摘要
In this paper, the boundary control problem of a distributed parameter system described by the Schrodinger equation posed on finite interval α≤x≤β:{ iyt +yzz+|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 (hi), (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 φ∈ 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 SchrSdinger equation posed on the whole line R. The discovered smoothing properties of Schrodinger 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 Schrodinger equation.
In this paper, the boundary control problem of a distributed parameter system described by the Schrodinger equation posed on finite interval α≤x≤β:{ iyt +yzz+|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 (hi), (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 φ∈ 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 SchrSdinger equation posed on the whole line R. The discovered smoothing properties of Schrodinger 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 Schrodinger equation.
基金
Supported by the NNSF of China(10371136).