no. 1
Controllability of Schrödinger equation with a nonlocal term
ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 23-41.

This paper is concerned with the internal distributed control problem for the 1D Schrödinger equation, i ut(x,t) = -uxx+α(x) u+m(u) u, that arises in quantum semiconductor models. Here m(u) is a non local Hartree-type nonlinearity stemming from the coupling with the 1D Poisson equation, and α(x) is a regular function with linear growth at infinity, including constant electric fields. By means of both the Hilbert Uniqueness Method and the contraction mapping theorem it is shown that for initial and target states belonging to a suitable small neighborhood of the origin, and for distributed controls supported outside of a fixed compact interval, the model equation is controllable. Moreover, it is shown that, for distributed controls with compact support, the exact controllability problem is not possible.

DOI : 10.1051/cocv/2013052
Classification : 93B05, 81Q93, 35Q55
Mots clés : nonlinear Schrödinger-Poisson, Hartree potential, constant electric field, internal controllability 
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     title = {Controllability of {Schr\"odinger} equation with a nonlocal term},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {23--41},
     publisher = {EDP-Sciences},
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De Leo, Mariano; Sánchez Fernández de la Vega, Constanza; Rial, Diego. Controllability of Schrödinger equation with a nonlocal term. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 23-41. doi : 10.1051/cocv/2013052. http://archive.numdam.org/articles/10.1051/cocv/2013052/

[1] T. Cazenave, Semilinear Schrödinger equations. AMS (2003). | Zbl

[2] M. De Leo, On the existence of ground states for nonlinear Schrödinger-Poisson equation. Nonlinear Anal. 73 (2010) 979-986. | Zbl

[3] M. De Leo and D. Rial, Well-posedness and smoothing effect of nonlinear Schrödinger -Poisson equation. J. Math. Phys. 48 (2007) 093509-1,15. | MR | Zbl

[4] G.K. Harkness, G.L. Oppo, E. Benkler, M. Kreuzer, R. Neubecker and T. Tschudi, Fourier space control in an LCLV feedback system. J. Optics B: Quantum and Semiclassical Optics 1 (1999) 177-182.

[5] R. Illner, H. Lange and H. Teismann, A note on vol. 33 of the Exact Internal Control of Nonlinear Schrödinger Equations, in Quantum Control: Mathematical and Numerical Challenges, vol. 33 of CRM Proc. Lect. Notes (2003) 127-136. | MR

[6] R. Illner, H. Lange and H. Teismann, Limitations on the Control of Schrödinger Equations. ESAIM: COCV 12 (2006) 615-635. | Numdam | MR | Zbl

[7] T. Kato, Perturbation Theory for Linear Operators. Springer (1995). | MR | Zbl

[8] P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor equations. Springer, Vienna (1990). | MR | Zbl

[9] G.S. Mcdonald and W.J. Firth, Spatial solitary-wave optical memory. J. Optical Soc. America B 7 (1990) 1328-1335.

[10] M. Reed and B. Simon, Methods of Modern Math. Phys. Vol. II: Fourier Analysis, Self-Adjointness. Academic Press (1975). | MR

[11] L. Rosier and B. Zhang, Exact boundary controllability of the nonlinear Schrödinger equation. J. Differ. Equ. 246 (2009) 4129-4153. | MR | Zbl

[12] B. Simon, Phase space analysis of simple scattering systems: extensions of some work of Enss. Duke Math. J. 46 (1979) 119-168. | MR | Zbl

[13] E. Zuazua, Remarks on the controllability of the Schrödinger equation, in Quantum Control: Mathematical and Numerical Challenges, vol. 33 of CRM Proc. Lect. Notes (2003) 193-211. | MR

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