American Institute of Mathematical Sciences

June  2014, 4(2): 203-259. doi: 10.3934/mcrf.2014.4.203

Carleman estimates for semi-discrete parabolic operators with a discontinuous diffusion coefficient and applications to controllability

 1 Université d'Orléans, Bâtiment de Mathématiques (MAPMO), B.P. 6759, 45067 Orléans cedex 2, France

Received  January 2013 Revised  July 2013 Published  February 2014

In the discrete setting of one-dimensional finite-differences we prove a Carleman estimate for a semi-discretization of the parabolic operator $\partial_t-\partial_x (c\partial_x )$ where the diffusion coefficient $c$ has a jump. As a consequence of this Carleman estimate, we deduce consistent null-controllability results for classes of semi-linear parabolic equations.
Citation: Thuy N. T. Nguyen. Carleman estimates for semi-discrete parabolic operators with a discontinuous diffusion coefficient and applications to controllability. Mathematical Control & Related Fields, 2014, 4 (2) : 203-259. doi: 10.3934/mcrf.2014.4.203
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References:
 [1] Damien Allonsius, Franck Boyer. Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries. Mathematical Control & Related Fields, 2020, 10 (2) : 217-256. doi: 10.3934/mcrf.2019037 [2] El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj, Lahcen Maniar. Carleman Estimates and null controllability of coupled degenerate systems. Evolution Equations & Control Theory, 2013, 2 (3) : 441-459. doi: 10.3934/eect.2013.2.441 [3] Genni Fragnelli. Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 687-701. doi: 10.3934/dcdss.2013.6.687 [4] Enrique Zuazua. Controllability of partial differential equations and its semi-discrete approximations. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 469-513. doi: 10.3934/dcds.2002.8.469 [5] Sylvie Benzoni-Gavage, Pierre Huot. Existence of semi-discrete shocks. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 163-190. doi: 10.3934/dcds.2002.8.163 [6] Ansgar Jüngel, Oliver Leingang. Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4755-4782. doi: 10.3934/dcdsb.2019029 [7] Abdeladim El Akri, Lahcen Maniar. Uniform indirect boundary controllability of semi-discrete $1$-$d$ coupled wave equations. Mathematical Control & Related Fields, 2020, 10 (4) : 669-698. doi: 10.3934/mcrf.2020015 [8] Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1 [9] Lingyang Liu, Xu Liu. Controllability and observability of some coupled stochastic parabolic systems. Mathematical Control & Related Fields, 2018, 8 (3&4) : 829-854. doi: 10.3934/mcrf.2018037 [10] Zimo Zhu, Gang Chen, Xiaoping Xie. Semi-discrete and fully discrete HDG methods for Burgers' equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021132 [11] Fabio Camilli, Francisco Silva. A semi-discrete approximation for a first order mean field game problem. Networks & Heterogeneous Media, 2012, 7 (2) : 263-277. doi: 10.3934/nhm.2012.7.263 [12] Zhaoqiang Ge. Controllability and observability of stochastic implicit systems and stochastic GE-evolution operator. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021009 [13] Piermarco Cannarsa, Genni Fragnelli, Dario Rocchetti. Null controllability of degenerate parabolic operators with drift. Networks & Heterogeneous Media, 2007, 2 (4) : 695-715. doi: 10.3934/nhm.2007.2.695 [14] Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020 [15] Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control & Related Fields, 2020, 10 (1) : 89-112. doi: 10.3934/mcrf.2019031 [16] Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control & Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001 [17] Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013 [18] Fengyan Yang. Exact boundary null controllability for a coupled system of plate equations with variable coefficients. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021036 [19] J. Carmelo Flores, Luz De Teresa. Null controllability of one dimensional degenerate parabolic equations with first order terms. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3963-3981. doi: 10.3934/dcdsb.2020136 [20] Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 61-102. doi: 10.3934/eect.2020052

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