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
[Inégalité d'observabilité pour les équations de Schrödinger stochastiques]

Présenté par : 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.

Résumé
Abstract

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.

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.

Reçu le : 2010-01-03
Accepté le : 2010-10-07
Publié le : 2010-10-27

DOI : 10.1016/j.crma.2010.10.016

Affiliations des auteurs :

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},
}

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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/

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