Internal controllability of the korteweg–de vries equation on a bounded domain

Capistrano–Filho, Roberto A.; Pazoto, Ademir F.; Rosier, Lionel

doi:10.1051/cocv/2014059

Capistrano–Filho, Roberto A.^1,² ; Pazoto, Ademir F.³ ; Rosier, Lionel ⁴

¹ Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68530, Cidade Universitária, Ilha do Fundão, 21941-909 Rio de Janeiro (RJ), Brazil.
² Institut Elie Cartan, UMR 7502 UHP/CNRS/INRIA, BP 70239, 54506 Vandœuvre-les-Nancy cedex, France.
³ Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68530, Cidade Universitária, Ilha do Fundão, 21941-909 Rio de Janeiro (RJ), Brazil.
⁴ Centre Automatique et Systèmes, MINES ParisTech, PSL Research University, 60 boulevard Saint-Michel, 75272 Paris cedex 06, France.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 1076-1107.

Résumé

This paper is concerned with the control properties of the Korteweg–de Vries (KdV) equation posed on a bounded interval $(0, L)$ with a distributed control. When the control region is an arbitrary open subdomain $(l_{1}, l_{2})$ , we prove the null controllability of the KdV equation by means of a new Carleman inequality. As a consequence, we obtain a regional controllability result, which roughly tells us that any target function arbitrarily chosen on $(0, l_{1})$ and null on $(l_{2}, L)$ is reachable. Finally, when the control region is a neighborhood of the right endpoint, an exact controllability result in a weighted $L^{2}$ -space is also established.

MR Zbl | 3 citations dans Numdam

DOI : 10.1051/cocv/2014059

Classification : 35Q53, 37K10, 93B05, 93D15
Mots-clés : KdV equation, Carleman estimate, null controllability, exact controllability

Affiliations des auteurs :

Capistrano–Filho, Roberto A. ^{1, 2} ; Pazoto, Ademir F. ³ ; Rosier, Lionel ⁴

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     title = {Internal controllability of the korteweg{\textendash}de vries equation on a bounded domain},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1076--1107},
     publisher = {EDP-Sciences},
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Capistrano–Filho, Roberto A.; Pazoto, Ademir F.; Rosier, Lionel. Internal controllability of the korteweg–de vries equation on a bounded domain. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 1076-1107. doi : 10.1051/cocv/2014059. http://archive.numdam.org/articles/10.1051/cocv/2014059/

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