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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.

##### MSC:
 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
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