Singular and Boundary Control of Multidimensional Diffusion

小行星9705017 建议的研究范围内的最优随机控制。 它 讨论了扩散过程的最优控制问题， 边界以及在反射边界处的最优控制， 随机过程 这种类型的控制自然会出现在 这是没有自然的限制率的控制和最佳的 控制率是无限的。(This一种控制模式也是一种良好的 近似的情况下，利率有界，但非常大）。 最优策略则是从先验未知数反映过程 边界这种类型的行动出现的自然模型与 添加剂输入的问题，其中没有先验限制， 控制率或作为一个近似的问题与最佳政策的 “砰砰”型的拟议的研究包括开发 相关的梯度约束偏微分方程的理论，研究了最优解 反映多维案例中的障碍，并将结果应用于 不同的机械、制造和金融模式。 本研究为开发新的生物技术提供了理论基础 导弹自动巡航控制的最优控制算法 受到不确定的风力条件或空间飞行器受到小的 机械单位的扰动。这项技术将使一个 以计算最大校正力应该为 以及该力的最佳方向。在“基层” 这一理论将提供一个工具，以获得最佳库存水平的复杂 灵活的制造系统，以及最佳的时间变化， 制造过程。在金融界，这一理论将提供 一种为金融公司寻找最佳资金水平的方法， 资产是随机波动的。例如，在大型 流动资产不断波动的保险公司， 不确定性的时间和数额的传入索赔，这将是可能的 计算应维持的最佳储备水平， 以在风险和利润潜力之间取得最佳平衡。

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.