On the nearest stable 2��2 matrix, dedicated to Prof. Sze-Bi Hsu in appreciation of his inspiring ideas

Kuo, Yeong-Chau; Cheng, Huan Chang; Syu, Jhih You; Shieh, Shih Feng

doi:10.3934/dcdsb.2020358

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2021

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“…After the present paper appeared as a preprint, a follow-up work has appeared in which the authors find a different formula for the minimizer that does not require comparing several candidates [32].…”

Section: Remark 57mentioning

confidence: 99%

Nearest $$\varOmega $$-stable matrix via Riemannian optimization

Noferini

Poloni

2021

Numer. Math.

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We study the problem of finding the nearest $$\varOmega $$ Ω -stable matrix to a certain matrix A, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set $$\varOmega $$ Ω . Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.

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Section: Remark 57mentioning

confidence: 99%