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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
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Abstract

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.

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

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