%0 Journal Article
%T 2D Grushin-type equations: minimal time and null controllable data
%+ Institut de Recherche Mathématique de Rennes (IRMAR)
%+ École normale supérieure - Rennes (ENS Rennes)
%+ Modélisation aléatoire de Paris X (MODAL'X)
%+ Institut de Mathématiques de Marseille (I2M)
%A Beauchard, Karine
%A Miller, Luc
%A Morancey, Morgan
%Z ANR-11-IDEX-0001-02, Agence Nationale de la Recherche
%< avec comité de lecture
%@ 0022-0396
%J Journal of Differential Equations
%I Elsevier
%V 259
%N 11
%P 5813-5845
%8 2015
%D 2015
%R 10.1016/j.jde.2015.07.007
%K minimal time
%K degenerate parabolic equations
%K null controllability
%K Grushin operator
%K transmutation
%K Carleman estimates
%K geometric control condition
%Z Mathematics [math]/Optimization and Control [math.OC]
%Z Mathematics [math]/Analysis of PDEs [math.AP]Journal articles
%X We study internal null controllability for degenerate parabolic equa-tions of Grushin-type Gγ = ∂_{xx} + |x|^2γ ∂_{yy} , (γ > 0), in the rectangle (x, y) ∈ Ω = (−1, 1) × (0, 1). Previous works proved that null controllability holds for weak degen-eracies (γ small), and fails for strong degeneracies (γ large). Moreover, in the transition regime and with strip shaped control domains, a positive minimal time is required. In this paper, we work with controls acting on two strips, symmetric with respect to the degeneracy. We give the explicit value of the minimal time and we characterize the initial data that can be steered to zero in time T (when the system is not null controllable): their regularity depends on the control domain and the time T . We also prove that, with a control that acts on one strip, touching the degeneracy line {x = 0}, then Grushin-type equations are null controllable in any time T > 0 and for any degeneracy γ > 0. Our approach is based on a precise study of the observability property for the one-dimensional heat equations satisfied by the Fourier coefficients in variable y. This precise study is done, through a transmutation process, on the resulting one-dimensional wave equations, by lateral propagation of energy method.
%G English
%2 https://hal.science/hal-01082175/document
%2 https://hal.science/hal-01082175/file/Tmin_Grushin_final.pdf
%L hal-01082175
%U https://hal.science/hal-01082175
%~ UNIV-RENNES1
%~ IRMAR
%~ UR2-HB
%~ CNRS
%~ UNIV-AMU
%~ UNIV-PARIS10
%~ INSA-RENNES
%~ EC-MARSEILLE
%~ UNAM
%~ IRMAR-AN
%~ ENS-RENNES
%~ I2M
%~ I2M-2014-
%~ CHL
%~ TDS-MACS
%~ UR1-HAL
%~ UPN
%~ UR1-MATH-STIC
%~ AGREENIUM
%~ UNIV-RENNES2
%~ MODALX
%~ TEST-UNIV-RENNES
%~ TEST-UR-CSS
%~ UNIV-RENNES
%~ INSA-GROUPE
%~ AMIDEX
%~ UNIV-PARIS-LUMIERES
%~ ANR
%~ UR1-MATH-NUM
%~ UNIV-PARIS-NANTERRE
%~ INSTITUT-AGRO
%~ IRMAR-ANM
%~ IRMAR-ANUM