We analyze a bilinear optimal control problem for the Stokes--Brinkman equations: the control variable enters the state equations as a coefficient. In two- and three-dimensional Lipschitz domains, we perform a complete continuous analysis that includes the existence of solutions and first- and second-order optimality conditions. We also develop two finite element methods that differ fundamentally in whether the admissible control set is discretized or not. For each of the proposed methods, we perform a convergence analysis and derive a priori error estimates; the latter under the assumption that the domain is convex. Finally, assuming that the domain is Lipschitz, we develop an a posteriori error estimator for each discretization scheme, obtain a global reliability bound, and investigate local efficiency estimates.

翻译：本文分析了Stokes-Brinkman方程的双线性最优控制问题：控制变量作为系数进入状态方程。在二维和三维Lipschitz区域中，我们进行了完整的连续分析，包括解的存在性以及一阶和二阶最优性条件。我们还开发了两种有限元方法，其根本区别在于是否对容许控制集进行离散化。针对所提出的每种方法，我们进行了收敛性分析并推导了先验误差估计；后者假设区域是凸的。最后，在区域为Lipschitz的假设下，我们为每种离散化方案开发了后验误差估计器，获得了全局可靠性界，并研究了局部效率估计。