Properties of the realization of inner functions

Birgit Jacob, Hans Zwart

    Research output: Contribution to journalArticleAcademicpeer-review

    8 Citations (Scopus)

    Abstract

    In this paper we investigate fundamental properties of state-space realizations for inner functions. We derive necessary and sufficient conditions for the inner function to have a realization such that the associated $C_0$-semigroup is exponentially stable. Furthermore, we give necessary and sufficient conditions on the inner function such that the $C_0$-semigroup is a group. Combining these results, we have that the $C_0$-semigroup is an exponentially stable $C_0$-group if and only if the inner function is the product of a constant of modulus one and a Blaschke product for which the zeros satisfy the Carleson-Newman condition and the zeros lie in a vertical strip bounded away from the imaginary axis.
    Original languageEnglish
    Pages (from-to)356-379
    Number of pages24
    JournalMathematics of control, signals and systems
    Volume15
    Issue number4
    DOIs
    Publication statusPublished - Nov 2002

    Keywords

    • MSC-93B15
    • Infinite-dimensional systems
    • Exponential stability
    • Semigroups of operators
    • Realization theory
    • Inner functions

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