Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T01:19:01.015Z Has data issue: false hasContentIssue false
  • English
  • Français

On the null-controllability of diffusion equations

Published online by Cambridge University Press:  23 August 2010

Gérald Tenenbaum  and
Marius Tucsnak
Gérald Tenenbaum
Affiliation:
Institut Élie Cartan, Nancy Université/CNRS/INRIA, BP 70239, 54506 Vandœuvre-lès-Nancy Cedex, France. gerald.tenenbaum@iecn.u-nancy.fr; tucsnak@iecn.u-nancy.fr
Marius Tucsnak
Affiliation:
Institut Élie Cartan, Nancy Université/CNRS/INRIA, BP 70239, 54506 Vandœuvre-lès-Nancy Cedex, France. gerald.tenenbaum@iecn.u-nancy.fr; tucsnak@iecn.u-nancy.fr
Get access

Abstract

This work studies the null-controllability of a class of abstract parabolic equations. The main contribution in the general caseconsists 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.

Keywords

Heat equationcontrollabilityspectral conditionTurán's method
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 4 , October 2011 , pp. 1088 - 1100
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Fattorini, H.O. and Russell, D.L., Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rational Mech. Anal. 43 (1971) 272–292. CrossRef
Fattorini, H.O. and Russell, D.L., 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. CrossRef
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).
Lebeau, G. and Robbiano, L., Contrôle exact de l'équation de la chaleur. Comm. Partial Diff. Eq. 20 (1995) 335–356. CrossRef
Lebeau, G. and Zuazua, E., Null-controllability of a system of linear thermoelasticity. Arch. Rational Mech. Anal. 141 (1998) 297–329. CrossRef
Micu, S. and Zuazua, E., On the controllability of a fractional order parabolic equation. SIAM J. Control Optim. 44 (2006) 1950–1972. CrossRef
Miller, L., On the controllability of anomalous diffusions generated by the fractional Laplacian. Math. Control Signals Systems 18 (2006) 260–271. CrossRef
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).
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).
Seidman, T.I., How violent are fast controls. III. J. Math. Anal. Appl. 339 (2008) 461–468. CrossRef
M. Tucsnak and G. Weiss, Observation and control for operator semigroups. Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel (2009).
Turán, P., On a theorem of Littlewood. J. London Math. Soc. 21 (1946) 268–275. CrossRef