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On the null-controllability of diffusion equations

Controllability of Partial Differential Equations is a very active research field in the recent decades, especially after the classical paper by J.-L. Lions (SIAM Rev., 1988). In particular, H.O. Fattorini and D.L. Russell (1972) addressed the null controllability of linear parabolic equations in one space dimension, while the same controllability result in several space dimensions has been independently established by G. Lebeau and L. Robbiano (1995), and by A.V. Fursikov and O.Yu. Imanuvilov (1996). This article introduces a new abstract version of the Lebeau and Robbiano approach. In the special case of the heat equation in high dimensional rectangular domains, the authors provide an impressive alternative way, based on an inequality of P. Turan (1946), to check the key Lebeau-Robbiano spectral condition so that the corresponding null controllbility can be proved without using the usual Carleman-type estimate.

On the null-controllability of diffusion equations Gérald Tenenbaum and Marius Tucsnak
ESAIM: COCV 17 (2011) 1088-1100
http://dx.doi.org/10.1051/cocv/2010035