Exact boundary controllability of 1-D nonlinear Schrödinger equation

Abstract

In this paper, the boundary control problem of a distributed parameter system described by the Schrödinger equation posed on finite interval α ≤ x ≤ β: $\left\{ \begin{gathered} iy_t + y_{xx} + |y|^2 y = 0, \hfill \\ y(\alpha ,t) = h_1 (t),y(\beta ,t) = h_2 (t)fort > 0 \hfill \\ \end{gathered} \right.(S)$ is considered. It is shown that by choosing appropriate control inputs (h _j), (j = 1, 2) one can always guide the system (S) from a given initial state φ ∈ H ^s(α, β), (s ∈ R) to a terminal state ψ ∈ H ^s(α, β), in the time period [0, T]. The exact boundary controllability is obtained by considering a related initial value control problem of Schrödinger equation posed on the whole line R. The discovered smoothing properties of Schrödinger equation have played important roles in our approach; this may be the first step to prove the results on boundary controllability of (semi-linear) nonlinear Schrödinger equation.