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This paper studies the feedback stabilization of abstract Cauchy problemswith unbounded output operators by finite-dimensional controllers. Bothnecessary conditions and sufficient conditions for feedback stabilizability arepresented. The proof of closed-loop stability is based on a novel input-outputgain introduced in this paper. For systems satisfying a property we callquasi-finite, an equivalent characterization of feedback stabilizability isobtained. Quasi-finiteness is verified for classes of parabolic and hyperbolicequations.

Exponential stabilization of infinite-dimensional systems by finite-dimensional controllers