Remarques sur l'observabilité pour l'équation de Laplace

Kim-Dang Phung

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 9, page 621-635
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Phung, Kim-Dang. "Remarques sur l'observabilité pour l'équation de Laplace." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 621-635. <http://eudml.org/doc/90714>.

@article{Phung2010,
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. },
author = {Phung, Kim-Dang},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Laplace equation; observability; Carleman inequalities; approximate controllability.; approximate controllability},
language = {eng},
month = {3},
pages = {621-635},
publisher = {EDP Sciences},
title = {Remarques sur l'observabilité pour l'équation de Laplace},
url = {http://eudml.org/doc/90714},
volume = {9},
year = {2010},
}

TY - JOUR
AU - Phung, Kim-Dang
TI - Remarques sur l'observabilité pour l'équation de Laplace
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 9
SP - 621
EP - 635
AB - 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.
LA - eng
KW - Laplace equation; observability; Carleman inequalities; approximate controllability.; approximate controllability
UR - http://eudml.org/doc/90714
ER -

References

top
  1. R. Dautray et J.-L. Lions, Analyse mathématique et calcul numérique. Masson (1988).  Zbl0652.45001
  2. C. Fabre et G. Lebeau, Prolongement unique des solutions de l'équation de Stokes. Comm. Partial Differential Equations21 (1996) 573-596.  Zbl0849.35098
  3. E. Fernández-Cara et E. Zuazua, The cost of approximate controllability for heat equations: The linear case. Adv. Differential Equations5 (2000) 465-514.  Zbl1007.93034
  4. L. Hörmander, Linear partial differential operators. Springer Verlag, Berlin (1963).  Zbl0108.09301
  5. F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Comm. Pure Appl. Math.13 (1960) 551-585.  Zbl0097.08101
  6. G. Lebeau, Contrôle analytique. I. Estimations a priori. Duke Math. J.68 (1992) 1-30.  Zbl0780.93053
  7. J.-L. Lions, Contrôlabilité exacte, stabilisation et perturbation des systèmes distribués, Vol. 1, Coll. RMA. Masson, Paris (1988).  Zbl0653.93002
  8. G. Lebeau et L. Robbiano, Contrôle exact de l'équation de la chaleur. Comm. Partial Differential Equations20 (1995) 335-356.  Zbl0819.35071
  9. J.-L. Lions et E. Magenes, Problèmes aux limites non homogènes, Vol. 1. Dunod (1968).  Zbl0165.10801
  10. K.-D. Phung, Stabilisation d'ondes électromagnétiques. Thèse de l'ENS Cachan (1998).  
  11. L. Robbiano, Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques. Comm. Partial Differential Equations16 (1991) 789-800.  Zbl0735.35086
  12. L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptot. Anal.10 (1995) 95-115.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.