We are going to prove the local exact bilinear controllability for a Schrödinger equation, set in a bounded regular domain, in a neighborhood of an eigenfunction corresponding to a simple eigenvalue in dimension . For a general domain we will require a non degeneracy condition of the normal derivative of the eigenfunction on a part of the boundary satisfying the Geometric Control Condition (see [G. Lebeau. J. Math. Pures Appl. 71 (1992) 267–291]) and for a rectangle when or an interval for no further condition. In the general case we will use real potentials concentrated in the neighborhood of and the linear controllability results with real and sufficiently regular controls.
Accepté le :
DOI : 10.1051/cocv/2016049
Mots clés : Schrödinger equation, bilinear control
@article{COCV_2016__22_4_1264_0,
author = {Puel, Jean-Pierre},
title = {Local exact bilinear control of the {Schr\"odinger} equation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1264--1281},
publisher = {EDP-Sciences},
volume = {22},
number = {4},
year = {2016},
doi = {10.1051/cocv/2016049},
zbl = {1354.35126},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2016049/}
}
TY - JOUR AU - Puel, Jean-Pierre TI - Local exact bilinear control of the Schrödinger equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 1264 EP - 1281 VL - 22 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2016049/ DO - 10.1051/cocv/2016049 LA - en ID - COCV_2016__22_4_1264_0 ER -
%0 Journal Article %A Puel, Jean-Pierre %T Local exact bilinear control of the Schrödinger equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 1264-1281 %V 22 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2016049/ %R 10.1051/cocv/2016049 %G en %F COCV_2016__22_4_1264_0
Puel, Jean-Pierre. Local exact bilinear control of the Schrödinger equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1264-1281. doi : 10.1051/cocv/2016049. http://www.numdam.org/articles/10.1051/cocv/2016049/
, Local controllability of a 1D Schrödinger equation. J. Math. Pures Appl. 84 (2005) 851–956. | DOI | Zbl
and , Local controllability of 1D linear and nonlinear Schrödinger equations. J. Math. Pures Appl. 94 (2010) 520–554. | DOI | Zbl
K. Beauchard and C. Laurent, Local exact controllability of the 2D Schrödinger-Poisson system. Preprint hal-01333627 (2016).
, and , Controllability for distributed bilinear systems. SIAM J. Cont. Optim. 20 (1982) 575–597. | DOI | Zbl
and , A systematic method for building smooth controls for smooth data. Discrete Contin. Dyn. Syst. Ser. B 14 (2010) 1375–1401. | MR | Zbl
, Contrôle interne exact des vibrations d’une plaque rectangulaire. Port. Math. 47 (1990) 423–429. | Zbl
, Contrôle de l’equation de Schrödinger. J. Math. Pures Appl. 71 (1992) 267–291. | Zbl
J.-L. Lions, Contrôlabilité exacte, perturbations et stabilization des systèmes distribués. Tome 1, Contrôlabilité exacte. Collection R.M.A 8, Masson (1988). | Zbl
, Exact controllability for the Schrödinger equation. SIAM J. Control Optim. 32 (1994) 24–34. | DOI | Zbl
, A regularity property for Schrödinger equations on bounded domains. Rev. Mat. Complut. 26 (2013) 183–192. | DOI | Zbl
, , Fast and strongly localized observation for the Schrödinger equation. Trans. Amer. Math. Soc. 361 (2009) 951–977. | DOI | Zbl
, Das asymptotisch Verteilungsgezetz der Eigenwerte linearer partieller Differentialgleichungen. Math. Ann. 71 (1912) 441–479. | DOI | JFM
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