A note on Riccati matrix difference equations

TL;DR

This paper studies discrete time Riccati matrix difference equations, introducing novel duality and Floquet-type representations, and provides explicit solutions and bounds for the time-varying case, which has been less explored.

Contribution

It introduces a Riccati semigroup duality formula and a Floquet-type representation for time-varying Riccati difference equations, filling a gap in the literature.

Findings

Derived a Riccati semigroup duality formula.

Developed a Floquet-type representation for time-varying equations.

Provided explicit solutions and bounds for Riccati difference equations.

Abstract

Discrete algebraic Riccati equations and their fixed points are well understood and arise in a variety of applications, however, the time-varying equations have not yet been fully explored in the literature. In this article we provide a self-contained study of discrete time Riccati matrix difference equations. In particular, we provide a novel Riccati semigroup duality formula and a new Floquet-type representation for these equations. Due to the aperiodicity of the underlying flow of the solution matrix, conventional Floquet theory does not apply in this setting and thus further analysis is required. We illustrate the impact of these formulae with an explicit description of the solution of time-varying Riccati difference equations and its fundamental-type solution in terms of the fixed point of the equation and an invertible linear matrix map, as well as uniform upper and lower bounds on the Riccati maps. These are the first results of this type for time varying Riccati matrix difference equations.