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: {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 term inal 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 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 boun dary controllability of (semi-linear) nonlinear Schrödinger equation. © Editorial Committee of Applied Mathematics 2007.