Local exact bilinear control of the Schrödinger equation
ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 4, pp. 1264-1281.

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 N3. For a general domain we will require a non degeneracy condition of the normal derivative of the eigenfunction on a part Γ 0 of the boundary satisfying the Geometric Control Condition (see [G. Lebeau. J. Math. Pures Appl. 71 (1992) 267–291]) and for a rectangle when N=2 or an interval for N=1 no further condition. In the general case we will use real potentials concentrated in the neighborhood of Γ 0 and the linear controllability results with real and sufficiently regular controls.

Received:
Accepted:
DOI: 10.1051/cocv/2016049
Classification: 35B65, 35Q41
Mots-clés : Schrödinger equation, bilinear control
Puel, Jean-Pierre 1

1 Laboratoire de Mathématiques de Versailles, Université de Versailles St Quentin, 78035 Versailles cedex, France.
@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://archive.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://archive.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://archive.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, Volume 22 (2016) no. 4, pp. 1264-1281. doi : 10.1051/cocv/2016049. http://archive.numdam.org/articles/10.1051/cocv/2016049/

K. Beauchard, Local controllability of a 1D Schrödinger equation. J. Math. Pures Appl. 84 (2005) 851–956. | DOI | Zbl

K. Beauchard and C. Laurent, 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).

J. Ball, J. Marsden and M. Slemrod, Controllability for distributed bilinear systems. SIAM J. Cont. Optim. 20 (1982) 575–597. | DOI | Zbl

S. Ervedoza and E. Zuazua, A systematic method for building smooth controls for smooth data. Discrete Contin. Dyn. Syst. Ser. B 14 (2010) 1375–1401. | MR | Zbl

S. Jaffard, Contrôle interne exact des vibrations d’une plaque rectangulaire. Port. Math. 47 (1990) 423–429. | Zbl

G. Lebeau, 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

E. Machtyngier, Exact controllability for the Schrödinger equation. SIAM J. Control Optim. 32 (1994) 24–34. | DOI | Zbl

J.-P. Puel, A regularity property for Schrödinger equations on bounded domains. Rev. Mat. Complut. 26 (2013) 183–192. | DOI | Zbl

G. Tenenbaum, M. Tucsnak, Fast and strongly localized observation for the Schrödinger equation. Trans. Amer. Math. Soc. 361 (2009) 951–977. | DOI | Zbl

H. Weyl, Das asymptotisch Verteilungsgezetz der Eigenwerte linearer partieller Differentialgleichungen. Math. Ann. 71 (1912) 441–479. | DOI | JFM

Cited by Sources: