Finding the nearest $\hat{I}\text{c◯}$-stable pencil with Riemannian optimization

Abstract

This paper considers the problem of finding the nearest $\hat{I}\text{c◯}$-stable pencil to a given square pencil $A+xB\in {\mathbb{C}}^{n\times n}$, where a pencil is called $\hat{I}\text{c◯}$-stable if it is regular and all of its eigenvalues belong to the closed set $\hat{I}\text{c◯}$. 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 $\hat{I}\text{c◯}$, 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.