Solvability of Matrix Riccati Inequalities

TL;DR

This paper establishes necessary and sufficient conditions for solving matrix Riccati inequalities in the general sign indefinite case, extending classical results and using Hamiltonian matrix representations.

Contribution

It provides a complete characterization of Riccati inequality solvability beyond sign definite cases, which was previously addressed mainly by the Kalman-Yakubovich lemma.

Findings

Necessary and sufficient conditions for Riccati inequality solvability.

Extension of classical results to indefinite sign cases.

Illustrative example demonstrating the theoretical results.

Abstract

We consider matrix Riccati inequality arising in the theory of absolute stability, $H_\infty$ control problem, $LQ$ problem, and optimal estimation problem. In the case of sign definite frequency domain function, the solvability of Riccati inequalities is a subject of the famous Kalman- Yakubovich lemma. This paper presents necessary and sufficient conditions for solvability of Riccati inequality in the general sign indefinite case. To this end we use special representations of Hamiltonian matrices. The results are illustrated by an example.