Observability estimate for stochastic Schrödinger equations

Qi Lü

doi:10.1016/j.crma.2010.10.016

Partial Differential Equations/Optimal Control

Observability estimate for stochastic Schrödinger equations

Presented by: Alain Bensousson

Qi Lü ^1,²

¹ School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China
² School of Mathematics, Sichuan University, Chengdu 610064, China

Comptes Rendus. Mathématique, Volume 348 (2010) no. 21-22, pp. 1159-1162.

Abstract
Résumé

In this Note, we present an observability estimate for stochastic Schrödinger equations with nonsmooth lower order terms. The desired inequality is derived by a global Carleman estimate which is based on a fundamental weighted identity for stochastic Schrödinger-like operator. As an interesting byproduct, starting from this identity, one can deduce all the known controllability/observability results for several stochastic and deterministic partial differential equations that are derived before via Carleman estimate in the literature.

Dans cette Note, nous établissons une inégalité d'observabilité pour les équations de Schrödinger stochastiques avec des termes d'ordre inférieur à coefficients non réguliers. Notre inégalité s'obtient à partir des inégalités de Carleman globales qui découlent d'une identité à poids pour les opérateurs de type Schrödinger stochastiques. Comme conséquence intéressante de cette identité, on retrouve tous les résultats connus de controlabilité/observabilité pour les équations aux dérivées partielles stochastiques et déterministes, de type Schrödinger et hyperboliques, où l'utilisation des inégalités de Carleman a joué un rôle.

Received: 2010-01-03
Accepted: 2010-10-07
Published online: 2010-10-27

DOI: 10.1016/j.crma.2010.10.016

Author's affiliations:

Qi Lü ^{1, 2}

¹ School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China
² School of Mathematics, Sichuan University, Chengdu 610064, China

@article{CRMATH_2010__348_21-22_1159_0,
     author = {Qi L\"u},
     title = {Observability estimate for stochastic {Schr\"odinger} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1159--1162},
     publisher = {Elsevier},
     volume = {348},
     number = {21-22},
     year = {2010},
     doi = {10.1016/j.crma.2010.10.016},
     language = {en},
}

TY  - JOUR
AU  - Qi Lü
TI  - Observability estimate for stochastic Schrödinger equations
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 1159
EP  - 1162
VL  - 348
IS  - 21-22
PB  - Elsevier
DO  - 10.1016/j.crma.2010.10.016
LA  - en
ID  - CRMATH_2010__348_21-22_1159_0
ER  -

%0 Journal Article
%A Qi Lü
%T Observability estimate for stochastic Schrödinger equations
%J Comptes Rendus. Mathématique
%D 2010
%P 1159-1162
%V 348
%N 21-22
%I Elsevier
%R 10.1016/j.crma.2010.10.016
%G en
%F CRMATH_2010__348_21-22_1159_0

Qi Lü. Observability estimate for stochastic Schrödinger equations. Comptes Rendus. Mathématique, Volume 348 (2010) no. 21-22, pp. 1159-1162. doi : 10.1016/j.crma.2010.10.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.10.016/

References
Cited by

[1] V. Barbu; A. Răscanu; G. Tessitore Carleman estimate and controllability of linear stochastic heat equations, Appl. Math. Optim., Volume 47 (2003), pp. 97-120

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

[3] X. Fu A weighted identity for partial differential operators of second order and its applications, C. R. Acad. Sci. Paris Sér. I, Volume 342 (2006), pp. 579-584

[4] X. Fu; J. Yong; X. Zhang Exact controllability for the multidimensional semilinear hyperbolic equations, SIAM J. Control Optim., Volume 46 (2007), pp. 1578-1614

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

[6] I. Lasiecka; R. Triggiani; X. Zhang Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates: Part I. $H^{1}$ -estimates, J. Inverse Ill-Posed Probl., Volume 11 (2004), pp. 43-123

[7] G. Lebeau Contrôle de l'équation de Schrödinger, J. Math. Pures Appl., Volume 71 (1992), pp. 267-291

[8] Q. Lü, Carleman and observability estimates for stochastic Schrödinger equations, and its applications, preprint.

[9] E. Machtyngier Exact controllability for the Schrödinger equation, SIAM J. Control Optim., Volume 32 (1994), pp. 24-34

[10] A. Mercado; A. Osses; L. Rosier Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights, Inverse Problems, Volume 24 (2008), p. 015017

[11] K.D. Phung Observability and control of Schrödinger equations, SIAM J. Control Optim., Volume 40 (2001), pp. 211-230

[12] S. Tang; X. Zhang Null controllability for forward and backward stochastic parabolic equations, SIAM J. Control Optim., Volume 48 (2009), pp. 2191-2216

[13] X. Zhang Exact controllability of the semilinear plate equations, Asymptot. Anal., Volume 27 (2001), pp. 95-125

[14] X. Zhang Carleman and observability estimates for stochastic wave equations, SIAM J. Math. Anal., Volume 40 (2008), pp. 851-868

[15] E. Zuazua Remarks on the controllability of the Schrödinger equation, Quantum Control: Mathematical and Numerical Challenges, CRM Proc. Lecture Notes, vol. 33, Amer. Math. Soc., Providence, RI, 2003, pp. 193-211

Cited by Sources:

Comments - Politique