The spectrum of delay-differential equations: numerical methods - KTH

Abstract
Three types of problems related to time-delay systems are treated in this thesis.
In our context, a time-delay system, or sometimes delay-differential equation
(DDE), is a generalization of an ordinary differential equation (ODE) with constant
coefficients. For DDEs, unlike ODEs, the derivative of the state at some
time-point is not only dependent on the state at that time-point, but also on one
or more previous states.
We first consider the problem of numerically computing the eigenvalues of
a DDE, i.e., finding solutions of the characteristic equation, here referred to as
the delay eigenvalue problem. Unlike standard ODEs, the characteristic equation
of a DDE contains an exponential term. Because of this nonlinear term, the
delay eigenvalue problem belongs to a class of problems referred to as nonlinear
eigenvalue problems. An important contribution of the first part of this thesis is
the application of a projection method for nonlinear eigenvalue problems, to the
author’s knowledge, previously not applied to the delay eigenvalue problem. We
compare this projection method with other methods, suggested in the literature,
and used in software packages. This includes methods based on discretizations of
the solution operator and discretizations of the equivalent boundary value problem
formulation of the DDE, i.e., the infinitesimal generator. We review discretizations
based on, but not limited to, linear multi-step, Runge-Kutta and spectral
collocation. The projection method is computationally superior to all of the other
tested method for the presented large-scale examples. We give interpretations of
the methods based on discretizations in terms of rational approximations of the
exponential function or the logarithm. Some notes regarding a special case where
the spectrum can be explicitly expressed are presented. The spectrum can be
expressed with a formula containing a matrix version of the Lambert W func-