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 language | English |
---|---|
Pages (from-to) | 341-350 |
Number of pages | 19 |
Journal | Systems and control letters |
Volume | 48 |
Issue number | 3-4 |
DOIs | |
Publication status | Published - 15 Mar 2003 |
Keywords
- $C_0$-semigroup
- Conditional basis
- Admissible control operator
- MSC-93C25
- Infinite-dimensional system