Disproof of an admissibility conjecture of Weiss

Heiko J. Zwart, B. Jacob, Birgit Jacob

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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 condition is satisfied. The examples given in this paper show that even for analytic semigroups the conjectures do not hold. In the last section we show that this example leads to a semigroup example showing that the first estimate in the Hille-Yosida Theorem is not sufficient to conclude boundedness of the semigroup.
    Original languageUndefined
    Place of PublicationEnschede
    PublisherUniversity of Twente, Department of Applied Mathematics
    Number of pages17
    Publication statusPublished - 2000

    Publication series

    NameMemorandum Faculteit TW
    PublisherUniversity of Twente, Department of Applied Mathematics
    No.1539
    ISSN (Print)0169-2690

    Keywords

    • IR-65726
    • EWI-3359
    • MSC-93A05
    • MSC-47D60
    • MSC-93C25
    • METIS-141340

    Cite this

    Zwart, H. J., Jacob, B., & Jacob, B. (2000). Disproof of an admissibility conjecture of Weiss. (Memorandum Faculteit TW; No. 1539). Enschede: University of Twente, Department of Applied Mathematics.