Algebraic Riccati equations (AREs) have been extensively applicable in linear optimal control problems and many efficient numerical methods were developed. The most attention of numerical solutions is the (almost) stabilizing solution in the past works. Nevertheless, it is an interesting and challenging issue in finding the extremal solutions of AREs which play a vital role in the applications. In this paper, based on the semigroup property, an accelerated fixed-point iteration (AFPI) is developed for solving the extremal solutions of the discrete-time algebraic Riccati equation. In addition, we prove that the convergence of the AFPI is at least R-suplinear with order $r>1$ under some mild assumptions. Numerical examples are shown to illustrate the feasibility and efficiency of the proposed algorithm.
翻译：代数里卡蒂方程式(AREs)广泛适用于线性最佳控制问题,并开发了许多高效的数字方法,数字解决方案的注意力最集中的是过去作品中(几乎)的稳定解决方案,然而,在寻找在应用中起重要作用的地区极端解决办法方面,这是一个有趣和具有挑战性的问题,在本文中,基于半组属性,制定了加速固定点迭代法(AFPI),用于解决离散代数里卡蒂方程式的极端解决办法。此外,我们还证明,在一些轻度假设下,AFPI的趋同至少是R-suplinear与$>1美元订单的趋同,并举例说明了拟议算法的可行性和效率。
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