Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations
2009 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 431, no 3-4, 369-380 p.Article in journal (Refereed) Published
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. (C) 2009 Elsevier Inc. All rights reserved.
Place, publisher, year, edition, pages
2009. Vol. 431, no 3-4, 369-380 p.
Delay-differential equations; Two-parameter eigenvalue problem; Multiparameter eigenvalue problem; Critical delays; Robustness; Stability; Asymptotic stability; Companion form; Quadratic eigenvalue problem; Polynomial eigenvalue problem; Quadratic two-parameter eigenvalue problem; Polynomial two-parameter eigenvalue problem
Computer and Information Science
IdentifiersURN: urn:nbn:se:kth:diva-53255DOI: 10.1016/j.laa.2009.02.008ISI: 000266824400006OAI: oai:DiVA.org:kth-53255DiVA: diva2:469691
QC 201202072011-12-262011-12-262012-02-07Bibliographically approved