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
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

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