For a class of optimal control problems constrained with certain time- and space-fractional diffusive equations, by making use of mixed discretizations of temporal finite-difference and spatial finite-element schemes along with Lagrange multiplier approach, we obtain specially structured block two-by-two linear systems. We demonstrate positive definiteness of the coefficient matrices of these discrete linear systems, construct rotated block-diagonal preconditioning matrices, and analyze spectral properties of the corresponding preconditioned matrices. Both theoretical analysis and numerical experiments show that the preconditioned Krylov subspace iteration methods, when incorporated with these rotated block-diagonal preconditioners, can exhibit optimal convergence property in the sense that their convergence rates are independent of both discretization stepsizes and problem parameters, and their computational workloads are linearly proportional with the number of discrete unknowns.

对于受某些时空分数扩散方程约束的一类最优控制问题，通过利用时间有限差分和空间有限元格式的混合离散化以及拉格朗日乘数法，我们得到了特殊的结构块-两个线性系统。我们证明了这些离散线性系统的系数矩阵的正定性，构造了旋转的块对角预条件矩阵，并分析了相应预条件矩阵的光谱特性。理论分析和数值实验均表明，与这些旋转的块对角前置条件预处理器结合使用的预处理Krylov子空间迭代方法，