Optimal Control Of Parameter Identification Problems For Two Types Of Evolutionary Equations

Parameter identification problems for two types of evolutionary equations are studied in this paper,which have important applications in stochastic control,financial mathematics,reservoir detection,earth science,etc.The first model of the article is an inverse problem of simultaneously reconstructing the two unknown parameters in the Kolmogorov-type equation.The main difficulty of this problem lies in the mutual influence between the parameters,especially the inversion corresponding to the two parameters is ill-posed,which are quite different.The second model is an inverse problem of reconstructing the zero-order term coefficients in a hyperbolic equation.The main difficulties of this problem are that the problem is seriously ill-posed and the zero-order term of the equation is a nonlinear function.For solving these difficulties,the two kinds of inverse problems are transformed into optimization problems respectively under the optimal control framework.Finally,the local well-posedness of the solutions for two optimization problems is mainly considered.The work consists of the following four chapters:The first chapter mainly focuses on the inverse problems of parabolic equations and hyperbolic equations,after summarizing the development history and the current research status of inverse problems,we explains the specific work to be done in each chapter of the paper.In the second chapter,an inverse problem of simultaneously reconstructing the initial value and first-order term coefficient in the Kolmogorov-type equation is studied.Firstly,since the problem is ill-posed,it is transformed into an optimal control problem to be studied.The cost functional is constructed from the terminal observation data,and then the existence of the optimal solution and the necessary conditions are deduced.Different from the single parameter problem,the cost functional constructed in this chapter is a binary functional containing two different variables and two independent regularization parameters.Finally,by assuming that the terminal time is relatively small,the obtained necessary conditions and energy estimates are used to prove the local uniqueness and stability of the optimal solution.In the third chapter,we consider an inverse problem of identifying the zero-order term coefficient in the second order nonlinear hyperbolic equation based on the optimization method.Firstly,from the point of view of optimal control,the problem is transformed into a nonlinear optimization problem,and then the existence of optimal solution is derived.Secondly,the necessary condition which is a couple systems of a parabolic equation and a variational inequality is deduced.Finally,the local uniqueness and stability of the optimal solution are derived by using relevant lemmas and theorems.In the fourth chapter,the research problems are summarized and conclusions of this paper are drawn,and some problems that can be considered in the follow-up work are analyzed.This paper mainly obtains the well-posedness of the optimal solution for the inverse problem from the theoretical point of view.In the follow-up,the numerical simulation can be carried out to detect the error between the actual value and the theoretical value.