  This paper studies the feedback stabilization of abstract Cauchy problems
with unbounded output operators by finite-dimensional controllers. Both
necessary conditions and sufficient conditions for feedback stabilizability are
presented. The proof of closed-loop stability is based on a novel input-output
gain introduced in this paper. For systems satisfying a property we call
quasi-finite, an equivalent characterization of feedback stabilizability is
obtained. Quasi-finiteness is verified for classes of parabolic and hyperbolic
equations.
