  The model reduction problem for semistable infinite-dimensional control
systems is studied in this paper. In relation to these systems, we study an
object we call the semistability Gramian, which serves as a generalization of
the ordinary controllability Gramian valid for semistable systems. This Gramian
is then given geometric as well as algebraic characterization via a Lyapunov
equation. We then proceed to show that under a commutativity assumption
relating the original and reduced systems, and as long as the semistability is
preserved, we may derive a priori error formulas in $\mathcal{H}_2$-norm in
terms of the trace of this Gramian.
