# Controllability of Schrödinger equation with a nonlocal term

Mariano De Leo; Constanza Sánchez Fernández de la Vega; Diego Rial

ESAIM: Control, Optimisation and Calculus of Variations (2014)

- Volume: 20, Issue: 1, page 23-41
- ISSN: 1292-8119

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topDe Leo, Mariano, Sánchez Fernández de la Vega, Constanza, and Rial, Diego. "Controllability of Schrödinger equation with a nonlocal term." ESAIM: Control, Optimisation and Calculus of Variations 20.1 (2014): 23-41. <http://eudml.org/doc/272756>.

@article{DeLeo2014,

abstract = {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.},

author = {De Leo, Mariano, Sánchez Fernández de la Vega, Constanza, Rial, Diego},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {nonlinear Schrödinger–Poisson; Hartree potential; constant electric field; internal controllability; nonlinear Schrödinger-Poisson},

language = {eng},

number = {1},

pages = {23-41},

publisher = {EDP-Sciences},

title = {Controllability of Schrödinger equation with a nonlocal term},

url = {http://eudml.org/doc/272756},

volume = {20},

year = {2014},

}

TY - JOUR

AU - De Leo, Mariano

AU - Sánchez Fernández de la Vega, Constanza

AU - Rial, Diego

TI - Controllability of Schrödinger equation with a nonlocal term

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2014

PB - EDP-Sciences

VL - 20

IS - 1

SP - 23

EP - 41

AB - 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.

LA - eng

KW - nonlinear Schrödinger–Poisson; Hartree potential; constant electric field; internal controllability; nonlinear Schrödinger-Poisson

UR - http://eudml.org/doc/272756

ER -

## References

top- [1] T. Cazenave, Semilinear Schrödinger equations. AMS (2003). Zbl1055.35003
- [2] M. De Leo, On the existence of ground states for nonlinear Schrödinger–Poisson equation. Nonlinear Anal.73 (2010) 979–986. Zbl1191.35251
- [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. Zbl1152.81652MR2355098
- [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. MR2043524
- [6] R. Illner, H. Lange and H. Teismann, Limitations on the Control of Schrödinger Equations. ESAIM: COCV 12 (2006) 615–635. Zbl1162.93316MR2266811
- [7] T. Kato, Perturbation Theory for Linear Operators. Springer (1995). Zbl0836.47009MR1335452
- [8] P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor equations. Springer, Vienna (1990). Zbl0765.35001MR1063852
- [9] G.S. McDonald and W.J. Firth, Spatial solitary-wave optical memory. J. Optical Soc. America B7 (1990) 1328–1335.
- [10] M. Reed and B. Simon, Methods of Modern Math. Phys. Vol. II: Fourier Analysis, Self-Adjointness. Academic Press (1975). Zbl0308.47002MR493420
- [11] L. Rosier and B. Zhang, Exact boundary controllability of the nonlinear Schrödinger equation. J. Differ. Equ.246 (2009) 4129–4153. Zbl1171.35015MR2514738
- [12] B. Simon, Phase space analysis of simple scattering systems: extensions of some work of Enss. Duke Math. J.46 (1979) 119–168. Zbl0402.35076MR523604
- [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. MR2043529

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