Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 5, pp. 1035-1078.

In arbitrary dimension, in the discrete setting of finite-differences we prove a Carleman estimate for a semi-discrete parabolic operator, in which the large parameter is connected to the mesh size. This estimate is applied for the derivation of a (relaxed) observability estimate, that yield some controlability results for semi-linear semi-discrete parabolic equations. Sub-linear and super-linear cases are considered.

DOI : 10.1016/j.anihpc.2013.07.011
Classification : 35K10, 35K58, 65M06, 93B05, 93B07
Mots-clés : Parabolic operator, Semi-discrete Carleman estimates, Observability, Null controllability, Semi-linear equations 
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     title = {Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1035--1078},
     publisher = {Elsevier},
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Boyer, Franck; Le Rousseau, Jérôme. Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 5, pp. 1035-1078. doi : 10.1016/j.anihpc.2013.07.011. http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.011/

[1] V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Optim. 42 (2000), 73 -89 | MR | Zbl

[2] F. Boyer, F. Hubert, J. Le Rousseau, Discrete Carleman estimates and uniform controllability of semi-discrete parabolic equations, J. Math. Pures Appl. 93 (2010), 240 -276 | MR | Zbl

[3] F. Boyer, F. Hubert, J. Le Rousseau, Discrete Carleman estimates for elliptic operators in arbitrary dimension and applications, SIAM J. Control Optim. 48 (2010), 5357 -5397 | MR | Zbl

[4] F. Boyer, F. Hubert, J. Le Rousseau, Uniform null-controllability properties for space/time-discretized parabolic equations, Numer. Math. 118 (2011), 601 -661 | MR | Zbl

[5] I. Bihari, A generalization of a lemma of bellman and its application to uniqueness problems of differential equations, Acta Math. Hung. 7 (1956), 81 -94 | MR | Zbl

[6] F. Boyer, On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems, http://hal.archives-ouvertes.fr/hal-00812964 | Zbl

[7] T. Carleman, Sur une problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes, Ark. Mat. Astron. Fys. 26B no. 17 (1939), 1 -9 | MR | Zbl

[8] T. Duyckaerts, X. Zhang, E. Zuazua, On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25 (2008), 1 -41 | EuDML | Numdam | MR | Zbl

[9] E. Fernández-Cara, S. Guerrero, Global Carleman inequalities for parabolic systems and application to controllability, SIAM J. Control Optim. 45 no. 4 (2006), 1395 -1446 | MR | Zbl

[10] E. Fernández-Cara, A. Münch, Numerical null controllability of semi-linear 1D heat equations: Fixed point, least squares and Newton methods, Math. Control Relat. Fields 2 no. 3 (2012), 217 -246 | MR | Zbl

[11] E. Fernández-Cara, E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 17 (2000), 583 -616 | EuDML | Numdam | MR | Zbl

[12] A. Fursikov, O.Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes vol. 34 , Seoul National University, Korea (1996) | MR | Zbl

[13] L. Hörmander, On the uniqueness of the Cauchy problem, Math. Scand. 6 (1958), 213 -225 | EuDML | MR | Zbl

[14] L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin (1963) | MR | Zbl

[15] L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. IV, Springer-Verlag (1985) | MR

[16] O.Yu. Imanuvilov, Controllability of parabolic equations, Mat. Sb. (N.S.) 186 (1995), 109 -132 | MR

[17] J. Le Rousseau, 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 | EuDML | Numdam | MR | Zbl

[18] G. Lebeau, L. Robbiano, Contrôle exact de l'équation de la chaleur, Commun. Partial Differ. Equ. 20 (1995), 335 -356 | MR | Zbl

[19] S. Labbé, E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control systems, Syst. Control Lett. 55 (2006), 597 -609 | MR | Zbl

[20] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Probl. 25 (2009), 123013 | MR | Zbl

[21] E. Zuazua, Control and numerical approximation of the wave and heat equations, International Congress of Mathematicians, III, Madrid, Spain (2006), 1389 -1417 | MR | Zbl

[22] C. Zuily, Uniqueness and Non Uniqueness in the Cauchy Problem, Prog. Math. , Birkhäuser (1983) | MR | Zbl

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