Numerical Methods Of A Class Of Discrete HJB Equation

The Hamilton-Jacobi-Bellman equation(in short HJB equation) are encoun-tered first solving optimal control problem using dynamic programming. Af-terward, the HJB equation have been widely used in sciences,engineering andeconomics. So, the numerical solutions for the HJB equation have attracted muchattention. The HJB equations are important problems in the numerical solutionsof partial di?erential equations, too. In the thesis, we discuss mainly the numer-ical solutions of the discrete problems of a kind of HJB equation. We constructmany new iterative algorithms and prove their convergence. The numerical re-sults show the e?ectiveness of algorithms.The discrete HJB equation may be approximated by a quasivariational in-equality system on the condition. We consider a relaxation algorithm for theapproximate quasivariational inequality system of HJB equation, the relaxationalgorithm is Gauss-Seidel type algorithm onω= 1. Then we construct a domaindecomposition method for the quasivariational inequality system based on therelaxation algorithm. We give the corresponding monotone convergence theoriesabout their algorithms. Numerical experiments show that the e?ectiveness of thealgorithms are improved using properω.Lions and Mercier have proposed two iterative schemes for the numericalsolution of the discrete HJB equation. At each iteration, a variational inequalityis solved in Scheme I. We propose, based on Scheme I, a relaxation scheme witha parameterω. We call it Lions-Mercier relaxation algorithm. The monotoneconvergence of the algorithm has been proved. In our numerical examples, thealgorithm is faster than Scheme I choosing a proper parameterω.A new successive relaxation iterative algorithm for discrete HJB equation isproposed. It does not solve a linear equation system or a linear complementarityproblem but carry out simple arithmetic operations at each iteration, and weuse the newest results at each iteration. Numerical tests show it is faster thantraditional algorithms. Monotone convergence has been proved for the algorithm.Finally, we construct a new multigrid method for discrete HJB equation.We choose a nonlinear smoother, that is a relation iterative algorithm in the lastparagraph, as a smoothing operator. Numerical experiments show it is faster than mulitgrid methods that they used to solve the discrete HJB equation.