Inverse Coefficient Problems For Three Kinds Of Evolutionary Equations

The thesis mainly investigates three inverse problems of reconstructing the unknown co-efficient in different evolution equations,such type problems have important application in a large field of engineering and science,particularly in probability theory,stochastic control,fi-nancial mathematics,electromagnetic metal forming and radio system and so on.On the basis of the optimal control framework,we discuss the well-posedness of the solution of the opti-mization problems corresponding to three inverse problems.The main difficulties with all such arguments are the ill-posedness,fully non-linear and the nonconvex of the control functional.Especially for the latter two problems,due to the lack of additional data,it's hard to obtain the uniqueness of optimal solution.However,in this article,we put forward some new methods for the prior estimates,and we detect that the optimal solution can be proved to be local unique and stable if the terminal time is relatively small,which is also the main contribution of this paper.This work includes the following five parts:First of all,the introduction makes a brief exposition for the research background and research status of the inverse problems.The first chapter deals with an inverse problem of identifying the convection coefficient in a class of Kolmogorov equation from some additional information mainly from the angle of theoretical analysis.After shortly introducing the problem P1 and proving which has an uniqueness solution,and considering the identification problem P1 is ill-posed,it's transformed into an optimal control problem P1'.Then the existence and the necessary condition of the minimizer for the cost functional are obtained.Finally,the uniqueness and stability of the minimizer are deduced by energy estimates and the necessary condition in this part.The second chapter studies a non-linear inverse problem of recovering the unknown dif-fusion coefficient in a coupled parabolic-elliptic system from the final measurement data by optimization methods.Being different from other works which are governed by single par-tial differential equations,the underlying mathematical model in this chapter is coupled with a parabolic equation and an elliptic equation which are defined in different domains,and the extra condition used in solving the identification problem is only imposed on the solution in the small domain rather than in the whole domain.On the basis of the optimal control framework,the in-verse coefficient problem P2 is translated into an optimization problem P2',then the existence,the uniqueness and the stability of the optimal solution are shown in turn.The third chapter discusses about a problem of reconstructing the zero-order coefficient in a Schrodinger equation using the final overspecified data.Problems of this type widely arise in radio system,and the unknown coefficient is refractive index.After briefly describing the problem,the existence of the solution of the optimal control problem is turned out at first,and then the necessary condition of the minimizer satisfied is deduced.Finally,the uniqueness and the stability of the optimal solution are obtained from the necessary condition,energy estimates and the properties of the complex function.The fourth chapter summarizes the full text and proposes the forecast to the further re-search.For the three problems investigated in the work,the further research can be focused on the following two parts.For one thing,try to deal with the more general things using the methods in this paper.For another,find out the numerical method corresponding to all kinds of problems in the thesis,so as to recover the unknown coefficient from the view of numerical simulation.