Inverse Source Problem For Degenerate Parabolic Equation Based On The Optimal Control Theory

As we all know, some classic inverse problems can be traced back to very early, but therise of the inverse problem is something in recent years. There is no exact definition of theinverse problem and it is relative to the direct problem, from a practical point of view, thedifficulty of research inverse problem is much greater than research the corresponding directproblem. Because we solve the direct problem is according to the existing information to getresults, but to solve the inverse problem is that according to some pieces of information whichare obtained by experiments or measurements to reverse the unknown original information.Obviously, it goes against the natural order of physical process, and it is no longer satisfiedwith many good natures of direct problem, which is a main difficulty for the research.We know that most of the inverse problem is ill-posed, i.e., the solution of the inverseproblem does not exist, not unique, or unstable. Because the known information that we canused when we solve the inverse problem is very little, which are obtained by experiments ormeasurements mostly, but it is inevitable that exists errors in actual operation, because ofill-posed of the inverse problems, a small perturbation in the input data may result in a bigchange in the solutions of inverse problems, in such case, the obtained solution will bemeaningless.The manuscript is divided into five parts:The first chapter introduces the research background and the development at home andabroad of the inverse problem.The second chapter studies a developmental inverse problem from the point of view ofthe theory. The problem P is proposed in Section2.1. In Section2.2, taking into account theill-posed of problem P, it is transformed into an optimal control problem, and proves theexistence of the minimum for the control problem. We get the necessary condition whichsatisfies the optimal solution in Section2.3. In Section2.4, according to the discrete observedates re-construct continuous function prove continuity which satisfies the optimal solution.Summarize this chapter in Section2.5.The third chapter mainly deals with a degenerate parabolic equitation. The problem Q isproposed in Section3.1, it is different from problem P, problem Q does not need to give theboundary conditions, in other word, it does not satisfies the uniformly elliptic condition. InSection3.2, the problem Q is transformed into an optimal control problem. We prove theexistence of the minimum for the control problem in Section3.3. In Section3.4, we get thenecessary condition which satisfies the optimal solution. We get the global uniqueness andstability of optimal solution from the necessary conditions which obtained in the previoussection in Section3.5. Summarize this chapter in Section3.6. The main content of the fourth chapter is numerical simulation for the model whichproposed in the previous chapter. In Section4.1, the conjugate gradient method and thecorresponding iterative steps are proposed. To test the effect of the gradient method, twospecific examples are proposed in Section4.2, from the results we can see that the iterativemethod has good stability, and the unknown heat source is recovered very well. Summarizethis chapter in Section4.3.The fifth chapter is the summery and expectation of this dissertation. For problem P, thesubsequent work is to study from a numerical point of view and look for a good iterativemethod can recover the unknown parameters. In addition, this work discusses problem Qfrom1-D, for the multidimensional case, we can use the similar method to research. Of course,it is conceivable that make numerical experiment is more complex than1-D situation; it isalso a major work in future.