Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T06:46:08.803Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarques sur l'observabilité pour l'équation de Laplace

Published online by Cambridge University Press:  15 September 2003

Kim-Dang Phung
Kim-Dang Phung*
Affiliation:
17 rue Léonard Mafrand, 92320 Châtillon, France; phung@cmla.ens-cachan.fr.
Get access

Abstract

We consider the Laplace equation in a smooth bounded domain. We prove logarithmic estimates, in the sense of John [5] of solutions on a part of the boundary or of the domain without known boundary conditions. These results are established by employing Carleman estimates and techniques that we borrow from the works of Robbiano [8,11]. Also, we establish an estimate on the cost of an approximate control for an elliptic model equation.

Keywords

Laplace equationobservabilityCarleman inequalitiesapproximate controllability.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 621 - 635
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R. Dautray et J.-L. Lions, Analyse mathématique et calcul numérique. Masson (1988).
Fabre, C. et Lebeau, G., Prolongement unique des solutions de l'équation de Stokes. Comm. Partial Differential Equations 21 (1996) 573-596. CrossRef
Fernández-Cara, E. et Zuazua, E., The cost of approximate controllability for heat equations: The linear case. Adv. Differential Equations 5 (2000) 465-514.
L. Hörmander, Linear partial differential operators. Springer Verlag, Berlin (1963).
John, F., Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Comm. Pure Appl. Math. 13 (1960) 551-585. CrossRef
Lebeau, G., Contrôle analytique. I. Estimations a priori. Duke Math. J. 68 (1992) 1-30. CrossRef
J.-L. Lions, Contrôlabilité exacte, stabilisation et perturbation des systèmes distribués, Vol. 1, Coll. RMA. Masson, Paris (1988).
Lebeau, G. et Robbiano, L., Contrôle exact de l'équation de la chaleur. Comm. Partial Differential Equations 20 (1995) 335-356. CrossRef
J.-L. Lions et E. Magenes, Problèmes aux limites non homogènes, Vol. 1. Dunod (1968).
K.-D. Phung, Stabilisation d'ondes électromagnétiques. Thèse de l'ENS Cachan (1998).
Robbiano, L., Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques. Comm. Partial Differential Equations 16 (1991) 789-800. CrossRef
Robbiano, L., Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptot. Anal. 10 (1995) 95-115.