Abstract
This paper investigates the admissibility of control and observation operators in UMD spaces. Necessary and/or sufficient conditions for unbounded control operators to be admissible and weakly admissible in the Salamon–Weiss sense are presented. This is illustrated by an example which shows that the UMD-property is essential. In particular, we get a direct proof of the known result of Driouich and and El-Mennaoui (Arch Math 72:56–63, 1999) on the validity of the inverse formula of the Laplace transform for C 0-semigroups on UMD-spaces and in Hilbert spaces, as proved earlier by Yao (SIAM J Math Anal 26(5):1331–1341, 1995). We outline how these results can be used to prove a partial validity of the inverse Laplace transform for semigroups in general Banach spaces. In particular, we obtain the result on the inverse Laplace transform due to Hille and Philllips (Am Math Soc Transl Ser 2, 1957).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amann, H.: Linear and quasilinear parabolic problems. In: Abstract Linear Theory, vol. 1. Birkhäuser, Basel (1995)
Arendt W.: Vector-valued Laplace transform and Cauchy problems. Isr. J. Math. 59, 327–352 (1987)
Arendt, W., Batty, C., Hieber, C., Neubrander, F.: Vector valued Laplace transforms and Cauchy problems. In: Monographs in Mathematics, vol. 96. Birkhäuser, Basel (2001)
Bounit H., Driouich A., El-Mennaoui O.: A direct approach to the Weiss conjecture for analytic semigroups. Czechoslovak Math. J. 60(2), 527–539 (2010)
Bounit, H., Driouich, A., El-Mennaoui, O.: A direct approach to the weighted admissibility of observation operators for bounded analytic semigroups. Semigroup Forum (under revision)
Burkholder, D.L.: Martingales and singular intergrals in Banach spaces. In: Handbook of the Geometry of Banach Spaces, vol. I, pp. 233–269. North- Holland, Amsterdam (2001)
Butzer P.L., Berens H.: Semigroups of operators and approximations. Die Grundlehren der math. Wiss. 145. Springer-Verlag, York (1967)
Curtain, R.F., Zwart, H.: An introduction to infinite-dimensional linear systems theory. In: Texts in Applied Mathematics, vol. 21. Springer-Verlag, New York (1995)
Driouich A., El-Mennaoui O.: On the inverse Laplace transform for C 0-semigroups in UMD-spaces. Arch. Math. 72, 56–63 (1999)
Diestel, J., Uhl, J.J.: Vector Measures. Math. Surveys, no. 15. Amer. Math. Soc., Providence (1977)
Engel K.J., Nagel R.: One-Parameter Semigroups for Linear Evolution Equations. Springer-Verlag, New York (2000)
Emirsajlow Z., Townley S.: From PDE with a boundary control to the abstract state equation with unbounded control operator: a tutorial. Eur. J. Control 7(1), 1–23 (2000)
Grabowski P.: Admissibility of observation functionals. Int. J. Control 62, 1163–1173 (1995)
Greiner G., Nagel R.: On the stability of strongly continuous semigroups of positive operators on L 2 (μ). Annali della Scuola Normale Superiore di Pisa - Classe di Scienze Sr. 4 10(2), 257–262 (1983)
Haak B.H., Haase M., Kunstmann P.C.: Perturbation, interpolation, and maximal regularity. Adv. Differ. Equ. 11(2), 201–240 (2006)
Hansen S., Weiss G.: The operator Carleson measure criterion for admissibility of control operators for diagonal semigroups on L 2. Syst. Control Lett. 16, 219–227 (1991)
Hille, E., Philllips, R.S.: Functional Analysis and Semigroups. Amer. Math. Soc. Transl. Ser. 2 (1957)
Jacob, B., Partington, J.R.: Admissibility of control and observation operators for semigroups: a survey. In: Ball, J.A., Helton, J.W., Klaus, M., Rodman, L. (eds.) Current Trends in Operator Theory and its Applications. Proceedings of the IWOTA 2002. Operator Theory: Advances and Applications, vol. 149, pp. 199–221. Birkhäuser Verlag
Jacob B., Partington J.R.: The Weiss conjecture on admissibility of observation operators for contraction semigroups. Integr. Equ. Oper. Theory 40, 231–243 (2001)
Jacob, B., Partington, J.R.: Admissibility of control and observation operators for semigroups: a survey. In: Current Trends in Operator Theory and its Applications, vol. 149 of Oper. Theory Adv. Appl., pp. 199–221. Birkhäuser, Basel (2004)
Jacob B., Zwart H.: Counterexamples concerning observation operators for C 0-semigroups. SIAM J. Control Optim. 43(1), 137–153 (2004)
Le Merdy C.: The Weiss conjecture for bounded analytic semigroups. J. Lond. Math. Soc. (2) 67(3), 715–738 (2003)
Nagel, R.: One-Parameter Semigroups of Positive Operators. LNM 1184, Berlin-Heidelberg-New York (1985)
Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, Berlin (1983)
Staffans, O.J.: Well-posed linear systems. In: Encyclopedia of Mathematics and its Applications, vol. 103. Cambridge University Press, Cambridge (2005)
Triebel H.: Interpolation Theory, Functions Spaces, Differential Operators. North-Holland, Amsterdam (1978)
Yao P.F.: On the inversion of the Laplace transform of C 0-semigroups and its applications. SIAM J. Math. Anal. 26(5), 1331–1341 (1995)
Weiss, G.: Two conjectures on the admissibility of control operators. In: Desch, W., Kappel, F. (eds.) Estimation and Control of Distributed Parameter Systems, pp. 367–378. Birkhäuser Verlag (1991)
Weiss G.: Admissibility of unbounded control operators. SIAM J. Control Optim. 27, 527–545 (1989)
Weiss G.: Admissibile observation operators for linear semigroups. Isr. J. Math. 65, 17–43 (1989)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was completed with the support of our \(\text{\TeX}\)-pert.
Rights and permissions
About this article
Cite this article
Bounit, H., Driouich, A. & El-Mennaoui, O. Admissibility of Control Operators in UMD Spaces and the Inverse Laplace Transform. Integr. Equ. Oper. Theory 68, 451–472 (2010). https://doi.org/10.1007/s00020-010-1838-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-010-1838-z