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

TL;DR

This paper characterizes dissipative mappings between matrices and applies these findings to analyze the stability of dissipative-Hamiltonian systems under structured perturbations, providing new tools for stability analysis.

Contribution

It introduces necessary and sufficient conditions for dissipative mappings, characterizes all such mappings, and applies these results to determine system stability margins.

Findings

Characterization of dissipative mappings between matrices.

Identification of the minimal Frobenius norm dissipative mapping.

Application to compute the distance to instability in dissipative-Hamiltonian systems.

Abstract

In this paper, we find necessary and sufficient conditions to identify pairs of matrices $X$ and $Y$ for which there exists $\Delta \in \mathbb C^{n,n}$ 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. Mat. Anal. Appl.\ 37 (4): 1625-1654, 2016).