论文标题
在能量空间中dirichlet边界控制的分析和近似
Analysis and approximations of Dirichlet boundary control of Stokes flows in the energy space
论文作者
论文摘要
我们研究2D多边形域中Stokes流的Dirichlet边界控制。我们考虑具有两个不同边界控制正则术语的成本功能：$ l^2 $ norm和一个能量空间半径。我们证明了两个问题的适应性和规律性结果，为这两个问题开发有限的元素离散化，并证明了后一个问题的有限元误差估计。研究能量空间问题的动机来自我们的分析：我们证明，控制空间的选择$ {\ bm l}^2（γ）$也会导致最佳控制，即使域是凸面，也可以在拐角处不连续。我们在数值实验中观察到了这种现象。这种行为不会发生在凸多边形域上泊松方程的dirichlet边界控制问题中，并且在实际应用中可能是不可取的。对于能量空间问题，我们得出一阶最佳条件，并证明控制问题的解决方案比$ {\ bm l}^2（γ）$正则化的问题更规则。我们还证明了能量规范中控制的先验误差估计值，并为凸面和非convex域上的控制问题提供了几个数值实验。
We study Dirichlet boundary control of Stokes flows in 2D polygonal domains. We consider cost functionals with two different boundary control regularization terms: the $L^2$ norm and an energy space seminorm. We prove well-posedness and regularity results for both problems, develop finite element discretizations for both problems, and prove finite element error estimates for the latter problem. The motivation to study the energy space problem follows from our analysis: we prove that the choice of the control space ${\bm L}^2(Γ)$ can lead to an optimal control with discontinuities at the corners, even when the domain is convex. We observe this phenomenon in numerical experiments. This behavior does not occur in Dirichlet boundary control problems for the Poisson equation on convex polygonal domains, and may not be desirable in real applications. For the energy space problem, we derive the first order optimality conditions, and show that the solution of the control problem is more regular than the solution of the problem with the ${\bm L}^2(Γ)$ regularization. We also prove a priori error estimates for the control in the energy norm, and present several numerical experiments for both control problems on convex and nonconvex domains.
