We present new Neumann-Neumann algorithms based on a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semi-discretization, the Lagrange multiplier approach provides a coupled forward-backward optimality system, which can be solved using a time domain decomposition. Due to the forward-backward structure of the optimality system, nine variants can be found for the Neumann-Neumann 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：本文提出基于时间域分解的新型Neumann-Neumann算法，用于求解无约束抛物型最优控制问题。在空间半离散化后，拉格朗日乘子法构建了耦合的前向后向最优性系统，该系统可通过时间域分解求解。由于最优性系统的前向后向结构，Neumann-Neumann算法可衍生出九种变体。我们分析了这些算法的收敛行为，并为每种算法确定了最优松弛参数。分析表明，最自然的算法实际上仅能作为良好的平滑器，而存在更优选择可形成高效求解器。数值实验验证了理论分析结果。