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Weak admissibility does not imply admissibility for analytic semigroups. (English) Zbl 1157.93421
Summary: 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.

93C25 Control/observation systems in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
47N70 Applications of operator theory in systems, signals, circuits, and control theory
93B05 Controllability
Full Text: DOI
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