Characterization of the dissipative mappings and their application to perturbations of dissipative‐Hamiltonian systems. Issue 6 (14th July 2021)

• Record Type: Journal Article
• Authors: Baghel, Mohit Kumar; Gillis, Nicolas; Sharma, Punit
• Is Part Of: Numerical linear algebra with applications. Volume 28:Issue 6(2021)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2021-07-14
• Subjects: dissipative‐Hamiltonian systems -- positive semidefinite matrix -- stability radius -- structured mapping problems -- structured stability radius; Algebras, Linear -- Periodicals; 512.5
• DOI: 10.1002/nla.2402
• Languages: English
• ISSNs: 1070-5325

Abstract

Abstract: In this article, we find necessary and sufficient conditions to identify pairs of matrices X and Y for which there exists Δ ∈ ℂ n, n such that Δ + Δ ∗ is positive semidefinite and Δ X = Y . Such a Δ is called a dissipative mapping taking X to Y . We also provide two different characterizations for the set of all dissipative mappings, and use them to characterize the unique dissipative mapping with minimal Frobenius norm. The minimal‐norm dissipative mapping is then used to determine the distance to asymptotic instability for dissipative‐Hamiltonian systems under general structure‐preserving perturbations. We illustrate our results over some numerical examples and compare them with those of Mehl, Mehrmann, and Sharma (Stability radii for linear Hamiltonian systems with dissipation under structure‐preserving perturbations, SIAM J Matrix Anal Appl, 37(4):1625–54, 2016).