The infinite Arnoldi method and an application to time-delay systems with distributed delays

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 ẋ(t) = A 0x(t) + A 1x(t-τ)+∫ -τ0F(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. © 2012 Springer-Verlag GmbH Berlin Heidelberg.