On the maximal solution of the conjugate discrete-time algebraic Riccati equation

In this paper we consider a class of conjugate discrete-time Riccati equations, arising originally from the linear quadratic regulation problem for discrete-time antilinear systems. Under some mild assumptions and the framework of the fixedpoint iteration, a constructive proof is given for the existence of the maximal solution to the conjugate discrete-time Riccati equation, in which the control weighting matrix is nonsingular and its constant term is Hermitian. Moreover, starting with a suitable initial matrix, we also show that the nonincreasing sequence generated by the fixed-point iteration converges at least linearly to the maximal solution of the Riccati equation. An example is given to demonstrate the correctness of our main theorem and provide considerable insights into the study of another meaningful solutions. (c) 2022 Elsevier Ltd. All rights reserved.

Automated summary

In this paper, we study a class of conjugate discrete-time Riccati equations that arise from the linear quadratic regulation problem for discrete-time antilinear systems. We provide a constructive proof for the existence of the maximal solution to the conjugate discrete-time Riccati equation under mild assumptions and the fixed-point iteration framework. The control weighting matrix in the maximal solution is nonsingular and has a Hermitian constant term. We also show that the fixed-point iteration generates a nonincreasing sequence that converges at least linearly to the maximal solution of the Riccati equation. An example is provided to demonstrate the correctness of the main theorem and offer insights into the study of other meaningful solutions.