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...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.展开更多
The purpose of this paper is to extend the notion of regional controllability for hyperbolic parameter dependent systems.The key idea is the characterization of the averaged regional control with minimal energy.This c...The purpose of this paper is to extend the notion of regional controllability for hyperbolic parameter dependent systems.The key idea is the characterization of the averaged regional control with minimal energy.This control steers the state average(with respect to such a parameter)towards the desired state only on a given part of the system evolution domain.In this paper,we give the precis definition and the properties of this new concept.Then,we use an approach based on an extension of the Hilbert uniqueness method devoted to the calculation of the control in two different cases:zone control and pointwise control.展开更多
In this paper,we show by Hilbert Uniqueness Method that the boundary value problem of fifth-order KdV equation{y_(t)-y_(5x)=0,(x,t)∈(0,2π)×(0,T),y(t,2π)-y(t,0)=h_(0)(t),y_(x)(t,2π)-y_(x)(t,0)=h_(1)(t),y_(2x)(...In this paper,we show by Hilbert Uniqueness Method that the boundary value problem of fifth-order KdV equation{y_(t)-y_(5x)=0,(x,t)∈(0,2π)×(0,T),y(t,2π)-y(t,0)=h_(0)(t),y_(x)(t,2π)-y_(x)(t,0)=h_(1)(t),y_(2x)(t,2π)-y_(2x)(t,0)=h_(2)(t),y_(3x)(t,2π)-y_(3x)(t,0)=h_(3)(t),y_(4x)(t,2π)-y_(4x)(t,0)=h_(4)(t),(with boundary data as control inputs)is exact controllability.展开更多
文摘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This work was supported by the Directorate-General for Scientific Research and Technological Development(DGRSDT)The authors wish to acknowledge the editor and anonymous reviewers for their insightful comments,which have improved the quality of this publication.The authors also acknowledge Prof.Carlos Castro,from Polytechnic University of Madrid(Spain),for a fruitful discussion and the referee for the remarks that have improved the final version of the paper.
文摘The purpose of this paper is to extend the notion of regional controllability for hyperbolic parameter dependent systems.The key idea is the characterization of the averaged regional control with minimal energy.This control steers the state average(with respect to such a parameter)towards the desired state only on a given part of the system evolution domain.In this paper,we give the precis definition and the properties of this new concept.Then,we use an approach based on an extension of the Hilbert uniqueness method devoted to the calculation of the control in two different cases:zone control and pointwise control.
基金supported by the Zhejiang Provincial Natural Science Foundation of China(No.LY18A010024)National Natural Science Foundation of China(No.12075208).
文摘In this paper,we show by Hilbert Uniqueness Method that the boundary value problem of fifth-order KdV equation{y_(t)-y_(5x)=0,(x,t)∈(0,2π)×(0,T),y(t,2π)-y(t,0)=h_(0)(t),y_(x)(t,2π)-y_(x)(t,0)=h_(1)(t),y_(2x)(t,2π)-y_(2x)(t,0)=h_(2)(t),y_(3x)(t,2π)-y_(3x)(t,0)=h_(3)(t),y_(4x)(t,2π)-y_(4x)(t,0)=h_(4)(t),(with boundary data as control inputs)is exact controllability.
