和渐近行为。这项研究是控制理论的一部分，在工程和计算机科学中具有潜在的应用前景。

The following topics will be investigated: (1) Some Singular Stochastic Control Problems: Two problems: One is concerned with the analysis of a class of singular control problems in several dimensions, and the second pertains to a adapted version of singular stochastic control in one dimension. (2) Impulse Control Problems: Two novel applications of impulse control theory to optimal maintenance and sequential scheduling problems will be studied. (3) Ergodic Control Problems with a class of stochastic control of diffusion with jumps with a long- term average criterion. The emphasis is on the study of the associated invariant measure. (4) Numerical Analysis of Stochastic Control Problems: Numerical algorithm for solving a class of singular control problems in several dimensions. Numerical approximation of some Hamilton-Jacobi-Bellman equations. (5) Some Problems in Stochastic Distributed- Parameter Sytems: The investigators plan to study the solution of an infinite-dimensional H.J.B. equation arising from optimal control of a non-linear stochastic distributed system. The stability and asymptotic behavior of such a system with possible degeneracy will be investigated. This research is part of control theory with potential applications in engineering and computer science.