Carleman estimates and controllability results for the one-dimensional heat equation with <i>BV</i> coefficients

Jérôme Le Rousseau

doi:10.1016/j.crma.2006.12.011

Partial Differential Equations

Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients

Presented by: Gilles Lebeau

Jérôme Le Rousseau ¹

¹ Laboratoire d'analyse topologie probabilités, CNRS UMR 6632, Université d'Aix-Marseille I, 13453 Marseille cedex 13, France

Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, pp. 175-180.

Abstract
Résumé

We derive global Carleman estimates for one-dimensional linear parabolic operators $\partial_{t} \pm \partial_{x} (c \partial_{x})$ with a coefficient c with bounded variations. These estimates are obtained by approximating c by piecewise regular coefficients, $c_{ε}$ , and passing to the limit in the Carleman estimates associated to the operators defined with $c_{ε}$ . 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 $\partial_{t} \pm \partial_{x} (c \partial_{x})$ avec un coefficient c à variations bornées. Ces inégalités sont obtenues par approximations de c par des coefficients réguliers par morceaux, $c_{ε}$ , et en passant à la limite dans les inégalités de Carleman associées aux opérateurs définis par $c_{ε}$ . De telles inégalités donnent des résultats de contrôlabilité aux trajectoires pour une classe d'équations semilinéaires paraboliques.

Received: 2006-06-21
Accepted: 2006-12-12
Published online: 2007-01-16

DOI: 10.1016/j.crma.2006.12.011

Author's affiliations:

Jérôme Le Rousseau ¹

¹ Laboratoire d'analyse topologie probabilités, CNRS UMR 6632, Université d'Aix-Marseille I, 13453 Marseille cedex 13, France

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     author = {J\'er\^ome Le Rousseau},
     title = {Carleman estimates and controllability results for the one-dimensional heat equation with {\protect\emph{BV}} coefficients},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {175--180},
     publisher = {Elsevier},
     volume = {344},
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     doi = {10.1016/j.crma.2006.12.011},
     language = {en},
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Jérôme Le Rousseau. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.12.011/

References
Cited by

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

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

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

[4] A. Benabdallah; Y. Dermenjian; J. Le Rousseau 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] A. Bressan Hyperbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem, Oxford University Press, 2000

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

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

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

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

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

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

[14] O.Yu. Imanuvilov; M. Yamamoto 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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