Previous |  Up |  Next

Article

Keywords:
infinite dimensional systems; analytic semigroups; unbounded observation operator; admissibility; fractional power
Summary:
We give a new proof of the Weiss conjecture for analytic semigroups. Our approach does not make any recourse to the bounded $H^{\infty }$-calculus and is based on elementary analysis.
References:
[1] Amann, H.: Linear and quasilinear Parabolic Problems. Vol. I. Birkhäuser, Basel (1995). MR 1345385
[2] Arendt, W., Batty, C., Hieber, C., Neubrander, F.: Vector Valued Laplace transforms and Cauchy Problems. Vol. 96 of Monographes in Mathematics. Birkäuser (2001). Zbl 0978.34001
[3] Balakrishnan, A. V.: Fractional powers of closed operators and the semigroups generated by them. Pacific J. Math. 10 (1960), 419-437. DOI 10.2140/pjm.1960.10.419 | MR 0115096 | Zbl 0103.33502
[4] Engel, K.: On the characterization of admissible control and observation operators. Systems and Control Letters (1998), 34 225-227. MR 1637265 | Zbl 0909.93034
[5] Faming, G.: Admissibility of linear systems in Banach spaces. Journal of electronic Science and Technology in China (2004), 75-78.
[6] Gao, M. C., Hou, J. C.: The infinite-time admissibility of observation operators and operator Lyapunov equation. Integral Equations and Operator Theory 35 (1999), 53-64. DOI 10.1007/BF01225527 | MR 1707930
[7] Grabowski, P.: Admissibility of observation functionals. Internat. J. Control 62 (1995), 1163-1173. DOI 10.1080/00207179508921589 | MR 1636622 | Zbl 0837.93005
[8] Grabowski, P., Callier, F. M.: Admissibility of observation operators: Semigroup criteria for admissibility. Integral Equations Operator Theory 25 (1996), 183-196. DOI 10.1007/BF01308629 | MR 1388679
[9] Haak, B., Kunstmann, P. C.: Weighted admissibility and wellposedness of linear systems in Banach spaces. SIAM J. Control Optimization 45 2094-2118 (2007). DOI 10.1137/060656139 | MR 2285716 | Zbl 1126.93021
[10] Hansen, S., Weiss, G.: The operator Carleson measure criterion for admissibility of control operators for diagonal semigroups on $L^2$. Systems Control Letters 16 219-227 (1991). DOI 10.1016/0167-6911(91)90051-F | MR 1098682
[11] Jacob, B., Partington, J. R.: The Weiss conjecture on admissibility of observation operators for contraction semigroups. Integral Equations and Operator Theory 40 (2001), 231-243. DOI 10.1007/BF01301467 | MR 1831828 | Zbl 1031.93107
[12] Jacob, B., Partington, J. R., Pott, S.: Admissible and weakly admissible observation operators for the right shift semigroup. Proc. Edinb. Math. Soc. 45 (2002), 353-362. DOI 10.1017/S0013091500001024 | MR 1912645 | Zbl 1176.47065
[13] Jacob, B., Zwart, H.: Counterexamples concerning observation operators for $C_0$-semigroups. SIAM J. Control Optim. 43 137-153 (2004). DOI 10.1137/S0363012903423235 | MR 2082696 | Zbl 1101.93042
[14] Komatsu, H.: Fractional power of operators. Pacific J. Math. 19 (1966), 285-346. DOI 10.2140/pjm.1966.19.285 | MR 0201985
[15] LeMerdy, C.: The Weiss conjecture for bounded analytic semigroups. J. London Math. Soc. 67 (2003), 715-738. DOI 10.1112/S002461070200399X | MR 1967702
[16] Partington, J. R., Pott, S.: Admissibility and exact observability of observation operators for semigroups. Irish Math. Soc. Bulletin 55 19-39 (2005). MR 2185647 | Zbl 1159.47302
[17] Partington, J. R., Weiss, G.: Admissible observation operators for the right-shift semigroup. Mathematics of Control, Signals and Systems (2000), 13 179-192. MR 1784262 | Zbl 0966.93033
[18] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, Berlin (1983). MR 0710486 | Zbl 0516.47023
[19] Staffans, O.: Well-posed linear systems. Cambridge University press, Cambridge (2005). MR 2154892 | Zbl 1057.93001
[20] Weiss, G.: Two conjectures on the admissibility of control operators. In Estimation and Control of Distributed Parameter Systems, Birkhäuser Verlag W. Desch, F. Kappel (1991), 367-378. MR 1155659 | Zbl 0763.93041
[21] Weiss, G.: Admissibility of unbounded control operators. SIAM Journal Control & Optimization 27 527-545 (1989). DOI 10.1137/0327028 | MR 0993285 | Zbl 0685.93043
[22] Weiss, G.: Admissibile observation operators for linear semigroups. Israel J. Math. 65 17-43 (1989). DOI 10.1007/BF02788172 | MR 0994732
[23] Zwart, H., Jacob, B., Staffans, O.: Weak admissibility does not imply admissibility for analytic semigroups. Systems Control Letters 48 (2003), 341-350. DOI 10.1016/S0167-6911(02)00277-3 | MR 2020649 | Zbl 1157.93421
[24] Zwart, H. J.: Sufficient Conditions for Admissibility. Systems and Control Letters 54 973-979 (2005). DOI 10.1016/j.sysconle.2005.02.009 | MR 2166288 | Zbl 1129.93422
Partner of
EuDML logo