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

TL;DR

This paper establishes the existence and convergence of the maximal solution for a class of conjugate discrete-time Riccati equations, relevant to control problems in antilinear systems, using fixed-point iteration methods.

Contribution

It provides a constructive proof for the maximal solution's existence and shows linear convergence of fixed-point iteration under mild assumptions.

Findings

Maximal solution exists under mild conditions.

Fixed-point iteration converges linearly to the maximal solution.

An example demonstrates the theoretical results.

Abstract

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 fixed-point 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.