The Numerical Solution Of Hamilton-Jacobi-Bellman Equations

The Hamilton-Jacobi-Bellman equations have been widely used in engineering and economy,and the theory as well as the numerical solutions for them have attracted much attention. In this paper,we discuss mainly the numerical solution of the discrete problem of a kind of Hamilton- Jacobi- Bellman equations.First, we consider an iterative algorithm of Jacobi type and the corresponding domain decomposition algorithm for a approximate quasivariational inequality system of the Hamilton- Jacobi- Bellman equation and the corresponding convergence problem. We introduce the iterative method of Jacobi type to solve the discrete problem of the quasivariational inequality system. A new proof for the monotone convergence of the iterative method is given under appropriate conditions. The domain decomposition method of Jacobi type for the quasivariational inequality system is proposed and the corresponding monotone convergence theory is established.Second, we propose an iterative algorithm of Gauss-seidel type and a domain decomposition algorithm. Based on the algorithms of Jacobi type, we construct algorithms of Gauss-Seidel type. One of the advantages of the algorithms of Gauss-Seidel type is that they,in each iteration,use the newest messages in the computation. Hence,the algorithms of Gauss-Seidel type are faster than the algorithms of Jacobi type generally. Under appropriate conditions the monotone convergence of the algorithms of Gauss-Seidel type is proved. The corresponding convergence theory is established also for domain decomposition method of Gauss-Seidel type.Finally , we propose a new iterative method for solving the discrete Hamilton-Jacobi-Bellman equations directly. Compared with the algorithms mentioned above,one of the advantages of the method is that it does not need to solve any linear equations or inequalities. So the implementation of the method is very simple and easy. Although its convergence rate is slow, namely it needs more iterative times,the procedure is very simple, moreover,it is convenient to be parallellized. Furthermore,corresponding theoretical researches on how to choose the initial value in the givenalgorithm have been studied in details. To show the effectiveness of the algorithms proposed here, numerical experiments have been performed.The numerical results show the effectivity of the algorithms.