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 and obtain a global reliability bound.

翻译：我们分析了Stokes-Brinkman方程的双线性最优控制问题：控制变量以系数形式进入状态方程。在二维和三维Lipschitz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。我们还发展了两种有限元方法，这两种方法的基本区别在于是否对容许控制集进行离散化。针对每种提出的方法，我们进行了收敛性分析并推导了先验误差估计；后者在假设域为凸的条件下成立。最后，在假设域为Lipschitz的条件下，我们为每种离散格式发展了一个后验误差估计器，并获得了全局可靠性界。