Finding the nearest $Ω$-stable pencil with Riemannian optimization
Vanni Noferini, Lauri Nyman
TL;DR
The paper addresses the problem of finding the nearest $\Omega$-stable pencil to a given pencil $A+xB$, where stability means regularity and eigenvalues lie in $\Omega$. It develops a Schur-form based reformulation and optimizes on the manifold $U(n)\times U(n)$ using efficient projections $p_{\Omega}$ for Hurwitz and Schur stability, with explicit gradient and Hessian expressions. The authors provide publicly available implementations and demonstrate, through numerical experiments, that their method outperforms existing approaches and reveals structural properties (e.g., long Jordan chains) of the minimizers. The work advances robust stability approximation in control and DAE contexts and lays groundwork for real-variant extensions. Overall, the approach integrates generalized Schur form, projection geometry, and Riemannian optimization to solve a challenging, structured pencil problem with practical impact.
Abstract
This paper considers the problem of finding the nearest $Ω$-stable pencil to a given square pencil $A+xB \in \mathbb{C}^{n \times n}$, where a pencil is called $Ω$-stable if it is regular and all of its eigenvalues belong to the closed set $Ω$. 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 $Ω$, 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.