Comptes Rendus
Théorie du contrôle
An inverse problem for a hyperbolic system in a bounded domain
[Un problème inverse pour un système hyperbolique dans un domaine borné]
Laure Cardoulis1
1 Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France.
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 653-665.

Dans cette Note, on considère un système hyperbolique de deux équations, défini dans un domaine borné. En utilisant la méthode des inégalités de Carleman, on obtient un résultat de stabilité Lipschitz pour les quatre coefficients dépendant de la variable d’espace de ce système, avec des mesures d’une seule composante de la solution et grâce à la donnée de deux ensembles de conditions initiales.

In this Note we consider a two-by-two hyperbolic system defined on a bounded domain. Using Carleman inequalities, we obtain a Lipschitz stability result for the four spatially varying coefficients with measurements of only one component, given two sets of initial conditions.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.431
Classification : 35R30
Laure Cardoulis 1

1 Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{CRMATH_2023__361_G3_653_0,
     author = {Laure Cardoulis},
     title = {An inverse problem for a hyperbolic system in a bounded domain},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {653--665},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.431},
     language = {en},
}
TY  - JOUR
AU  - Laure Cardoulis
TI  - An inverse problem for a hyperbolic system in a bounded domain
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 653
EP  - 665
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.431
LA  - en
ID  - CRMATH_2023__361_G3_653_0
ER  - 
%0 Journal Article
%A Laure Cardoulis
%T An inverse problem for a hyperbolic system in a bounded domain
%J Comptes Rendus. Mathématique
%D 2023
%P 653-665
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.431
%G en
%F CRMATH_2023__361_G3_653_0
Laure Cardoulis. An inverse problem for a hyperbolic system in a bounded domain. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 653-665. doi : 10.5802/crmath.431. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.431/

[1] Lucie Baudouin; Jean-Pierre Puel Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Probl., Volume 18 (2002) no. 6, pp. 1537-1554 | DOI | Zbl

[2] Larisa Beilina; Michel Cristofol; Shumin Li; Masahiro Yamamoto Lipschitz stability for an inverse hyperbolic problem of determining two coefficients by a finite number of observations, Inverse Probl., Volume 34 (2018) no. 1, 015001, 27 pages | MR | Zbl

[3] Mourad Bellassoued; Masahiro Yamamoto Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer Monographs in Mathematics, Springer, 2018

[4] Assia Benabdallah; Michel Cristofol; Patricia Gaitan; Masahiro Yamamoto Inverse problem for a parabolic system with two components by measurements of one component, Appl. Anal., Volume 88 (2009) no. 5, pp. 683-709 | DOI | MR | Zbl

[5] Alexandre L. Bukhgeim; Michael V. Klibanov Global uniqueness of a class of multidimensional inverse problems, Sov. Math., Dokl., Volume 17 (1981), pp. 244-247

[6] Laure Cardoulis An inverse problem for a parabolic system in an unbounded guide (submitted)

[7] Laure Cardoulis Applications of Carleman inequalities for a two-by-two parabolic system in an unbounded guide, Rostocker Math. Kolloq., Volume 72 (2020), pp. 49-71 | MR

[8] Laure Cardoulis; Michel Cristofol An inverse problem for a generalized FitzHug–Nagumo system (in preparation)

[9] Laure Cardoulis; Michel Cristofol An inverse problem for the heat equation in an unbounded guide, Appl. Math. Lett., Volume 62 (2016), pp. 63-68 | DOI | MR | Zbl

[10] Laure Cardoulis; Michel Cristofol; Patricia Gaitan Inverse problem for the Schrödinger operator in an unbounded strip, J. Inverse Ill-Posed Probl., Volume 16 (2008), pp. 127-146

[11] Laure Cardoulis; Michel Cristofol; Morgan Morancey A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide, Math. Control Relat. Fields, Volume 11 (2021) no. 4, pp. 965-985 | DOI | MR | Zbl

[12] Michel Cristofol; Patricia Gaitan; Hichem Ramoul Inverse problems for a 2×2 reaction diffusion system using a Carleman estimate with one observation, Inverse Probl., Volume 22 (2006) no. 5, pp. 1561-1573 | DOI | MR | Zbl

[13] Michel Cristofol; Shumin Li; Eric Soccorsi Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary, Math. Control Relat. Fields, Volume 6 (2016) no. 3, pp. 407-427 | DOI | MR | Zbl

[14] Michel Cristofol; Eric Soccorsi Stability estimate in an inverse problem for non autonomous magnetic Schrödinger equations, Appl. Anal., Volume 90 (2011) no. 9-10, pp. 1499-1520 | DOI | Zbl

[15] Xinchi Huang; Oleg Imanuvilov; Masahiro Yamamoto Stability for inverse source problems by Carleman estimates, Inverse Probl., Volume 36 (2020) no. 12, 125006, 20 pages | MR | Zbl

[16] Oleg Imanuvilov; Victor Isakov; Masahiro Yamamoto An inverse problem for the dynamical Lamé system with two sets of boundary data, Commun. Pure Appl. Math., Volume 56 (2003), pp. 1366-1383 | DOI | Zbl

[17] Oleg Imanuvilov; Masahiro Yamamoto Global Lipschitz stability in an inverse hyperbolic problem by interior observations, Inverse Probl., Volume 17 (2001) no. 4, pp. 717-728 | DOI | MR | Zbl

[18] Oleg Imanuvilov; Masahiro Yamamoto Global uniqueness and stability in determining coefficients of wave equations, Commun. Partial Differ. Equations, Volume 26 (2001) no. 7-8, pp. 1409-1425 | DOI | Zbl

[19] Oleg Imanuvilov; Masahiro Yamamoto Determination of a coefficient in an acoustic equation with a single measurement, Inverse Probl., Volume 19 (2003) no. 1, pp. 157-171 | DOI | Zbl

[20] Oleg Imanuvilov; Masahiro Yamamoto Carleman estimates for the non-stationary Lamé system and the application to an inverse problem, ESAIM, Control Optim. Calc. Var., Volume 11 (2005), pp. 1-56 | DOI | Numdam | Zbl

[21] Michael V. Klibanov Inverse problems and Carleman estimates, Inverse Probl., Volume 8 (1992) no. 4, pp. 575-596 | DOI | MR | Zbl

[22] Michael V. Klibanov Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse Ill-Posed Probl., Volume 21 (2013) no. 4, pp. 477-560 | Zbl

[23] Masahiro Yamamoto Carleman estimates for parabolic equations and applications, Inverse Probl., Volume 25 (2009) no. 12, 123013, 75 pages | MR | Zbl

[24] Ganghua Yuan; Masahiro Yamamoto Lipschitz stability in the determination of the principal part of a parabolic equation, ESAIM, Control Optim. Calc. Var., Volume 15 (2009) no. 3, pp. 525-554 | DOI | Numdam | MR | Zbl

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Identification of two independent coefficients with one observation for the Schrödinger operator in an unbounded strip

Laure Cardoulis; Patricia Gaitan

C. R. Math (2010)

Inverse problem for the Schrödinger operator in an unbounded strip

Laure Cardoulis; Michel Cristofol; Patricia Gaitan

C. R. Math (2008)

An inverse problem for a time-dependent Schrödinger operator in an unbounded strip

Laure Cardoulis

C. R. Math (2012)