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

Chun-Yueh Chiang

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.

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

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.

Paper Structure

This paper contains 14 sections, 26 theorems, 47 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some useful definitions, theory and notations
  4. Two subsets $\mathbb{T}$ and $\mathbb{O}$
  5. Two auxiliary matrix equations
  6. The dual-CDARE
  7. The transformed-DARE
  8. The FPI \ref{['fpi']} with its convergence property
  9. Extremal solutions: $\mathbf{X_m}$, $\mathbf{Y_M}$, $\mathbf{Z_M}$, $\mathbf{Z_m}$, $\mathbf{W_M}$ and $\mathbf{W_m}$
  10. The existence of $\mathbf{X_m}$ and $\mathbf{Y_M}$
  11. The existence of $\mathbf{Z_M}$, $\mathbf{Z_m}$, $\mathbf{W_M}$ and $\mathbf{W_m}$
  12. The definiteness of the matrix $R_X$
  13. (Almost) stabilizing and anti-stabilizing solutions
  14. Concluding remarks

Key Result

Lemma 2.1

Bernstein2009 Let $A$ be an arbitrary matrix of size $n$. $X$ and $Y$ are two $n\times n$ positive definite matrices. Then,

Theorems & Definitions (56)

  • Example 1.1
  • Definition 2.1
  • Lemma 2.1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 46 more