Memorandum 1539
H.J. Zwart & B. Jacob
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 Î 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.
Keywords:
Infinite-dimensional system, admissible control
operator, conditional basis, C0-semigroup
Mathematics Subject Classification: 93C25, 93A05, 47D60