# The Infinite Arnoldi Method and an Application to Time-Delay Systems with Distributed Delays

Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 423)

## Abstract

The Arnoldi method, which is a well-established numerical method for standard and generalized eigenvalue problems, can conceptually be applied to standard but infinite-dimensional eigenvalue problems associated with an operator. In this work, we show how such a construction can be used to compute the eigenvalues of a time-delay system with distributed delays, here given by $$\dot{x}(t)=A_0x(t)+A_1x(t-\tau)+\int_{-\tau}^{0}F(s)x(t+s)ds$$, where A 0,A 1,F(s) ∈ ℂ n×n . The adaption is based on formulating a more general problem as an eigenvalue problem associated with an operator and showing that the action of this operator has a finite-dimensional representation when applied to polynomials. This allows us to implement the infinite-dimensional algorithm using only (finite-dimensional) operations with matrices and vectors of size n. We show, in particular, that for the case of distributed delays, the action can be computed from the Fourier cosine transform of a function associated with F, which in many cases can be formed explicitly or computed efficiently.

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© Springer-Verlag GmbH Berlin Heidelberg 2012

## Authors and Affiliations

• Elias Jarlebring
• 1
Email author
• Wim Michiels
• 1
• Karl Meerbergen
• 1
1. 1.KU LeuvenHeverleeBelgium