  We consider linear dynamical systems consisting of ordinary differential
equations with high dimensionality. The aim of model order reduction is to
construct an approximating system of a much lower dimension. Therein, the
reduced system may be unstable, even though the original system is
asymptotically stable. We focus on projection-based model order reduction of
Galerkin-type. A transformation of the original system guarantees an
asymptotically stable reduced system. This transformation requires the
numerical solution of a high-dimensional Lyapunov equation. We specify an
approximation of the solution, which allows for an efficient iterative
treatment of the Lyapunov equation under a certain assumption. Furthermore, we
generalise this strategy to preserve the asymptotic stability of stationary
solutions in model order reduction of nonlinear dynamical systems. Numerical
results for high-dimensional examples confirm the computational feasibility of
the stability-preserving approach.
