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 and Related Fields, 2014, 4 (2) : 203-259. doi: 10.3934/mcrf.2014.4.203
References:
[1]

A. Benabdallah, Y. Dermenjian and J. Le Rousseau, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem, J. Math. Anal. Appl., 336 (2007), 865-887. doi: 10.1016/j.jmaa.2007.03.024.

[2]

F. Boyer, F. Hubert and J. Le Rousseau, Discrete Carleman estimates for the elliptic operators and uniform controllability of semi discretized parabolic equations, J. Math. Pur. Appl., 93 (2010), 240-276. doi: 10.1016/j.matpur.2009.11.003.

[3]

F. Boyer, F. Hubert and J. Le Rousseau, Discrete Carleman estimates for the elliptic operators in arbitrary dimension and applications, SIAM J. Control Optim, 48 (2010), 5357-5397. doi: 10.1137/100784278.

[4]

F. Boyer and J. Le Rousseau, Carleman Estimates for Semi-Discrete Parabolic Operators and Application to the Controllability of Semi-Linear Semi-Discrete Parabolic Equations, prep. (2012).

[5]

Y. Chitour and E. Trélat, Controllability of partial differential equations, Advanced topics in control systems theory, 171-198, Lecture Notes in Control and Inform. Sci., 328, Springer, London, (2006). doi: 10.1007/11583592_5.

[6]

E. Fernández-Cara and S. Guerro, Global Carleman inequalities for parabolic systems and application to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446. doi: 10.1137/S0363012904439696.

[7]

E. Fernández-Cara and E. Zuazua, On the null controllability of the one-dimensional heat equation with BV coefficients, Comput. Appl. Math., 21 (2002), 167-190.

[8]

A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[9]

T. Nguyen, The uniform controllability property of semidiscrete approximations for the parabolic distributed parameter systems in Banach, prep. (2012).

[10]

J. Le Rousseau, Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients, J. Differential Equations, 233 (2007), 417-447. doi: 10.1016/j.jde.2006.10.005.

[11]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747. doi: 10.1051/cocv/2011168.

[12]

G. Lebeau and L. Robbiano, Contrôle exact de léquation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356. doi: 10.1080/03605309508821097.

[13]

J. Le Rousseau and L. Robbiano, Carleman estimate for elliptic operators with coefficents with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations, Arch. Rational Mech. Anal., 195 (2010), 953-990. doi: 10.1007/s00205-009-0242-9.

[14]

S. Labbé and E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control system, Systems and Control Letters, 55 (2006), 597-609. doi: 10.1016/j.sysconle.2006.01.004.

[15]

A. Lopez and E. Zuazua, Some new results related to the null controllability of the 1-D heat equation, Sem. EDP, Ecole Polytechnique, VIII (1998), 1-22.

[16]

E. Zuazua, Control and numerical approximation of the wave and heat equations, International Congress of Mathematicians, Madrid, III (2006), 1389-1417.

show all references

References:
[1]

A. Benabdallah, Y. Dermenjian and J. Le Rousseau, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem, J. Math. Anal. Appl., 336 (2007), 865-887. doi: 10.1016/j.jmaa.2007.03.024.

[2]

F. Boyer, F. Hubert and J. Le Rousseau, Discrete Carleman estimates for the elliptic operators and uniform controllability of semi discretized parabolic equations, J. Math. Pur. Appl., 93 (2010), 240-276. doi: 10.1016/j.matpur.2009.11.003.

[3]

F. Boyer, F. Hubert and J. Le Rousseau, Discrete Carleman estimates for the elliptic operators in arbitrary dimension and applications, SIAM J. Control Optim, 48 (2010), 5357-5397. doi: 10.1137/100784278.

[4]

F. Boyer and J. Le Rousseau, Carleman Estimates for Semi-Discrete Parabolic Operators and Application to the Controllability of Semi-Linear Semi-Discrete Parabolic Equations, prep. (2012).

[5]

Y. Chitour and E. Trélat, Controllability of partial differential equations, Advanced topics in control systems theory, 171-198, Lecture Notes in Control and Inform. Sci., 328, Springer, London, (2006). doi: 10.1007/11583592_5.

[6]

E. Fernández-Cara and S. Guerro, Global Carleman inequalities for parabolic systems and application to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446. doi: 10.1137/S0363012904439696.

[7]

E. Fernández-Cara and E. Zuazua, On the null controllability of the one-dimensional heat equation with BV coefficients, Comput. Appl. Math., 21 (2002), 167-190.

[8]

A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[9]

T. Nguyen, The uniform controllability property of semidiscrete approximations for the parabolic distributed parameter systems in Banach, prep. (2012).

[10]

J. Le Rousseau, Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients, J. Differential Equations, 233 (2007), 417-447. doi: 10.1016/j.jde.2006.10.005.

[11]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747. doi: 10.1051/cocv/2011168.

[12]

G. Lebeau and L. Robbiano, Contrôle exact de léquation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356. doi: 10.1080/03605309508821097.

[13]

J. Le Rousseau and L. Robbiano, Carleman estimate for elliptic operators with coefficents with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations, Arch. Rational Mech. Anal., 195 (2010), 953-990. doi: 10.1007/s00205-009-0242-9.

[14]

S. Labbé and E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control system, Systems and Control Letters, 55 (2006), 597-609. doi: 10.1016/j.sysconle.2006.01.004.

[15]

A. Lopez and E. Zuazua, Some new results related to the null controllability of the 1-D heat equation, Sem. EDP, Ecole Polytechnique, VIII (1998), 1-22.

[16]

E. Zuazua, Control and numerical approximation of the wave and heat equations, International Congress of Mathematicians, Madrid, III (2006), 1389-1417.

[1]

Damien Allonsius, Franck Boyer. Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries. Mathematical Control and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1

[9]

Zimo Zhu, Gang Chen, Xiaoping Xie. Semi-discrete and fully discrete HDG methods for Burgers' equation. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2021132

[10]

Lingyang Liu, Xu Liu. Controllability and observability of some coupled stochastic parabolic systems. Mathematical Control and Related Fields, 2018, 8 (3&4) : 829-854. doi: 10.3934/mcrf.2018037

[11]

Fabio Camilli, Francisco Silva. A semi-discrete approximation for a first order mean field game problem. Networks and Heterogeneous Media, 2012, 7 (2) : 263-277. doi: 10.3934/nhm.2012.7.263

[12]

Carlo Bardaro, Ilaria Mantellini. Boundedness properties of semi-discrete sampling operators in Mellin–Lebesgue spaces. Mathematical Foundations of Computing, 2022, 5 (3) : 219-229. doi: 10.3934/mfc.2021031

[13]

Zhaoqiang Ge. Controllability and observability of stochastic implicit systems and stochastic GE-evolution operator. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 339-351. doi: 10.3934/naco.2021009

[14]

Piermarco Cannarsa, Genni Fragnelli, Dario Rocchetti. Null controllability of degenerate parabolic operators with drift. Networks and Heterogeneous Media, 2007, 2 (4) : 695-715. doi: 10.3934/nhm.2007.2.695

[15]

Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations and Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020

[16]

Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control and Related Fields, 2020, 10 (1) : 89-112. doi: 10.3934/mcrf.2019031

[17]

Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control and Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001

[18]

Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013

[19]

Fengyan Yang. Exact boundary null controllability for a coupled system of plate equations with variable coefficients. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021036

[20]

J. Carmelo Flores, Luz De Teresa. Null controllability of one dimensional degenerate parabolic equations with first order terms. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3963-3981. doi: 10.3934/dcdsb.2020136

2020 Impact Factor: 1.284

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]