Mathematical Sciences: Singular and Boundary Control of Multidimensional Diffusion Processes

建议的研究属于最佳领域 随机控制它涉及扩散的最优控制 过程的自由边界，以及最优控制 在随机过程的反射边界处。这种类型的 控制自然出现在没有控制的问题中， 自然限制的控制率和最佳的 控制速率ID无穷大。最优策略是反映 从先验未知边界的过程。这种类型的 在添加剂控制的问题中， 控制率没有先验限制。其旨在 发展了相关的偏微分方程， 梯度约束，研究最佳反射屏障， 多维的情况下，并调查其他问题 针对排队、库存和柔性的奇异控制， 制造模型。 建议的研究属于最佳领域 随机控制这种类型的控制问题出现在模型中 涉及飞机的自动巡航控制， 不确定的风的条件下，空间飞行器受到小 机械单位的扰动。这个问题的解决方案 建议最佳时机的纠正过程中， 太空飞行器的飞机。奇异控制也出现了 在排队/生产/制造模型中。在一个典型的 一个柔性制造系统寻求最优调度 打开和关闭生产单元，购买的时间 新设备或增加现有设备的容量， 寻找最佳库存水平，以备供应等。 在许多情况下，生产过程面临不确定性， 需求波动或由于工艺不可靠 本身，它是可能的，使用奇异控制理论， 解决上述问题。

The proposed research lies within the area of optimal stochastic control. It deals with optimal control of diffusion processes by means of a free boundary, as well as optimal control at a reflection boundary of the stochastic process. This type of control naturally appears in problems in which there is no natural restriction on the rates of control and the optimal control rate id infinite. The optimal policy is then to reflect the process from the a priori unknown boundary. Such type of action arises in problems with additive control, where there are no a priori limits on the control rates. It is intended to develop of the related partial differential equations with gradient constraints, to study optimal reflection barriers in multi-dimensional cases and to investigate other problems pertinent to singular control in queuing, inventory and flexible manufacturing models. The proposed research lies within the area of optimal stochastic control. This type of control problem arises in models involving automatic cruise control of an airplane subject to uncertain wind conditions, of a space vehicle subject to small perturbations in mechanical units. The solution to this problem suggests the optimal timing of the correction of the course of the airplane of the space vehicle. Singular control also appears in queuing/production/manufacturing models. In a typical model of a flexible manufacturing system one seeks the optimal scheduling of turning on and off the production unit, the timing of buying new equipment or increasing the capacity of the existing unit, finding optimal levels of inventory stocked for supply, etc. In many cases when the production process faces uncertainties due to fluctuation of demand or due to unreliability in the process itself, it is possible to use the theory of singular control to address the questions posed above.