  We consider the characterization and computation of H-infinity norms for a
class of time-delay systems. It is well known that in the finite dimensional
case the H-infinity norm of a transfer function can be computed using the
connections between the corresponding singular value curves and the imaginary
axis eigenvalues of a Hamiltonian matrix, leading to the established level set
methods. We show a similar connection between the transfer function of a
time-delay system and the imaginary axis eigenvalues of an infinite dimensional
linear operator $\mathcal{L}_{\xi}^N$. Based on this result, we propose a
predictor-corrector algorithm for the computation of the H-infinity norm. In
the prediction step, a finite-dimensional approximation of the problem, induced
by a spectral discretization of the operator $\mathcal{L}_{\xi}$, and an
adaptation of the algorithms for finite-dimensional systems, allow to obtain an
approximation of the H-infinity norm of the transfer function of the time-delay
system. In the next step the approximate results are corrected to the desired
accuracy by solving a set of nonlinear equations which are obtained from the
reformulation of the eigenvalue problem for the linear infinite-dimensional
operator $\mathcal{L}_{\xi}^N$ as a finite dimensional nonlinear eigenvalue
problem. These equations can be interpreted as characterizations of peak values
in the singular value plot. The effects of the discretization in the predictor
step are fully characterized and the choice of the number of discretization
points is discussed. The paper concludes with a numerical example and the
presentation of the results of extensive benchmarking.
