Optimal Control Of Free Boundary Problems

Free boundary problems are very popular. They occur mostly in heat-flow problems with phase changes and in certain diffusion processes. In mathematics finance problems, there is an important problem, called pricing American options, which is also a free boundary problem. Some of the boundary of the free boundary problems is often unknown, and sometimes is moving. It needs to be solved with the solution of the equation. When we study a free boundary problem, we always think to rewrite the problem in order to make the free boundary disappear. The thesis introduced the variational inequality, so we can study the problem in some"weak"sense. For the parabolic obstacle problems, the thesis studied the optimal control of the obstacle. And the obstacle is a condition which the original problem should be satisfied with. Because only a little of free boundary problems can be given their analytical solutions, how to get accurate solutions with a sophisticated high-quality numerical algorithm becomes very important. From rewriting the free boundary problem, we naturally consider to solve the problem with variational inequality method. In other words, the problem becomes how to solve the variational inequality. Today there are many methods to solve it. The most popular one is Finite Element method. The thesis solved the oxygen diffusion problem by the method, and got a conclusion that there is a great error in the solution of the free boundary. So for a problem whose purpose is to get the free boundary, the method must not be the first choice. For the above reasons the thesis introduced Chebyshev spectral method, which is tested by the oxygen problem. And we know spectral method is an accurate method for the free boundary. In addition, we studied a useful problem called pricing American options. First of all, we rewrite the problem into a parabolic variational inequality, then studied the price of the option by maximum principle, finally got the numerical solutions and drew the picture of the optimal exercise boundary. Because the initial data of the pricing American options is weak singular, the thesis use smoothed function to approach the initial data, and proved that the solution of the approximative problem is convergent to the solution of the original problem.