EXACT BOUNDARY CONTROLLABILITY OF 1-D NONLINEAR SCHRODINGER EQUATION

In this paper, the boundary control problem of a distributed parameter system described by the Schr(o)dinger equation posed on finite interval α≤ x ≤β:{iyt yxx |y|2y = 0,y(α,t) = h1(t),y(β,t) = h2(t) for t ＞ 0 (S)is considered. It is shown that by choosing appropriate control inputs (hj), (j = 1,2) one can always guide the system (S) from a given initial state ψ∈ Hs(α,β),(s ∈ R) to a terminal state ψ∈ Hs(α,β), in the time period 0, T]. The exact boundary controllability is obtained by considering a related initial value control problem of Schr(o)dinger equation posed on the whole line R. The discovered smoothing properties of Schr(o)dinger 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 Schr(o)dinger equation.

Exact boundary controllability of 1-D nonlinear Schrödinger equation

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.