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

TL;DR

This paper investigates the existence of multiple extremal solutions for conjugate discrete-time Riccati equations, extending previous work by identifying additional Hermitian solutions and analyzing their properties in control contexts.

Contribution

The work introduces new meaningful Hermitian solutions to CDAREs and analyzes their extremal properties, which were previously understudied.

Findings

Existence of additional Hermitian solutions to CDAREs.

Certain extremal solutions cannot be attained simultaneously.

Almost stabilizing solutions coincide with some extremal solutions.

Abstract

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.