Nonlinear Eigenvalue Approach to Differential Riccati Equations for Contraction Analysis

TL;DR

This paper introduces a nonlinear eigenvalue method for solving differential Riccati equations in contraction analysis, linking solutions to eigenvectors of the differential Hamiltonian matrix and characterizing solution properties.

Contribution

It extends the eigenvalue approach from algebraic to differential Riccati equations, providing new insights into solution properties via nonlinear eigenvalues and eigenvectors.

Findings

Solutions expressed as functions of nonlinear eigenvectors.

Characterization of solution properties under specific assumptions.

Extension of eigenvalue methods to differential Riccati equations.

Abstract

In this paper, we extend the eigenvalue method of the algebraic Riccati equation to the differential Riccati equation (DRE) in contraction analysis. One of the main results is showing that solutions to the DRE can be expressed as functions of nonlinear eigenvectors of the differential Hamiltonian matrix. Moreover, under an assumption for the differential Hamiltonian matrix, real symmetricity, regularity, and positive semidefiniteness of solutions are characterized by nonlinear eigenvalues and eigenvectors.