On the bilinear control of the Gross-Pitaevskii equation
Chambrion, Thomas1 ; Thomann, Laurent2
1 Université Bourgogne Franche-Comté, CNRS, IMB, F-21000 Dijon, France
2 Université de Lorraine, CNRS, IECL, F-54000 Nancy, France
Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 605-626.
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

In this paper we study the bilinear-control problem for the linear and non-linear Schrödinger equation with harmonic potential. By the means of different examples, we show how space-time smoothing effects (Strichartz estimates, Kato smoothing effect) enjoyed by the linear flow, can help to prove obstructions to controllability.

DOI : 10.1016/j.anihpc.2020.01.001
Classification : 35Q93, 35L05
Mots-clés : Control theory, Bilinear control, Obstructions, Non-linear Schrödinger equation
Chambrion, Thomas 1 ; Thomann, Laurent 2

1 Université Bourgogne Franche-Comté, CNRS, IMB, F-21000 Dijon, France
2 Université de Lorraine, CNRS, IECL, F-54000 Nancy, France 
@article{AIHPC_2020__37_3_605_0,
     author = {Chambrion, Thomas and Thomann, Laurent},
     title = {On the bilinear control of the {Gross-Pitaevskii} equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {605--626},
     publisher = {Elsevier},
     volume = {37},
     number = {3},
     year = {2020},
     doi = {10.1016/j.anihpc.2020.01.001},
     mrnumber = {4093621},
     zbl = {1437.35681},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2020.01.001/}
}
TY  - JOUR
AU  - Chambrion, Thomas
AU  - Thomann, Laurent
TI  - On the bilinear control of the Gross-Pitaevskii equation
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2020
SP  - 605
EP  - 626
VL  - 37
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2020.01.001/
DO  - 10.1016/j.anihpc.2020.01.001
LA  - en
ID  - AIHPC_2020__37_3_605_0
ER  - 
%0 Journal Article
%A Chambrion, Thomas
%A Thomann, Laurent
%T On the bilinear control of the Gross-Pitaevskii equation
%J Annales de l'I.H.P. Analyse non linéaire
%D 2020
%P 605-626
%V 37
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2020.01.001/
%R 10.1016/j.anihpc.2020.01.001
%G en
%F AIHPC_2020__37_3_605_0
Chambrion, Thomas; Thomann, Laurent. On the bilinear control of the Gross-Pitaevskii equation. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 605-626. doi : 10.1016/j.anihpc.2020.01.001. http://www.numdam.org/articles/10.1016/j.anihpc.2020.01.001/

[1] Ball, J.; Marsden, J.; Slemrod, M. Controllability for distributed bilinear systems, SIAM J. Control Optim., Volume 20 (1982) no. 4, pp. 575–597 | DOI | MR | Zbl

[2] Beauchard, K. Local controllability of a 1-D Schrödinger equation, J. Math. Pures Appl. (9), Volume 84 (2005) no. 7, pp. 851–956 | DOI | MR | Zbl

[3] Beauchard, K. Local controllability and non controllability of a 1D wave equation, J. Differ. Equ., Volume 250 (2011), pp. 2064–2098 | DOI | MR | Zbl

[4] Beauchard, K.; Lange, H.; Teismann, H. Local exact controllability of a one-dimensional nonlinear Schrödinger equation, SIAM J. Control Optim., Volume 53 (2015) no. 5, pp. 2781–2818 | DOI | MR | Zbl

[5] Beauchard, K.; Laurent, C. Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control, J. Math. Pures Appl. (9), Volume 94 (2010) no. 5, pp. 520–554 | DOI | MR | Zbl

[6] Beauchard, K.; Laurent, C. Bilinear control of high frequencies for a 1D Schrödinger equation, Math. Control Signals Syst., Volume 29 (2017) no. 2 | DOI | MR | Zbl

[7] Boussaïd, N.; Caponigro, M.; Chambrion, T. Weakly coupled systems in quantum control, IEEE Trans. Autom. Control, Volume 58 (2013) no. 9, pp. 2205–2216 | DOI | MR | Zbl

[8] Boussaïd, N.; Caponigro, M.; Chambrion, T. On the Ball–Marsden–Slemrod obstruction in bilinear control systems, Proceedings of the 58th IEEE Conference on Decision and Control (CDC), 2019 (to appear)

[9] Burq, N.; Thomann, L.; Tzvetkov, N. Remarks on the Gibbs measures for nonlinear dispersive equations, Ann. Fac. Sci. Toulouse Math. (6), Volume 27 (2018) no. 3, pp. 527–597 | DOI | Numdam | MR | Zbl

