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Controllability of Schrödinger equation with a nonlocal term

Published online by Cambridge University Press:  29 August 2013

Mariano De Leo ,
Constanza Sánchez Fernández de la Vega  and
Diego Rial
Mariano De Leo
Affiliation:
Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M. Gutiérrez 1150 (1613) Los Polvorines, Buenos Aires, Argentina. [email protected]
Constanza Sánchez Fernández de la Vega
Affiliation:
IMAS – CONICET and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I (1428) Buenos Aires, Argentina; [email protected]; [email protected]
Diego Rial
Affiliation:
IMAS – CONICET and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I (1428) Buenos Aires, Argentina; [email protected]; [email protected]
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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.

Keywords

Nonlinear Schrödinger–PoissonHartree potentialconstant electric fieldinternal controllability
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 1 , January 2014 , pp. 23 - 41
Copyright
© EDP Sciences, SMAI, 2013

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References

T. Cazenave, Semilinear Schrödinger equations. AMS (2003).
De Leo, M., On the existence of ground states for nonlinear Schrödinger–Poisson equation. Nonlinear Anal. 73 (2010) 979–986. Google Scholar
De Leo, M. and Rial, D., Well-posedness and smoothing effect of nonlinear Schrödinger –Poisson equation. J. Math. Phys. 48 (2007) 093509-1,15. Google Scholar
Harkness, G.K., Oppo, G.L., Benkler, E., Kreuzer, M., Neubecker, R. and Tschudi, T., Fourier space control in an LCLV feedback system. J. Optics B: Quantum and Semiclassical Optics 1 (1999) 177–182. Google Scholar
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. CrossRef
Illner, R., Lange, H. and Teismann, H., Limitations on the Control of Schrödinger Equations. ESAIM: COCV 12 (2006) 615–635. Google Scholar
T. Kato, Perturbation Theory for Linear Operators. Springer (1995).
P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor equations. Springer, Vienna (1990).
McDonald, G.S. and Firth, W.J., Spatial solitary-wave optical memory. J. Optical Soc. America B 7 (1990) 1328–1335. Google Scholar
M. Reed and B. Simon, Methods of Modern Math. Phys. Vol. II: Fourier Analysis, Self-Adjointness. Academic Press (1975).
Rosier, L. and Zhang, B., Exact boundary controllability of the nonlinear Schrödinger equation. J. Differ. Equ. 246 (2009) 4129–4153. Google Scholar
Simon, B., Phase space analysis of simple scattering systems: extensions of some work of Enss. Duke Math. J. 46 (1979) 119–168. Google Scholar
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. CrossRef