Characterization of the dissipative mappings and their application to perturbations of dissipative-Hamiltonian systems

In this article, we find necessary and sufficient conditions to identify pairs of matrices X and Y for which there exists Delta is an element of Double-struck capital Cn,n such that Delta+Delta* is positive semidefinite and Delta X=Y. Such a Delta 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).

Automated summary

This article examines the necessary and sufficient conditions for identifying matrix pairs X and Y, as well as the unique dissipative mapping with minimal Frobenius norm that maps X to Y. These findings are then used to determine the distance to asymptotic instability for dissipative-Hamiltonian systems under general structure-preserving perturbations.