We are concerned with efficient solutions of the time-periodic parabolic optimal control problems. By using the multiharmonic FEM, the linear algebraic equations characterizing the first-order optimality conditions can be decoupled into a series of parallel solvable block 4 × 4 linear systems with respect to the cosine and sine Fourier coefficients of the state and scaled control variables for different frequencies. Parameter robust preconditioners are proposed for solving these linear systems along with information on practical algorithm implementation and detailed spectral analysis. Problem independent eigenvalue bounds and upper bound approximations of the condition numbers of the eigenvector matrices are obtained for the preconditioned matrices. Such results ensure efficient Krylov subspace acceleration methods and a parameter-free Chebyshev acceleration method, which are both robust in view of all discretization and model parameters. Numerical experiments are presented to demonstrate the robustness and effectiveness of the proposed preconditioners within both Krylov subspace and Chebyshev accelerations compared with some already available preconditioned Krylov subspace methods.

我们关注的是时间周期抛物线最优控制问题的有效解决方案。通过使用多谐 FEM，表征一阶最优性条件的线性代数方程可以解耦为一系列并行可解块 4 × 4 线性系统，这些线性系统关于状态的余弦和正弦傅里叶系数以及不同的缩放控制变量频率。提出了用于求解这些线性系统的参数稳健预处理器以及有关实际算法实现和详细频谱分析的信息。对于预处理矩阵，获得了与问题无关的特征值边界和特征向量矩阵条件数的上界近似值。这样的结果确保了有效的 Krylov 子空间加速方法和无参数的 Chebyshev 加速方法，它们在所有离散化和模型参数方面都是稳健的。数值实验证明了所提出的预处理器在 Krylov 子空间和 Chebyshev 加速度内的鲁棒性和有效性，与一些已经可用的预处理 Krylov 子空间方法相比。