The Optimal Control Problems Of Nonlinear Schr(?)dinger System

This dissertation considers the optimal control problems governed by nonlinear Schr(?)dinger systems by using the variational method,our start points are the local existence,global existence and regularity of solutions for the corresponding nonlinear Schr(?)dinger system.Following the frame established in the literature [54],for the models we discussed,the existence of the minima of the objective functional is obtained,and the Fr′echet differentiability with respect to the control parameter for the objective functional is also proved.In addition,the first order optimality conditions is rigorously derived.We mainly deal with two patterns of control,one is the bilinear control via external potential,another is nonlinearity control via the nonlinear interaction.In Chapter 1,we give an introduction to the background of our study and the notations we used,and introduce some useful inequalities and lemmas.In Chapter 2,we study the optimal bilinear control problem governed by the logarithmic Schr(?)dinger equation.Firstly,we define an approximate system,and discuss the existence and regularity of the approximate system.And then by a convergence argument,we get the existence of the original system.Secondly,we study the optimal control problem for the approximate system,we prove the existence of a minima for the approximate objective functional(approximate minimizers),and derive the control equation(approximate control equation).Finally,by a rigorously discussion,we show that the approximate minimizers are convergence,and the limit,satisfies the limit equation of approximate control equation,is a minimizer of the original optimal control problem.In Chapter 3,we study the optimal control problem governed by the weakly coupled Schr(?)dinger system.Without the subquadratic assumption on the potential,we consider the problem with the control potential 1/||.The key problem is to overcome the difficulties caused by the invalidity of the Aubin-Lions lemma to prove the existence of the minimizer for the objective functional.In addition,due to the loss of local Lipschitz continuity caused by the coupled terms,for the objective functional,we can not get the Fr′echet differentiability with respect to the control parameter following directly the method in literatures,so we find another way,i.e.,with the aid of the higher regularity of the solution,to get the continuity of the solution with respect to the control,and then derive the control equation.Based on the idea developed in Chapter 3,in Chapter 4,we extend the study in literature [47].Finally,in Chapter 5,we deal with a kind of optimal problem of nonlinearity control originated from the experiment in Bose-Einstein condensate.