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.

翻译：本文研究一类源于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（CDARE）。近期，我们已在构造性方法框架下证明了控制权重矩阵非奇异时CDARE最大解的存在性。本工作的主要贡献在于发现了另一类具有实际意义的Hermitian解，该解在此研究领域中尚未得到充分关注。此外，我们证明了某些极值解无法同时获得，且几乎（反）镇定解与部分极值解具有一致性。预期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。