The discrete-time algebraic Riccati equation (DARE) have extensive applications in optimal control problems. We provide new theoretical supports to the stability properties of solutions to the DARE and reduce the convergence conditions under which the accelerated fixed-point iteration (AFPI) can be applied to compute the numerical solutions of DARE. In particular, we verify that the convergence of AFPI is R-superlinear when the spectral radius of the closed-loop matrix is greater than 1, which is shown by mild assumption and only using primary matrix theories. Numerical examples are shown to illustrate the consistency and effectiveness of our theoretical results.

翻译：离散代数测代里卡蒂方程式(DARE)在最佳控制问题上有着广泛的应用。我们为DARE解决方案的稳定性特性提供了新的理论支持,并减少了加速固定点迭代(AFPI)用于计算DARE的数值解决方案的趋同条件。特别是,我们核实,当闭环矩阵的光谱半径大于1时,AFPI的趋同是R-超级线性,而光环矩阵的光谱半径大于1时,这种半径以轻度假设和仅使用原始矩阵理论来显示。我们用数字例子来说明我们的理论结果的一致性和有效性。