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 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.

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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.
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