Counterexamples concerning observation operators for C0- semigroups

SIAM Journal on Control and Optimization

Volumn 43, Issue 1, 2005, Pages 137-153

  Jacob, Birgit ^a   Zwart, Hans ^b  

^a TU DORTMUND UNIVERSITY   (Germany)

^b UNIVERSITY OF TWENTE   (Netherlands)

Author keywords

Admissible observation operator; C0 semigroup; Conditional basis; Exact observability; Infinite dimensional system; Left invertibility

Indexed keywords

DIFFERENTIAL EQUATIONS; EIGENVALUES AND EIGENFUNCTIONS; INTEGRATION; LINEAR SYSTEMS; MATHEMATICAL OPERATORS; MATRIX ALGEBRA; SET THEORY;

ADMISSIBLE OBSERVATION OPERATOR; C0-SEMIGROUP; CONDITIONAL BASIS; EXACT OBSERVABILITY; INFINITE-DIMENSIONAL SYSTEMS; LEFT-INVERTIBILITY;

LINEAR ALGEBRA;

EID: 14244261729     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012903423235     Document Type: Article

References (26)

1. N.-E. BENAMARA AND N. NIKOLSKI, Resolvent test for similarity to a normal operator, Proc. London Math. Soc., 78 (1999), pp. 585-626.
2. J. A. VAN CASTEREN, Operators similar to unitary or selfadjoint ones, Pacific J. Math., 104 (1983), pp. 241-255.
3. J. A. VAN CASTEREN, Boundedness properties of resolvents and semigroups of operators, in Linear Operators (Warsaw, 1994), Banach Center Publ. 38, Polish Acad. Sci., Warsaw, 1997, pp. 59-74.
4. R. F. CURTAIN AND H. ZWART, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts Appl. Math. 21, Springer-Verlag, New York, 1995.
5. P. GRABOWSKI AND F. M. CALLIER, Admissible observation operators, semigroup criteria of admissibility, Integral Equations Operator Theory, 25 (1996), pp. 182-198.
6. B. JACOB AND J. R. PARTINGTON, The Weiss conjecture on admissibility of observation operators for contraction semigroups, Integral Equations Operator Theory, 40 (2001), pp. 231-243.
7. B. JACOB AND J. R. PARTINGTON, Admissibility of control and observation operators for semigroups: A survey, in Proceedings of the IWOTA 2002, J. A. Ball, J. W. Helton, M. Klaus, and L. Rodman, eds., Birkhäuser Verlag, 2004, to appear.
8. B. JACOB, J. R. PARTINGTON, AND S. POTT, Admissible and weakly admissible observation operators for the right shift semigroup, Proc. Edinburgh Math. Soc., 45 (2002), pp. 353-362.
9. B. JACOB AND H. ZWART, Exact observability of diagonal systems with a finite-dimensional output operator, Systems Control Lett., 43 (2001), pp. 101-109.
10. B. JACOB AND H. ZWART, Exact observability of diagonal systems with a one-dimensional output operator, Appl. Math. Comput. Sci., 11 (2001), pp. 1277-1283.
11. B. JACOB AND H. ZWART, A Hautus test for infinite-dimensional systems, in Unsolved Problems in Mathematical Systems and Control Theory, V. Blondel and A. Megretski, eds., Princeton University Press, Princeton, NJ, 2004, to appear.
12. C. LE MERDY, The Weiss conjecture for bounded analytic semigroups, J. London Math. Soc., 67 (2003), pp. 715-738.
13. J. VAN NEERVEN, The Asymptotic Behaviour of Semigroups of Linear Operators, Oper. Theory Adv. Appl. 88, Birkhäuser, Basel, 1996.
14. N. K. NIKOL'SKIǏ AND B. S. PAVLOV, Bases of eigenvectors of completely nonunitary contractions and the characteristic function, Math. USSR-Izvestija, 4 (1970), pp. 91-134.
15. J. R. PARTINGTON AND G. WEISS, Admissible observation operators for the right shift semigroup, Math. Control Signals Systems, 13 (2000), pp. 179-192.
16. A. PAZY, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983.
17. D. L. RUSSELL AND G. WEISS, A general necessary condition for exact observability, SIAM J. Control Optim., 32 (1994), pp. 1-23.
18. I. SINGER, Bases in Banach Spaces I, Springer-Verlag, Berlin, 1970.
19. 2, Systems Control Lett., 10 (1988), pp. 79-82.
20. G. WEISS, Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), pp. 17-43.
21. G. WEISS, Two conjectures on the admissibility of control operators, in Estimation and Control of Distributed Parameter Systems, F. Kappel and W. Desch, eds., Birkhäuser Verlag, Basel, 1991, pp. 367-378.
22. G. WEISS, A powerful generalization of the Carleson measure theorem?, in Open Problems in Mathematical Systems Theory and Control, V. Blondel, E. Sontag, M. Vidyasagar, and J. Willems, eds., Springer-Verlag, London, 1999, pp. 267-272.
23. Q. ZHOU AND M. YAMAMOTO, Hautus condition on the exact controllability of conservative systems, Internat. J. Control, 67 (1997), pp. 371-379.
24. H. ZWART, Sufficient Conditions for Admissibility, Memorandum 1547, Department of Applied Mathematics, University of Twente, 2000, available online from http://www.math.utwente.nl/publications.
25. 0-semigroups, Semigroup Forum, 63 (2001), pp. 153-160.
26. H. ZWART, B. JACOB, AND O. STAFPANS, Weak admissibility does not imply admissibility for analytic semigroups, Systems Control Lett., 48 (2003), pp. 341-350.