Logvinenko-Sereda Theorems and their application to control theory of the heat equation

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Résumé : The Logvinenko-Sereda Theorem on R^d is usually classified as an uncertainty principle for functions with compactly supported Fourier Transform. However, it can be also interpreted as a spectral inequality for the same class of functions. In this talk we present the statement of this (classical) theorem and its analogous on the d-dimensional torus T^d_L=[0,2\pi L]^d, L>0 and we discuss their use in the realm of control theory. In particular, we show null-controllability for the heat equation on R^d and on T^d_L with a measurable control set and we give consistent explicit control cost estimates for these two domains depending on the geometric parameters of the problem.