Linear algebra properties of dissipative Hamiltonian descriptor systems

TL;DR

This paper investigates the eigenvalue properties and structural characteristics of dissipative Hamiltonian descriptor systems, providing insights into their stability and minimal indices, and introduces a method for perturbing systems to achieve Lyapunov stability.

Contribution

It offers a comprehensive analysis of eigenvalues and minimal indices for dissipative Hamiltonian systems and proposes a structure-preserving perturbation method for stability enhancement.

Findings

Eigenvalues lie in the closed left half plane.

Nonzero finite eigenvalues on the imaginary axis are semisimple.

A structure-preserving perturbation method for zero eigenvalues is developed.

Abstract

A wide class of matrix pencils connected with dissipative Hamiltonian descriptor systems is investigated. In particular, the following properties are shown: all eigenvalues are in the closed left half plane, the nonzero finite eigenvalues on the imaginary axis are semisimple, the index is at most two, and there are restrictions for the possible left and right minimal indices. For the case that the eigenvalue zero is not semisimple, a structure-preserving method is presented that perturbs the given system into a Lyapunov stable system.