Mathematical Sciences: Singular and Boundary Control of Multidimensional Diffusion Processes

所提出的研究属于最优随机控制领域。它利用自由边界来研究扩散过程的最优控制，以及在随机过程的反射边界上的最优控制。这种类型的控制自然会出现在对控制率没有自然限制且最优控制率为无穷大的问题中。然后，最优策略是从先验未知边界反映该过程。这种类型的行动出现在附加控制的问题中，其中对控制率没有先验的限制。建立了相关的梯度约束偏微分方程组，研究了多维情形下的最优反射障碍，并研究了排队、库存和柔性制造模型中与奇异控制有关的其他问题。所提出的研究属于最优随机控制领域。这种类型的控制问题出现在涉及自动巡航控制的模型中，该模型涉及在不确定的风条件下的飞机的自动巡航控制，以及在机械单元中受到微小扰动的航天器。这一问题的解决方案提出了航天器飞机航向修正的最佳时机。奇异控制也出现在排队/生产/制造模型中。在柔性制造系统的典型模型中，人们寻求开启和关闭生产单元的最优调度、购买新设备或增加现有单元的能力的时机、寻找用于供应的最优库存水平等。在许多情况下，当生产过程由于需求波动或由于过程本身的不可靠性而面临不确定性时，可以使用奇异控制理论来解决上面提出的问题。

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.