On the null-controllability of diffusion equations

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

Received: 29 August 2009
Revised: 21 April 2010

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.