Singular and Boundary Control of Multidimensional Diffusion

9705017塔克萨建议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出现在对控制率没有自然限制且最优控制率是无穷大的问题中。(对于速率有限但非常大的情况，这种类型的控制模型也是一个很好的近似值。)然后，最优策略是从先验未知边界反映这一过程。出现这类行为的自然模型与加性输入问题有关，其中对控制率没有先验限制，或者作为具有“bang-bang”类型的最优策略问题的近似。提出的研究包括发展相关的梯度约束偏微分方程理论，研究多维情形下的最优反射障碍，并将结果应用于不同的机械、制造和金融模型。这项研究应为开发自动巡航控制算法提供可靠的理论基础，以便在不确定的风力条件下对导弹进行巡航控制，或在机械单元中对航天器进行微小扰动。开发的技术将使人们能够计算出应该施加最大校正力的位置以及该力的最佳方向。在“基层”，这一理论将为复杂的柔性制造系统提供一种工具来得出最优库存水平，并为制造过程中的变化提供最优的时间安排。在金融界，这一理论将为资产受到随机波动影响的金融公司提供一种寻找最佳资金水平的方法。例如，对于一家大型保险公司，其流动资产由于收到索赔的时间和金额的不确定性而不断波动，将有可能计算出应保持的准备金的最佳水平，以便在风险和潜在利润之间取得最佳平衡。

9705017 Taksar 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 the problems in which there is no natural restriction on the rates of control and the optimal control rates are infinite. (This type of control model serves also as a good approximation for the situations when the rates are bounded but very large.) The optimal policy is then to reflect the process from an a priori unknown boundary. Natural models where such type of action arises are related to the problems with additive input, where there are no a priori limits on the control rates or as an approximation for the problems with optimal policy of a "bang-bang" type. The proposed research consists of developing the theory of the related PDE with gradient constraints, studying optimal reflecting barriers in multidimensional cases, and applying the results to different mechanical, manufacturing and financial models. This research should provide a sound theoretical base for developing optimal control algorithms for an automatic cruise control of a missile subject to uncertain wind conditions or a space vehicle subject to small perturbations in mechanical units. The developed technique will enable one to calculate the position when the maximal correction force should be applied as well as the optimal direction of this force. At the "ground level" this theory will give a tool to derive optimal inventory levels for complex flexible manufacturing systems as well as optimal timings for changes in the manufacturing processes. In the financial world this theory will provide a method for finding optimal levels of funds for financial companies whose assets are subject to random fluctuations. For example, in the case of a large insurance company whose liquid assets are constantly fluctuating due to uncertainty in the times and amounts of incoming claims, it will be possible to calculate the optimal level of the reserve which should be maintained in order to have the optimal balance between the risk and profit potential.