This article considers an optimal control problem for the stationary Stokes system in a three-dimensional domain with a highly oscillating boundary. The controls are acting on the state through the Neumann data on the oscillating part of the boundary with appropriate scaling parameters ${ϵ}^{\alpha }$ with $\alpha \ge 1$. The periodic unfolding operators are used to characterize the optimal controls. Using the unfolding operators, we analyse the asymptotic behaviour of the optimal control problem under consideration. For $\alpha =1$, the limit optimal control problem has both boundary and interior controls. For $\alpha >1$, the limit optimal control problem has only boundary controls.

中文翻译：

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由斯托克斯方程控制的边界最优控制问题的均匀化

本文考虑了具有高度振荡边界的三维域中静止斯托克斯系统的最优控制问题。控件通过具有适当缩放参数的边界振荡部分上的 Neumann 数据作用于状态${\epsilon }^{\alpha }$和$\alpha \ge 1$. 周期性展开算子用于表征最优控制。使用展开算子，我们分析了正在考虑的最优控制问题的渐近行为。为了$\alpha =1$，极限最优控制问题既有边界控制又有内部控制。为了$\alpha >1$，极限最优控制问题只有边界控制。