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arXiv · 2608.09794

A Singular Control Problem for Data Center Electricity Cost Minimization

Rene Carmona·Xiaoyu Cui
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Abstract

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.

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Rene Carmona, Xiaoyu Cui. 2026-08-10. A Singular Control Problem for Data Center Electricity Cost Minimization. https://arxiv.org/abs/2608.09794

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