A Cone-preserving Solution to a Nonsymmetric Riccati Equation

TL;DR

This paper establishes a simple criterion for the existence of a stabilizing cone-preserving solution to a nonsymmetric Riccati equation, extending previous results for nonnegative matrices and providing convergence conditions for related matrix sequences.

Contribution

It introduces a necessary and sufficient condition based on matrix stability for solutions to the nonsymmetric Riccati equation, generalizing prior work to broader cone-preserving contexts.

Findings

A stable associated coefficient matrix guarantees a cone-preserving solution.

Boundedness in a single vectorial direction ensures convergence of matrix sequences.

The results extend existing conditions from nonnegative matrices to general cones.

Abstract

In this paper, we provide the following simple equivalent condition for a nonsymmetric Algebraic Riccati Equation to admit a stabilizing cone-preserving solution: an associated coefficient matrix must be stable. The result holds under the assumption that said matrix be cross-positive on a proper cone, and it both extends and completes a corresponding sufficient condition for nonnegative matrices in the literature. Further, key to showing the above is the following result which we also provide: in order for a monotonically increasing sequence of cone-preserving matrices to converge, it is sufficient to be bounded above in a single vectorial direction.