Controllability of Schrödinger equation with a nonlocal term.

MARIANO DE LEO; DIEGO RIAL; CONSTANZA SÁNCHEZ FERNÁNDEZ DE LA VEGA

ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS

EDP SCIENCES S A

Lugar: Paris; Año: 2012

1262-3377

This paper is concerned with the internal distributed control problem for the 1D Schödinger equation, i u_t (x, t) = −u_xx + α(x) u + m(u) u, that arises in quantum semi- conductor 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.