2018/04/13 by Karine Beauchard, Philippe Jaming, Beauchard, Karine +3 · 2 citations
Engineering · Mathematics · #35H10 #42C05 #93B05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1804.04895
openalex publication_date 2018/04/13 · openalex created_date 2022/03/02 · openalex updated_date 2026/07/28
Some recent works have shown that the heat equation posed on the whole\nEuclidean space is null-controllable in any positive time if and only if the\ncontrol subset is a thick set. This necessary and sufficient condition for\nnull-controllability is linked to some uncertainty principles as the\nLogvinenko-Sereda theorem which give limitations on the simultaneous\nconcentration of a function and its Fourier transform. In the present work, we\nprove new uncertainty principles for finite combinations of Hermite functions\nand establish an analogue of the Logvinenko-Sereda theorem with an explicit\ncontrol of the constant with respect to the energy level of the Hermite\nfunctions as eigenfunctions of the harmonic oscillator for thick control\nsubsets. This spectral inequality allows to derive the null-controllability in\nany positive time from thick control regions for parabolic equations associated\nwith a general class of hypoelliptic non-selfadjoint quadratic differential\noperators. More generally, the spectral inequality for finite combinations of\nHermite functions is actually shown to hold for any measurable control subset\nof positive Lebesgue measure, and some quantitative estimates of the constant\nwith respect to the energy level are given for two other classes of control\nsubsets including the case of non-empty open control subsets.\n