Optimal Control
Carleman inequalities and inverse problems for the Schrödinger equation
Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 53-58.

In this Note, we derive new Carleman inequalities for the evolution Schrödinger equation under a weak pseudoconvexity condition, which allows us to use weights with a linear spatial dependence. As a result, less restrictive boundary or internal observation regions may be used to obtain the stability for the inverse problem consisting in retrieving a stationary potential in the Schrödinger equation from a single boundary or internal measurement, respectively.

Dans cette Note, nous établissons de nouvelles inégalités de Carleman pour l'équation d'évolution de Schrödinger sous une hypothèse de pseudoconvexité faible, qui permet d'utiliser des poids affines en la variable d'espace. Comme application, nous pouvons définir des régions d'observabilité moins restrictives dans le problème inverse consistant à retrouver un potentiel stationnaire dans l'équation de Schrödinger à partir d'une mesure simple effectuée au bord ou à l'intérieur du domaine.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.11.014
Mercado, Alberto 1; Osses, Axel 1; Rosier, Lionel 1, 2

1 Centro de Modelamiento Matemático (CMM) and Departamento de Ingeniería Matemática (DIM), Universidad de Chile (UMI CNRS 2807), Avenida Blanco Encalada 2120, Casilla 170-3, Correo 3, Santiago, Chile
2 Institut Elie-Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 239, 54506 Vandœuvre-lès-Nancy cedex, France
@article{CRMATH_2008__346_1-2_53_0,
     author = {Mercado, Alberto and Osses, Axel and Rosier, Lionel},
     title = {Carleman inequalities and inverse problems for the {Schr\"odinger} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {53--58},
     publisher = {Elsevier},
     volume = {346},
     number = {1-2},
     year = {2008},
     doi = {10.1016/j.crma.2007.11.014},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2007.11.014/}
}
TY  - JOUR
AU  - Mercado, Alberto
AU  - Osses, Axel
AU  - Rosier, Lionel
TI  - Carleman inequalities and inverse problems for the Schrödinger equation
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 53
EP  - 58
VL  - 346
IS  - 1-2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2007.11.014/
DO  - 10.1016/j.crma.2007.11.014
LA  - en
ID  - CRMATH_2008__346_1-2_53_0
ER  - 
%0 Journal Article
%A Mercado, Alberto
%A Osses, Axel
%A Rosier, Lionel
%T Carleman inequalities and inverse problems for the Schrödinger equation
%J Comptes Rendus. Mathématique
%D 2008
%P 53-58
%V 346
%N 1-2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2007.11.014/
%R 10.1016/j.crma.2007.11.014
%G en
%F CRMATH_2008__346_1-2_53_0
Mercado, Alberto; Osses, Axel; Rosier, Lionel. Carleman inequalities and inverse problems for the Schrödinger equation. Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 53-58. doi : 10.1016/j.crma.2007.11.014. http://www.numdam.org/articles/10.1016/j.crma.2007.11.014/

[1] Baudouin, L.; Puel, J.-P. Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems, Volume 18 (2002), pp. 1537-1554

[2] Bukhgeim, A.; Klibanov, M. Global uniqueness of a class of inverse problems, Sov. Math. Dokl., Volume 24 (1982), pp. 244-247

[3] Burq, N.; Zworski, M. Geometric control in the presence of a black box, J. Amer. Math. Soc., Volume 17 (2004) no. 2, pp. 443-471

[4] Cardoulis, L.; Cristofol, M.; Gaitan, P. Inverse problem for the Schrödinger operator in an unbounded strip | arXiv

[5] Isakov, V. Carleman type estimates in an anisotropic case and applications, J. Differential Equations, Volume 105 (1993), pp. 217-238

[6] Lasiecka, I.; Triggiani, R.; Zhang, X. Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part I: H1-estimates, J. Inv. Ill-Posed Problems, Volume 11 (2004) no. 1, pp. 43-123

[7] Liu, K. Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., Volume 35 (1997) no. 5, pp. 1574-1590

[8] A. Mercado, A. Osses, L. Rosier, Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights, Inverse Problems, in press

[9] L. Rosier, B.-Y. Zhang, Null controllability of the complex Ginzburg–Landau equation, submitted for publication

Cited by Sources: