We present new Dirichlet-Neumann and Neumann-Dirichlet algorithms with a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semi-discretization, we use the Lagrange multiplier approach to derive a coupled forward-backward optimality system, which can then be solved using a time domain decomposition. Due to the forward-backward structure of the optimality system, three variants can be found for the Dirichlet-Neumann and Neumann-Dirichlet algorithms. We analyze their convergence behavior and determine the optimal relaxation parameter for each algorithm. Our analysis reveals that the most natural algorithms are actually only good smoothers, and there are better choices which lead to efficient solvers. We illustrate our analysis with numerical experiments.

翻译：本文针对无约束抛物型最优控制问题，提出了基于时间域分解的Dirichlet-Neumann与Neumann-Dirichlet新算法。在空间半离散化后，采用拉格朗日乘子法导出耦合的前向-后向最优性系统，进而通过时间域分解求解。鉴于最优性系统前向-后向结构的特殊性，Dirichlet-Neumann与Neumann-Dirichlet算法可衍生出三种变体。我们分析了各算法的收敛特性，并确定了最优松弛参数。研究表明，最直观的算法实际上仅能作为良好光滑子，而更优的选择能构造出高效求解器。最后通过数值实验验证了理论分析结果。