We present a stationary iteration method, namely Alternating Symmetric positive definite and Scaled symmetric positive semidefinite Splitting (ASSS), for solving the system arisen from finite element discretization of a distributed optimal control problem with time-periodic parabolic equations. An upper bound for the spectral radius of the iteration method is given which is always less than 1. So convergence of the ASSS iteration method is guaranteed. The induced ASSS preconditioner is applied to accelerate the convergence speed of the GMRES method for solving the system. Numerical results are presented to demonstrate the effectiveness of both the ASSS iteration method and the ASSS preconditioner.

翻译：我们提出了一个固定迭代方法,即对称正对数确定和缩放正对称半确定分解(ASSS),用于解决因时间周期性抛物线方程式的分布最佳控制问题产生的系统分散式最佳控制问题的有限元素分解,给迭代法的光谱半径设定一个上限,该上限总是小于1。因此保证了ASSS迭代法的趋同。由ASSS导引导的先决条件用于加速GMRES解决系统方法的趋同速度。提供了数字结果,以证明ASSS迭代法和ASSS预设法的有效性。