TY - JOUR

T1 - Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations

AU - Jarlebring, E.

AU - Hochstenbach, M.E.

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

U2 - 10.1016/j.laa.2009.02.008

DO - 10.1016/j.laa.2009.02.008

M3 - Article

SN - 0024-3795

VL - 431

SP - 369

EP - 380

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 3-4

ER -