Weak admissibility does not imply admissibility for analytic semigroups

Hans Zwart (Corresponding Author), Birgit Jacob, Olof Staffans

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    17 Citations (Scopus)
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    Abstract

    Two conjectures on admissible control operators by George Weiss are disproved in this paper. One conjecture says that an operator $B$ defined on an infinite-dimensional Hilbert space $U$ is an admissible control operator if for every element $u \in U$ the vector $Bu$ defines an admissible control operator. The other conjecture says that $B$ is an admissible control operator if a certain resolvent estimate is satisfied. The examples given in this paper show that even for analytic semigroups the conjectures do not hold. In the last section we construct a semigroup example showing that the first estimate in the Hille-Yosida theorem is not sufficient to conclude boundedness of the semigroup.
    Original languageEnglish
    Pages (from-to)341-350
    Number of pages19
    JournalSystems and control letters
    Volume48
    Issue number3-4
    DOIs
    Publication statusPublished - 15 Mar 2003

    Keywords

    • $C_0$-semigroup
    • Conditional basis
    • Admissible control operator
    • MSC-93C25
    • Infinite-dimensional system

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