Abstract
For a wide class of infinite-dimensional linear systems it is shown that if the state-space realization is exponentially stabilizable and detectable then exponential stability is equivalent to input-output stability of the transfer function. Exponential stabilizability (or detectability) is a strong assumption as it implies that the system operator satisfies the spectrum decomposition assumption and has finitely many unstable eigenvalues of finite multiplicity. In addition, the finite-dimensional unstable projection of the system is controllable (or observable).
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Curtain, R.F. Equivalence of input-output stability and exponential stability for infinite-dimensional systems. Math. Systems Theory 21, 19–48 (1988). https://doi.org/10.1007/BF02088004
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DOI: https://doi.org/10.1007/BF02088004