Initial Value Problems And Exact Boundary Controllability Of Several Classes Of Schr?dinger Equation

Schr?dinger type equation is an important kind of development equation.In this paper,we mainly use Banach fixed point theorem to prove the existence and uniqueness of solutions for anisotropic Schr?dinger type equation,and use HUM to prove the exact controllability of Schr?dinger type equation.This paper is divided into two chapters:In chapter 1,we study the solutions to the initial value problems of two types of anisotropic Schr?dinger equations.First of all,we study the existence and uniqueness of the global solution in Sobolev space H(?)(Rn-d)and the continuous dependence of the solution on the initial value for the anisotropic fourth-order Schr?dinger equation:(?)R,1≤d<n under the initial value condition u(x,0)=φ(x),x ∈Rn.Secondly,we study the existence and uniqueness of the local solution in Sobolev space(?)(Rn-d)for the anisotropic sixth-order Schr?dinger equation:(?)0,x ∈Rn,t ∈ R,1 ≤d<n under the initial value condition u(x,0)=φ(x),x∈ Rn.In particular,when d= 1,n=2,the global solution of the anisotropic sixth-order Schr?dinger equation is discussed.In chapter 2,we study the exact controllability of Schr?dinger type equations.First of all,we consider the anisotropic fourth-order Schr?dinger equation:iyt+△y-gx1x1x1x1=g(x,t,),x=(x1,X2,...,xn)∈Ω,t ∈R under the initial condition:y(x,0)=y0(x),x∈Ω,where g(x,t)is a nonlinear function.When g(x,t)=0 and the boundary condition y=0,yx1=v,x∈Γ0;y=0,gX1=0,x ∈Γ*0,we study the boundary exact controllability of the anisotropic fourth-order Schr?dinger equation.When g(x,t)=hχω and the boundary condition satisfies y=0,gx1=0,(x,t)∈Γ×(0,T),we study the internal exact controllability of the anisotropic fourth-order Schr?dinger equation.Secondly,we study the exact controllability of the quadratic nonlinear Schr?dinger equa-tion:iut+uxx+u2= 0.When u(α,t)=h1(t),u(β,t)=h2(t),x∈(α,β),t>0,we study the boundary control problem of the quadratic nonlinear Schr?dinger equation.When the condition satisfies:u(x,0)=h(x),x∈R,t∈R,we study the initial control problem of the quadratic nonlinear Schr?dinger equation.