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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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[1] Benamara, N.-E.; Nikolski, N., Resolvent test for similarity to a normal operator, Proc. London math. soc., 78, 585-626, (1999) · Zbl 1028.47500
[2] R.F. Curtain, H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, Vol. 21, Springer, New York, 1995. · Zbl 0839.93001
[3] Engel, K.-J., On the characterization of admissible control- and observation operators, Systems control lett., 34, 225-227, (1998) · Zbl 0909.93034
[4] Garnett, J.B., Bounded analytic functions, (1981), Academic Press New York · Zbl 0469.30024
[5] Grabowski, P.; Callier, F.M., Admissible observation operators, semigroup criteria of admissibility, Integral equation oper. theory, 25, 182-198, (1996) · Zbl 0856.93021
[6] Hansen, S.; Weiss, G., The operator Carleson measure criterion for admissibility of control operators for diagonal semigroups on ℓ_2, Systems control lett., 16, 219-227, (1991) · Zbl 0728.93047
[7] Ho, L.F.; Russell, D.L., Admissible input elements for systems in Hilbert space and a Carleson measure criterion, SIAM J. control optim., 21, 4, 614-640, (1983) · Zbl 0512.93044
[8] B. Jacob, J.R. Partington, S. Pott, Admissible and weakly admissible observation operators for the right shift semigroup, Proc. Edinburgh Math. Soc. (2) 45 (2) (2002) 353-362. · Zbl 1176.47065
[9] I. Lasiecka, R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Vols. I-II, Cambridge University Press, Cambridge, 2000. · Zbl 0942.93001
[10] Le Merdy, C., A bounded compact semigroup on Hilbert space not similar to a contraction one, Progr. nonlinear differential equations appl., 42, 213-216, (2000) · Zbl 0982.47030
[11] C. Le Merdy, The Weiss conjecture for bounded analytic semigroups, J. London Math. Soc., to appear. · Zbl 1064.47045
[12] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer New York · Zbl 0516.47023
[13] Salamon, D., Infinite dimensional linear systems with unbounded control and observationa functional analysis approach, Trans. amer. math. soc., 300, 2, 383-431, (1987) · Zbl 0623.93040
[14] Singer, I., Bases in Banach spaces I, (1970), Springer Berlin · Zbl 0198.16601
[15] O.J. Staffans, Well-Posed Linear Systems, Book manuscript (available at ).
[16] Weiss, G., Admissibility of input elements for diagonal semigroups on l2, Systems control lett., 10, 79-82, (1988) · Zbl 0634.93046
[17] Weiss, G., Admissibility of unbounded control operators, SIAM J. control optim., 27, 527-545, (1989) · Zbl 0685.93043
[18] Weiss, G., Two conjectures on the admissibility of control operators, (), 367-378 · Zbl 0763.93041
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