On the existence of extremal solutions for the conjugate discrete-time Riccati equation

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 proven the existence of the maximal solution to the CDARE with a nonsingular control weighting matrix within the framework of a constructive method. Our contribution to the work is to find another meaningful Hermitian solution, which has received little attention in the existing literature. Moreover, we show that certain extremal solutions cannot be attained simultaneously, and almost (anti-)stabilizing solutions coincide with some extremal solutions. We expect that our theoretical results presented in this paper will play an important role in optimal control problems for discrete-time antilinear systems.