Finding the nearest $\Omega$-stable pencil with Riemannian optimization

TL;DR

This paper introduces a Riemannian optimization-based method to find the nearest $$-stable matrix pencil, improving accuracy and efficiency over existing techniques, especially for Hurwitz and Schur stability cases.

Contribution

It develops a novel Riemannian optimization approach on the manifold $U(n) imes U(n)$ for nearest $$-stable pencils, with efficient implementations for common stability sets.

Findings

The proposed algorithm outperforms existing methods in numerical experiments.

Efficient implementations are provided for Hurwitz and Schur stability cases.

The method is applicable to any closed set $$ in the complex plane.

Abstract

This paper considers the problem of finding the nearest $\Omega$-stable pencil to a given square pencil $A+xB \in \mathbb{C}^{n \times n}$, where a pencil is called $\Omega$-stable if it is regular and all of its eigenvalues belong to the closed set $\Omega$. We propose a new method, based on the Schur form of a matrix pair and Riemannian optimization over the manifold $U(n) \times U(n)$, that is, the Cartesian product of the unitary group with itself. While the developed theory holds for any closed set $\Omega$, we focus on two cases that are the most common in applications: Hurwitz stability and Schur stability. For these cases, we develop publicly available efficient implementations. Numerical experiments show that the resulting algorithm outperforms existing methods.