AI 中文总结
本文针对数据中心电费最小化问题，研究含运行最大值的奇异随机控制问题，证明值函数性质、HJB 方程解的存在唯一性，通过数值结果阐释相关激励的权衡关系。
AI 中文摘要
本文旨在对一类随机控制问题进行完整分析，其中待最小化的成本包含受控状态过程的运行最大值。该模型的动机来自于面临重合峰值电价、且可能从需求响应项目中获益的数据中心的电费最小化问题。从数学角度来看，主要挑战在于运行最大值被纳入系统状态，迫使原问题成为平面上闭合楔形区域内的奇异随机控制问题，且在部分边界上存在反射。我们通过耦合反射负载的间隙单调性估计，以及两个运行最大值之差的确定性总变差界，证明了值函数在空间上是局部 Lipschitz 连续的。随后针对普通渐进可测控制推导了动态规划原理和粘性形式。对应的 HJB 方程包含负载方向下边界上的反射 Neumann 条件，以及由于运行最大值过程的影响，在区域边界对角部分的斜边界条件。我们证明了粘性解的存在性，以及在对应粘性类中提供条件唯一性的比较定理。最后，我们给出数值结果，阐释构成数据中心预期成本的两个相互冲突的激励之间的权衡关系。
英文摘要
The goal of the paper is to provide a complete analysis of a stochastic control problem when the cost to minimize involves the running maximum of the underlying controlled state process. The model is motivated by the electricity cost minimization of a data center facing coincident peak charges and possibly benefiting from participation in a demand-response program. From a mathematical standpoint, the main challenge comes from the inclusion of the running maximum in the state of the system forcing the original problem into a singular stochastic control problem on a closed wedge in the plane with reflection on parts of the boundary. We prove that the value function is locally Lipschitz continuous in space using a gap-monotonicity estimate for coupled reflected loads and a deterministic total-variation bound for the difference of two running maxima. The dynamic programming principle and viscosity formulation are then derived for ordinary progressively measurable controls. The corresponding HJB equation includes a reflecting Neumann condition on the lower boundary for the load direction as well as an oblique boundary condition on the diagonal part of the boundary of the domain due to the effect of the running-maximum process. We prove existence of a solution in the viscosity sense as well as a comparison theorem providing conditional uniqueness in the corresponding viscosity class. Finally we provide numerical results illustrating the trade-off between the two conflicting incentives comprising the expected cost of the data center.
Comments27 pages, 3 figures
