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Stability perserving is an important topic in approximation of systems, e.g.\model reduction. If the original system is stable, we often want theapproximation to be stable. But even if an algorithm preserves stability theresulting system could be unstable in practice because of round-off errors. Ourapproach is approximating this unstable reduced system by a stable system. Moreprecisely, we consider the following problem. Given an unstable lineartime-invariant continuous-time descriptor system with transfer function $G$,find a stable one whose transfer function is the best approximation of $G$ inthe spaces $RH_2$ and $RH_\infty$, respectively. Explicit optimal solutions arepresented under consideration of numerical issues.

Optimal $RH_2$-- and $RH_\infty$--Approximation of Unstable Descriptor Systems