Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations

TL;DR

This paper introduces a unifying polynomial two-parameter eigenvalue problem framework to analyze the stability of delay-differential equations, revealing relations between existing methods and proposing new variants for improved computation.

Contribution

It presents a novel unifying framework for matrix pencil methods using polynomial two-parameter eigenvalue problems, clarifying their relations and introducing new variants.

Findings

Unified framework links different matrix pencil methods

New matrix pencil variants for DDE stability analysis

Potential for more efficient stability computation

Abstract

Several recent methods used to analyze asymptotic stability of delay-differential equations (DDEs) involve determining the eigenvalues of a matrix, a matrix pencil or a matrix polynomial constructed by Kronecker products. Despite some similarities between the different types of these so-called matrix pencil methods, the general ideas used as well as the proofs differ considerably. Moreover, the available theory hardly reveals the relations between the different methods. In this work, a different derivation of various matrix pencil methods is presented using a unifying framework of a new type of eigenvalue problem: the polynomial two-parameter eigenvalue problem, of which the quadratic two-parameter eigenvalue problem is a special case. This framework makes it possible to establish relations between various seemingly different methods and provides further insight in the theory of matrix pencil methods. We also recognize a few new matrix pencil variants to determine DDE stability. Finally, the recognition of the new types of eigenvalue problem opens a door to efficient computation of DDE stability.