On the null-controllability of diffusion equations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 4, p. 1088-1100

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.

DOI : https://doi.org/10.1051/cocv/2010035
Classification:  93C25,  93B07,  93C20
Keywords: heat equation, controllability, spectral condition, Turán's method
     author = {Tenenbaum, G\'erald and Tucsnak, Marius},
     title = {On the null-controllability of diffusion equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {4},
     year = {2011},
     pages = {1088-1100},
     doi = {10.1051/cocv/2010035},
     zbl = {1236.93025},
     mrnumber = {2859866},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2011__17_4_1088_0}
Tenenbaum, Gérald; Tucsnak, Marius. On the null-controllability of diffusion equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 4, pp. 1088-1100. doi : 10.1051/cocv/2010035. http://www.numdam.org/item/COCV_2011__17_4_1088_0/

[1] H.O. Fattorini and D.L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rational Mech. Anal. 43 (1971) 272-292. | MR 335014 | Zbl 0231.93003

[2] H.O. Fattorini and D.L. Russell, 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 510972 | Zbl 0281.35009

[3] A.V. Fursikov and O.Y. Imanuvilov, Controllability of Evolution Equations, Lect. Notes Ser. 34. Seoul National University Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996). | MR 1406566 | Zbl 0862.49004

[4] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur. Comm. Partial Diff. Eq. 20 (1995) 335-356. | MR 1312710 | Zbl 0819.35071

[5] G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity. Arch. Rational Mech. Anal. 141 (1998) 297-329. | MR 1620510 | Zbl 1064.93501

[6] S. Micu and E. Zuazua, On the controllability of a fractional order parabolic equation. SIAM J. Control Optim. 44 (2006) 1950-1972. | MR 2248170 | Zbl 1116.93022

[7] L. Miller, On the controllability of anomalous diffusions generated by the fractional Laplacian. Math. Control Signals Systems 18 (2006) 260-271. | MR 2272076 | Zbl 1105.93015

[8] L. Miller, 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 1219.93017

[9] H.L. Montgomery, 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 1297543 | Zbl 0814.11001

[10] T.I. Seidman, How violent are fast controls. III. J. Math. Anal. Appl. 339 (2008) 461-468. | MR 2370666 | Zbl 1129.93025

[11] M. Tucsnak and G. Weiss, Observation and control for operator semigroups. Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel (2009). | MR 2502023 | Zbl 1188.93002

[12] P. Turán, On a theorem of Littlewood. J. London Math. Soc. 21 (1946) 268-275. | MR 21568 | Zbl 0061.06504