On algebraic Riccati equations associated with M-Matrices

TL;DR

This paper investigates the algebraic Riccati equation linked to M-matrices, especially reducible singular cases, establishing conditions for solutions and their properties, and discussing solution methods.

Contribution

It extends known results to reducible singular M-matrices, showing existence of minimal nonnegative solutions under regularity assumptions and analyzing their properties.

Findings

Existence of minimal nonnegative solutions for reducible singular M-matrices.

Properties of the solutions under regularity assumptions.

Compatibility of existing methods with the solutions in this context.

Abstract

We consider the algebraic Riccati equation for which the four coefficient matrices form an M-matrix K. When K is a nonsingular M-matrix or an irreducible singular M-matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M-matrix. Under a regularity assumption on the M-matrix K, we show that the Riccati equation still has a minimal nonnegative solution. We also study the properties of this particular solution and explain how the solution can be found by existing methods.