Critical delays and Polynomial Eigenvalue Problems
2009 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, Vol. 224, no 1, 296-306 p.Article in journal (Refereed) Published
In this work we present a new method to compute the delays of delay-differential equations (DDEs), such that the DDE has a purely imaginary eigenvalue. For delay-differential equations with multiple delays, the critical curves or critical surfaces in delay space (that is, the set of delays where the DDE has a purely imaginary eigenvalue) are parameterized. We show how the method is related to other works in the field by treating the case where the delays are integer multiples of some delay value, i.e., commensurate delays. The parameterization is done by solving a quadratic eigenvalue problem which is constructed from the vectorization of a matrix equation and hence typically of large size. For commensurate delay-differential equations, the corresponding equation is a polynomial eigenvalue problem. As a special case of the proposed method, we find a closed form for a parameterization of the critical surface for the scalar case. We provide several examples with visualizations where the computation is done with some exploitation of the structure of eigenvalue problems. (c) 2008 Elsevier B.V. All rights reserved.
Place, publisher, year, edition, pages
2009. Vol. 224, no 1, 296-306 p.
Delay-differential equations; Quadratic eigenvalue problems; Critical delays; Robustness; Stability
Computer and Information Science
IdentifiersURN: urn:nbn:se:kth:diva-53254DOI: 10.1016/j.cam.2008.05.004ISI: 000261980200028OAI: oai:DiVA.org:kth-53254DiVA: diva2:469690
QC 201202072011-12-262011-12-262012-02-07Bibliographically approved