Spectral estimates for finite combinations of Hermite functions and null-controllability of hypoelliptic quadratic equations

Some recent works have shown that the heat equation on the whole Euclidean space is null-controllable in any positive time if and only if the control subset is a thick set. This necessary and sufficient condition for null-controllability is linked to some uncertainty principles, such as the Logvinenko-Sereda theorem, which give limitations on the simultaneous concentration of a function and its Fourier transform. In the present work, we prove new uncertainty principles for finite combinations of Hermite functions. We establish an analogue of the Logvinenko-Sereda theorem with an explicit control of the constant with respect to the energy level of the Hermite functions as eigenfunctions of the harmonic oscillator for thick control subsets. This spectral inequality allows us to derive null-controllability in any positive time from thick control regions for parabolic equations associated with a general class of hypoelliptic non-selfadjoint quadratic differential operators. More generally, the spectral estimate for finite combinations of Hermite functions is actually shown to hold for any measurable control subset of positive Lebesgue measure, and some quantitative estimates of the constant with respect to the energy level are given for another two classes of control subsets including the case of non-empty open control subsets.