The Research On Optimal Control Problems For Some Classes Of Nonlinear Schr(?)dinger Systems

Because of the wide application of Schr(?)dinger equation in natural science,many different types of optimal control problems determined by Schr(?)dinger equation have emerged.As a result,the optimal control problems of Schr(?)dinger systems have got a lot of attention from scholars at home and abroad.Compared to the optimal control problems of linear Schr(?)dinger systems,the optimal control problems of nonlinear Schr(?)dinger systems have more extensive background,and need more professional knowledge of mathematical theory for the complexity.Therefore,the research about the optimal control problems of nonlinear is very necessary and significant on theory and in practice.In this doctoral thesis,we mainly study the optimal control problems of three classes of nonlinear Schr(?)dinger systems with Hartree term by using the variational method.In addition,based on the integral order nonlinear Schr(?)dinger system,in order to further study optimal control problems of the fractional nonlinear Schr(?)dinger system,we also study sign-changing solutions for a class of Kirchhoff-type fractional Schr(?)dinger equation in order to provide a nonempty admissible set and mathematical tools for nonlinear Schr(?)dinger optimal control problems.This part is the theoretical need and beneficial supplement of the follow-up research on the optimal control problems of nonlinear Schr(?)dinger systems.This thesis is divided into six chapters.In Chapter 1,we first introduce the historical background and research status at home and abroad about optimal control problems of three classes of nonlinear Schr(?)dinger systems and a class of Kirchhoff-type fractional Schr(?)dinger equation.Then,we briefly give the main results of this doctoral thesis.In Chapter 2,some marks,definitions and related preparatory knowledge used in this doctoral thesis are introduced.In Chapter 3,we study the optimal control problem for a class of Hartree type Schr(?)dinger system with Coulomb potential.Firstly,the local existence,global existence and regularity of solutions of the Schr(?)dinger systems are obtained by using the contractive mapping principle.Secondly,the existence of optimal control element of optimal control problem is proved by discussing the properties of minimization sequence.Finally,we prove the local Lipschitz continuous dependence on control parameter of the solution and the first-order Fr′echet differentiability of the objective function with respect to the control parameter,then obtain the first-order optimal conditions for the control problem.In Chapter 4,we consider the optimal control problem for a class of Schr(?)dinger system with singular term.First of all,the global existence and regularity of solutions for the related Schr(?)dinger system are obtained.Then,the existence of optimal control element of the optimal control problem is proved.Finally,by discussing the local Lipschitz continuous dependence on control parameter of the solution of the system,we prove the first-order Fr′echet differentiability of the objective function with respect to the control parameter.As a result,we successfully obtain the firstorder optimal conditions of the control problem.In Chapter 5,we study the optimal control problem for a class of Hartree type nonlinear Schr(?)dinger system with general singular potential.Because the equation involves the general singular potential,Hardy inequality is no longer applicable,so more detailed and complex estimates have to be used to discuss the existence of the optimal control element of the optimal control problem and get the first optimal conditions of the control problem.In Chapter 6,by using the restricted variational method,topological degree theory and deformation lemma,we study the existence of sign-changing solutions for a class of Kirchhoff-type fractional Schr(?)dinger equation,the energy doubling characteristics and the control of parameter on the asymptotic behavior of sign-changing solutions.The results obtained in this chapter are helpful to the further study of the optimal control problem of nonlinear fractional order Schr(?)dinger equations.