An explicit Floquet-type representation of Riccati aperiodic exponential semigroups

TL;DR

This paper introduces a novel Floquet-type representation for a class of nonlinear Riccati equations with aperiodic solutions, enabling new stability analysis and contraction inequalities.

Contribution

It develops a Floquet-type representation for aperiodic Riccati equations and applies it to derive refined stability and contraction results.

Findings

Floquet-type representation for aperiodic Riccati equations

Refined global and local contraction inequalities

First results of this kind for this class of models

Abstract

The article presents a rather surprising Floquet-type representation of time-varying transition matrices associated with a class of nonlinear matrix differential Riccati equations. The main difference with conventional Floquet theory comes from the fact that the underlying flow of the solution matrix is aperiodic. The monodromy matrix associated with this Floquet representation coincides with the exponential (fundamental) matrix associated with the stabilizing fixed point of the Riccati equation. The second part of this article is dedicated to the application of this representation to the stability of matrix differential Riccati equations. We provide refined global and local contraction inequalities for the Riccati exponential semigroup that depend linearly on the spectral norm of the initial condition. These refinements improve upon existing results and are a direct consequence of the Floquet-type representation, yielding what seems to be the first results of this type for this class of models.