<p>This paper considers the problem of finding the nearest <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>-stable pencil to a given square pencil <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A+xB \in \mathbb {C}^{n \times n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>+</mo> <mi>x</mi> <mi>B</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, where a pencil is called <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>-stable if it is regular and all of its eigenvalues belong to the closed set <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>. We propose a new method, based on the Schur form of a matrix pair and Riemannian optimization over the manifold <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$U(n) \times U(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, that is, the Cartesian product of the unitary group with itself. While the developed theory holds for any closed set <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>, 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.</p>

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

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2025年7月4日

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Vanni Noferini,
Lauri Nyman

摘要

This paper considers the problem of finding the nearest $\Omega $ $Ω$ -stable pencil to a given square pencil $A+xB \in \mathbb {C}^{n \times n}$ $A + x B \in C^{n \times n}$ , where a pencil is called $\Omega $ $Ω$ -stable if it is regular and all of its eigenvalues belong to the closed set $\Omega $ $Ω$ . We propose a new method, based on the Schur form of a matrix pair and Riemannian optimization over the manifold $U(n) \times U(n)$ $U (n) \times U (n)$ , that is, the Cartesian product of the unitary group with itself. While the developed theory holds for any closed set $\Omega $ $Ω$ , 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.

Finding the nearest \(\Omega \)-stable pencil with Riemannian optimization

摘要