We present the Dirichlet-Neumann (DN) and Neumann-Neumann (NN) methods applied to the optimal control problems arising from elliptic partial differential equations (PDEs) under the $H^{-1}$ regularization. We use the Lagrange multiplier approach to derive a forward-backward optimality system with the $L^2$ regularization, and a singular perturbed Poisson equation with the $H^{-1}$ regularization. The $H^{-1}$ regularization thus avoids solving a coupled bi-Laplacian problem, yet the solutions are less regular. The singular perturbed Poisson equation is then solved by using the DN and NN methods, and a detailed analysis is given both in the one-dimensional and two-dimensional case. Finally, we provide some numerical experiments with conclusions.

翻译：在$H^{-1}$正则化下，我们提出了应用于椭圆偏微分方程（PDEs）最优控制问题的Dirichlet-Neumann（DN）与Neumann-Neumann（NN）方法。利用拉格朗日乘子法，我们推导出了$L^2$正则化下的前向-后向最优性系统，以及$H^{-1}$正则化下的奇异摄动泊松方程。$H^{-1}$正则化由此避免了求解耦合的双拉普拉斯问题，但解的正则性较低。随后，采用DN与NN方法求解该奇异摄动泊松方程，并在一维与二维情形下给出了详细分析。最后，我们提供了若干数值实验及其结论。