The preconditioned modified Hermitian/skew-Hermitian splitting (PMHSS) iteration method and the corresponding preconditioning technique can achieve satisfactory results for solving optimal control problems governed by Poisson’s equation. We explore the feasibility of such a method and preconditioner for solving optimization problems constrained by the more complicated Stokes system. Theoretical results demonstrate that the PMHSS iteration method is convergent because the spectral radius of the iterative matrix is less than \(\frac {\sqrt {2}}{2}\). Additionally, the PMHSS preconditioner still clusters eigenvalues on a unitary segment. It guarantees that the convergence of the PMHSS iteration method and preconditioning is independent of not only discretizing mesh size, but also of the Tikhonov regularization parameter. A more effective preconditioner is proposed based on the PMHSS preconditioner. The proposed preconditioner avoids the inner iterations when solving saddle point systems appearing in the generalized residual equations. Furthermore, it is still convergent and maintains its independence of parameter and mesh size.

预处理的改进的Hermitian / skew-Hermitian分裂（PMHSS）迭代方法和相应的预处理技术可以解决由Poisson方程控制的最优控制问题，取得令人满意的结果。我们探索了这种方法和预处理器解决由更复杂的Stokes系统约束的优化问题的可行性。理论结果表明，PMHSS迭代方法是收敛的，因为迭代矩阵的谱半径小于\（\ frac {\ sqrt {2}} {2} \）。此外，PMHSS预处理器仍将特征值聚集在单一分段上。它保证了PMHSS迭代方法和预处理的收敛不仅与离散化网格大小无关，而且与Tikhonov正则化参数无关。提出了一种基于PMHSS预处理器的更有效的预处理器。当求解鞍点系统出现在广义残差方程中时，提出的预处理器避免了内部迭代。此外，它仍然收敛，并保持其参数和网格大小的独立性。