Optimal nonlinearity control of Schrödinger equation

We study the optimal nonlinearity control problem for the nonlinear Schrodinger equation \begin{document} $iu_{t} = -\triangle u+V(x)u+h(t)|u|^α u$ \end{document} , which is originated from the Fechbach resonance management in Bose-Einstein condensates and the nonlinearity management in nonlinear optics. Based on the global well-posedness of the equation for \begin{document} $0 , we show the existence of the optimal control. The Frechet differentiability of the objective functional is proved, and the first order optimality system for \begin{document} $N≤ 3$ \end{document} is presented.