E-Delay Systems Letter

The spectrum of delay-differential equations: numerical methods, stability and perturbation

Elias Jarlebring, T.U. Braunschweig (now at K.U. Leuven)
http://www.cs.kuleuven.be/~elias

Supervisor:
Prof. Dr. Heike Fassbender, T.U. Braunschweig
http://www.icm.tu-bs.de/~hfassben

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 function. The formula is not new. We clarify its range of applicability, and, by counter-example, show that it does not hold in general.

The second part of this thesis is related to exact stability conditions of the DDE. We find exact conditions on the delays such that the DDE has a purely imaginary eigenvalue. All those combinations of the delays such that there is a purely imaginary eigenvalue (called critical delays) are parameterized. That is, we construct a mapping consisting of computable expressions, from a simple mathematical object onto the set of all subsets of the critical delays. The mapping can be expressed explicitly with trigonometric functions for scalar DDEs. We find some new formulas and verify some formulas in the literature. In general, an evaluation of the map consists of solving a quadratic eigenvalue problem of squared dimension for non-scalar problems. The constructed eigenvalue problem is large, even for DDEs of moderate size. For that reason, we show how the computational cost for one evaluation of the map can be reduced. In particular we show that the matrix-vector product corresponding to the companion linearization of the quadratic eigenvalue problem can be computed by solving a Lyapunov equation. Most of the results in the chapter on critical delays are derived for retarded DDEs as well as neutral DDEs with an arbitrary number of delays.

The third and last part of this thesis is about perturbation results for nonlinear eigenvalue problems. We discuss some results in eigenvalue perturbation theory which can be generalized to nonlinear eigenvalue problems. A sensitivity formula for the movement of the eigenvalues extends nicely to nonlinear eigenvalue problems. We introduce a fixed point form for the nonlinear eigenvalue problem, and show that some methods in the literature can be interpreted as set-valued fixed point iterations. The convergence order of these types of iterations can be determined from an expression containing the left and right eigenvectors. We also use some results from fixed point theory. The famous result in perturbation theory referred to as the Bauer-Fike theorem, can be generalized to the nonlinear eigenvalue problem if we assume that the set-valued fixed point problem has a certain contraction property.

The PDF can be downloaded from the permanent link:

http://www.digibib.tu-bs.de/?docid=00023961