On numerical methods for the spectrum of delay-differential equations Elias Jarlebring, T.U. Braunschweig
We consider delay-differential equations (DDEs) with constant coefficients and show some ways to analyze numerically properties of such equations. Many properties of such DDEs, e.g., stability,
oscillation and asymptotic behavior in general, can be determined from the spectrum, i.e., the solutions of the characteristic equation. The characteristic equation of a DDE is, unlike the characteristic equation for ordinary differential equations, a transcendental equation with an infinite number of solutions. We give an overview of the current numerical methods aiming to find roots of the characteristic equation.
Numerical methods to compute the delay margin, i.e., minimum delay which causes the DDE to turn unstable, are also discussed. In particular, we show that the critical delays, i.e., those valued of the delays for which the DDE to have a purely imaginary eigenvalue, can be computed from the solutions of a polynomial eigenvalue problem.
Finally, we mention some examples where the current numerical methods do not work efficiently, such as problems of large dimension, DDEs with a large number delays or distributed delays and badly conditioned neutral and differential-algebraic problems.
Two OPTEC professors have been awarded three "Gouden Krijtjes", the yearly teaching awards given by the organization of engineering students (vtk). Prof. Lombaert was awarded the prize for the best course in civil engineering, and Prof. Diehl the prizes for the best professor and the best course in mathematical engineering (where he teaches numerical optimization). They received these awards at the yearly "proffentap" where experienced students taught them how to draft beer professionally.
