Convergence Analysis For Inverse Problems In Two Kings Of Evolutionary Equations

The inverse problem of parabolic equations is an important branch of the inverse prob-lem of mathematical physics,and it is also an ancient issue in industrial applications.Such as the determination of the material diffusion coefficient(heat transfer coefficient).The inver-sion of source/sink items and the determination of exchange coefficients at the boundary are all practical problems in industrial applications.Since the 1970s,not a few experts have done a lot of research on such issues,such as Cannon,Rundell,DuChateau,Isakov,and Beck,which have caused more and more attention on the research of inverse problems related to parabol-ic equations.This paper mainly considers the convergence of optimal solutions for two kinds of evolutionary equations.This paper studies the existence and convergence of solutions to regularization problems under appropriate additional conditions,and studies the existence,u-niqueness,and stability of solutions using the full variation regularization method.This article is mainly divided into the following five chapters:The first chapter is the introduction part,which briefly describes some research back-grounds and valuable domestic and international research status of the inverse problems of partial differential equations,especially the research on the inverse problems of parabolic equa-tions.The second chapter mainly considers the convergence of a class of optimization problems.The underlying equation of the problem is a second-order non-divergence degenerate parabolic equation.That is,the principal term coefficient in the equation degenerates into zero on the boundary of the region.The major difficulty in this paper is that the principal term coefficient is unknown.In fact,the degree of degeneration of the equation is usually determined by the properties of the principal term coefficients.On the other hand,in order to discuss degenerate equations,some weighted Sobolev spaces are employed,and these spaces are also related to the principal term coefficients.In order to overcome these difficulties,some new source conditions are introduced,and a strong regularity condition is added to the admissible function set of the principal term.Finally,the convergence of the optimal solution is proved successfully.In the third chapter,we mainly study the inverse problem of reconstructing zero-order term coefficients of second-order Schrodinger equations using terminal observations.The biggest feature of the Schrodinger equation is that its solution is a complex-valued function and the positive problem of the model is linear,but the inverse problem is nonlinear and seriously ill-posed.Based on the framework of optimal control theory,the original problem is transformed into an optimal control problem.The convergence of the input data minimal element with error perturbation is obtained by using Gteaux derivative theory and Poincare inequality.In chapter four,we investigates an inverse problem of using subsidiary condition to recon-struct the radiation coefficient of the second order degenerate parabolic equation.Based on the optimal control framework and total variation regularization,the problem is transformed into an optimization problem.Subsidiary condition in this paper is an average sense of observation,rather than a general terminal observation.Due to the total variation term is non-differential,so it is hard to work out the uniqueness of optimal solution.In order to overcome this difficulty,a polished total variation regularization term is introduced.Furthermore,the existence and the necessary conditions of the optimal solution are discussed.Finally,it is proved that the unique-ness and stability of the minimizer is proved successfully when the terminal time is assumed to be relatively small.The fifth chapter is summary and outlook.Subsequent work mainly considers from two aspects:On the one hand,the research object of this paper is one-dimensional,and later we will consider the high-dimensional situation.On the other hand,the problem of this article can be considered from the direction of TV-type penalties.