[10] Carles, R. Nonlinear Schrödinger equation with time dependent potential, Commun. Math. Sci., Volume 9 (2011) no. 4, pp. 937–964 | DOI | MR | Zbl

[11] Chambrion, T.; Mason, P.; Sigalotti, M.; Boscain, U. Controllability of the discrete-spectrum Schrödinger equation driven by an external field, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 26 (2009) no. 1, pp. 329–349 | DOI | Numdam | MR | Zbl

[12] Chambrion, T.; Thomann, L. A topological obstruction to the controllability of nonlinear wave equations with bilinear control term, SIAM J. Control Optim., Volume 57 (2019) no. 4, pp. 2315–2327 | DOI | MR | Zbl

[13] Feng, B.; Zhao, D. Optimal bilinear control of Gross-Pitaevskii equations with Coulombian potentials, J. Differ. Equ., Volume 260 (2016) no. 3, pp. 2973–2993 | DOI | MR | Zbl

[14] Feng, B.; Zhao, D.; Chen, P. Optimal bilinear control of nonlinear Schrödinger equations with singular potentials, Nonlinear Anal., Volume 107 (2014), pp. 12–21 | DOI | MR | Zbl

[15] Fujiwara, D. A construction of the fundamental solution for the Schrödinger equation, J. Anal. Math., Volume 35 (1979), pp. 41–96 | DOI | MR | Zbl

[16] Hintermüller, M.; Marahrens, D.; Markowich, P.; Sparber, C. Optimal bilinear control of Gross-Pitaevskii equations, SIAM J. Control Optim., Volume 51 (2013) no. 3, pp. 2509–2543 | DOI | MR | Zbl

[17] Illner, R.; Lange, H.; Teismann, H. Limitations on the control of Schrödinger equations, ESAIM Control Optim. Calc. Var., Volume 12 (2006) no. 4, pp. 615–635 | DOI | Numdam | MR | Zbl

[18] Khapalov, A. Controllability of Partial Differential Equations Governed by Multiplicative Controls, Lecture Notes in Mathematics, vol. 1995, Springer-Verlag, Berlin, 2010 (xvi+284 pp.) | DOI | MR | Zbl

[19] Mason, P.; Sigalotti, M. Generic controllability properties for the bilinear Schrödinger equation, Partial Differ. Equ., Volume 35 (2010) no. 4, pp. 685–706 | MR | Zbl

[20] Mirrahimi, M.; Rouchon, P. Controllability of quantum harmonic oscillators, IEEE Trans. Autom. Control, Volume 49 (2004) no. 5, pp. 745–747 | DOI | MR | Zbl

[21] Naibo, V.; Stefanov, A. On some Schrödinger and wave equations with time dependent potentials, Math. Ann., Volume 334 (2006) no. 2, pp. 325–338 | DOI | MR | Zbl

[22] Nersesyan, V.; Nersisyan, H. Global exact controllability in infinite time of Schrödinger equation, J. Math. Pures Appl. (9), Volume 97 (2012) no. 4, pp. 295–317 | DOI | MR | Zbl

[23] Poiret, A. Solutions globales pour des équations de Schrödinger sur-critiques en toutes dimensions (Preprint) | arXiv

[24] Poiret, A. Solutions globales pour l'équation de Schrödinger cubique en dimension 3 (Preprint) | arXiv

[25] Puel, J.-P. Local exact bilinear control of the Schrödinger equation, ESAIM Control Optim. Calc. Var., Volume 22 (2016) no. 4, pp. 1264–1281 | Numdam | MR | Zbl

[26] Sarychev, A. Controllability of the cubic Schroedinger equation via a low-dimensional source term, Math. Control Relat. Fields, Volume 2 (2012) no. 3, pp. 247–270 | DOI | MR | Zbl

[27] Shirikyan, A. Euler equations are not exactly controllable by a finite-dimensional external force, Physica D, Volume 237 (2008) no. 10–12, pp. 1317–1323 | MR | Zbl

[28] Taylor, M.E. Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, vol. 81, American Mathematical Society, Providence, RI, 2000 | MR | Zbl

[29] Turinici, G. On the controllability of bilinear quantum systems, Mathematical Models and Methods for Ab Initio Quantum Chemistry, Lecture Notes in Chem., vol. 74, Springer, Berlin, 2000, pp. 75–92 | MR | Zbl

[30] Yajima, K.; Zhang, G. Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity, J. Differ. Equ. (2004) no. 1, pp. 81–110 | DOI | MR | Zbl

Cité par Sources :