This work studies the null-controllability of a class of abstract parabolic equations. The main contribution in the general case consists in giving a short proof of an abstract version of a sufficient condition for null-controllability which has been proposed by Lebeau and Robbiano. We do not assume that the control operator is admissible. Moreover, we give estimates of the control cost. In the special case of the heat equation in rectangular domains, we provide an alternative way to check the Lebeau-Robbiano spectral condition. We then show that the sophisticated Carleman and interpolation inequalities used in previous literature may be replaced by a simple result of Turán. In this case, we provide explicit values for the constants involved in the above mentioned spectral condition. As far as we are aware, this is the first proof of the null-controllability of the heat equation with arbitrary control domain in a n-dimensional open set which avoids Carleman estimates.
Mots-clés : heat equation, controllability, spectral condition, Turán's method
@article{COCV_2011__17_4_1088_0,
author = {Tenenbaum, G\'erald and Tucsnak, Marius},
title = {On the null-controllability of diffusion equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1088--1100},
publisher = {EDP-Sciences},
volume = {17},
number = {4},
year = {2011},
doi = {10.1051/cocv/2010035},
mrnumber = {2859866},
zbl = {1236.93025},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2010035/}
}
TY - JOUR AU - Tenenbaum, Gérald AU - Tucsnak, Marius TI - On the null-controllability of diffusion equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2011 SP - 1088 EP - 1100 VL - 17 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2010035/ DO - 10.1051/cocv/2010035 LA - en ID - COCV_2011__17_4_1088_0 ER -
%0 Journal Article %A Tenenbaum, Gérald %A Tucsnak, Marius %T On the null-controllability of diffusion equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 1088-1100 %V 17 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2010035/ %R 10.1051/cocv/2010035 %G en %F COCV_2011__17_4_1088_0
Tenenbaum, Gérald; Tucsnak, Marius. On the null-controllability of diffusion equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 1088-1100. doi : 10.1051/cocv/2010035. http://archive.numdam.org/articles/10.1051/cocv/2010035/
[1] and , Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rational Mech. Anal. 43 (1971) 272-292. | MR | Zbl
[2] and , Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations. Quart. Appl. Math. 32 (1974) 45-69. | MR | Zbl
[3] and , Controllability of Evolution Equations, Lect. Notes Ser. 34. Seoul National University Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996). | MR | Zbl
[4] and , Contrôle exact de l'équation de la chaleur. Comm. Partial Diff. Eq. 20 (1995) 335-356. | MR | Zbl
[5] and , Null-controllability of a system of linear thermoelasticity. Arch. Rational Mech. Anal. 141 (1998) 297-329. | MR | Zbl
[6] and , On the controllability of a fractional order parabolic equation. SIAM J. Control Optim. 44 (2006) 1950-1972. | MR | Zbl
[7] , On the controllability of anomalous diffusions generated by the fractional Laplacian. Math. Control Signals Systems 18 (2006) 260-271. | MR | Zbl
[8] , A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. Preprint, available at http://hal.archives-ouvertes.fr/hal-00411846/en/ (2009). | Zbl
[9] , Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics 84. Published for the Conference Board of the Mathematical Sciences, Washington (1994). | MR | Zbl
[10] , How violent are fast controls. III. J. Math. Anal. Appl. 339 (2008) 461-468. | MR | Zbl
[11] and , Observation and control for operator semigroups. Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel (2009). | MR | Zbl
[12] , On a theorem of Littlewood. J. London Math. Soc. 21 (1946) 268-275. | MR | Zbl
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