Partial Differential Equations
Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients
Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, pp. 175-180.

We derive global Carleman estimates for one-dimensional linear parabolic operators t±x(cx) with a coefficient c with bounded variations. These estimates are obtained by approximating c by piecewise regular coefficients, cε, and passing to the limit in the Carleman estimates associated to the operators defined with cε. Such estimates yield results of controllability to the trajectories for a class of semilinear parabolic equations.

On obtient des inégalités de Carleman pour des opérateurs paraboliques linéaires t±x(cx) avec un coefficient c à variations bornées. Ces inégalités sont obtenues par approximations de c par des coefficients réguliers par morceaux, cε, et en passant à la limite dans les inégalités de Carleman associées aux opérateurs définis par cε. De telles inégalités donnent des résultats de contrôlabilité aux trajectoires pour une classe d'équations semilinéaires paraboliques.

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DOI: 10.1016/j.crma.2006.12.011
Le Rousseau, Jérôme 1

1 Laboratoire d'analyse topologie probabilités, CNRS UMR 6632, Université d'Aix-Marseille I, 13453 Marseille cedex 13, France
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Le Rousseau, Jérôme. Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients. Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, pp. 175-180. doi : 10.1016/j.crma.2006.12.011. http://www.numdam.org/articles/10.1016/j.crma.2006.12.011/

[1] G. Alessandrini, L. Escauriaza, Null-controllability of one-dimensional parabolic equations, ESAIM: Control Optim. Calc. Var., in press

[2] Barbu, V. Exact controllability of the superlinear heat equation, Appl. Math. Optim., Volume 42 (2000), pp. 73-89

[3] Benabdallah, A.; Dermenjian, Y.; Le Rousseau, J. Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications, C. R. Mecanique, Volume 334 (2006), pp. 582-586

[4] Benabdallah, A.; Dermenjian, Y.; Le Rousseau, J. Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem, 2006 http://www.cmi.univ-mrs.fr/~jlerous/publications.html (Preprint: LATP, Université d'Aix-Marseille I)

[5] Bressan, A. Hyperbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem, Oxford University Press, 2000

[6] Doubova, A.; Fernandez-Cara, E.; Gonzales-Burgos, M.; Zuazua, E. On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim., Volume 41 (2002), pp. 798-819

[7] Doubova, A.; Osses, A.; Puel, J.-P. Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients, ESAIM: Control Optim. Calc. Var., Volume 8 (2002), pp. 621-661

[8] Fernández-Cara, E.; Zuazua, E. Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincaré Analyse Non Linéaire, Volume 17 (2000), pp. 583-616

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

[10] Fursikov, A.; Imanuvilov, O.Yu. Controllability of Evolution Equations, Lecture Notes, vol. 34, Seoul National University, Korea, 1996

[11] Kolmogorov, A.; Fomin, S.V. Eléments de la théorie des fonctions et de l'analyse fonctionnelle, Editions Mir, 1974

[12] Lebeau, G.; Robbiano, L. Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, Volume 20 (1995), pp. 335-356

[13] Russell, D.L. A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Stud. Appl. Math., Volume 52 (1973), pp. 189-221

[14] Imanuvilov, O.Yu.; Yamamoto, M. Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications, Lecture Notes in Pure and Applied Mathematics, vol. 218, Dekker, New York, 2001 (pp. 113–137)

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