In the first part of this talk, I will present in details the so-called Lebeau-Robbiano strategy for the null-controllability of the heat equation on a bounded domain $Omega$ of R^n, the control being distributed on an open subset $/omega$ of $Omega$. This strategy relies on two ingredients: 1) A ``spectral inequality'' comparing the L^2-norm of finite packets of eigenfunctions on $/Omega$ and and their L^2-norm in $/omega$, the constants in the inequality depending in a crucial way on the size of the packets; 2) An iterative construction of a control, which relies on an ``active part'' where low frequencies are controlled with a certain cost depending on the frequency, and a ``passive part'', where high frequencies are dissipated.
In a second part (joint work with Enrique Zuazua), I will present  an applications to obtain a necessary and sufficient condition for the controllability of systems of heat equations with constant coefficients and non-diagonalizable diffusion matrices. The proof relies on the use of the Lebeau-Robbiano strategy together with a precise study of the cost of controllability for linear ordinary differential equations, and allows to treat the case where each component of the system is controlled in a different subdomain.

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