  A delay Lyapunov matrix corresponding to an exponentially stable system of
linear time-invariant delay differential equations can be characterized as the
solution of a boundary value problem involving a matrix valued delay
differential equation. This boundary value problem can be seen as a natural
generalization of the classical Lyapunov matrix equation. We present a general
approach for computing delay Lyapunov matrices and H2 norms for systems with
multiple discrete delays, whose applicability extends towards problems where
the matrices are large and sparse, and the associated positive semidefinite
matrix (the ``right-hand side' for the standard Lyapunov equation), has a low
rank. In contract to existing methods that are based on solving the boundary
value problem directly, our method is grounded in solving standard Lyapunov
equations of increased dimensions. It combines several ingredients: i) a
spectral discretization of the system of delay equations, ii) a targeted
similarity transformation which induces a desired structure and sparsity
pattern and, at the same time, favors accurate low rank solutions of the
corresponding Lyapunov equation, and iii) a Krylov method for large-scale
matrix Lyapunov equations. The structure of the problem is exploited in such a
way that the final algorithm does not involve a preliminary discretization
step, and provides a fully dynamic construction of approximations of increasing
rank. Interpretations in terms of a projection method directly applied to a
standard linear infinite-dimensional system equivalent to the original
time-delay system are also given. Throughout the paper two didactic examples
are used to illustrate the properties of the problem, the challenges and
methodological choices, while numerical experiments are presented at the end to
illustrate the effectiveness of the algorithm.
