关于共轭离散时间Riccati方程极值解的存在性
On the existence of extremal solutions for the conjugate discrete-time Riccati equation
AI总结：
本文针对源自离散时间反线性系统线性二次调节问题的共轭离散时间Riccati方程，在已有最大解存在性结论基础上，找到一类受关注较少的厄米特解，证明部分极值解无法共存且几乎所有（反）镇定解与部分极值解重合，为相关最优控制问题提供理论支撑。
摘要：
本文研究一类共轭离散时间Riccati方程（CDARE），这类方程最初源自离散时间反线性系统的线性二次调节问题。近期，我们在构造性方法的框架下，证明了控制加权矩阵非奇异时CDARE最大解的存在性。本文的贡献在于找到另一类有意义的厄米特解，该类解在本研究领域中受到的关注较少。此外，我们证明部分极值解无法同时存在，且几乎所有（反）镇定解都与某些极值解重合。预计本文提出的理论结果将在离散时间反线性系统的最优控制问题中发挥重要作用。
英文摘要：
In this paper we consider a class of conjugate discrete-time Riccati equations (CDARE), arising originally from the linear quadratic regulation problem for discrete-time antilinear systems. Recently, we have proved the existence of the maximal solution to the CDARE with a nonsingular control weighting matrix under the framework of the constructive method. Our contribution in the work is to finding another meaningful Hermitian solutions, which has received little attention in this topic. Moreover, we show that some extremal solutions cannot be attained at the same time, and almost (anti-)stabilizing solutions coincide with some extremal solutions. It is to be expected that our theoretical results presented in this paper will play an important role in the optimal control problems for discrete-time antilinear systems.
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