Several Numerical Algorithms For A Class Of Hamilton-Jacobi-Bellman Equations

The Hamilton-Jacobi-Bellman equations have been widely used in engineering andeconomy, and the theory as well as the numerical solutions for them have attracted muchattention. In this paper, we discuss mainly the numerical solution of the discrete problemof a kind of Hamilton-Jacobi-Bellman equations.The discrete HJB equation studied in this paper is as follows: max1≤j≤k{Aju ? fj} = 0,where Aj∈Rn×n, fj∈Rn, j = 1,···,k. We also request that Aj meet appropriateconditions, which can be satisfied in practice. Under these conditions, we have provedthe existence of discrete HJB equation. We give the concept of the supersolution andsubsolution of the discrete HJB equation, and study their properties.This paper presents a class of Jacobi-type iterative method for HJB equations. Themethod is based on the supersolution of HJB equations. Convergence theorem of the al-gorithm is proved under appropriate conditions. For the varied coe?cients case of ellipticoperators, an improved Jacobi-type iterative method has been constructed and the cor-responding convergence theorem is proved. Numerical example shows that the improvedmethod is faster than the original one. This Paper proposes a Cascadic multigrid for HJBequations based on a new smoothing operator which is nonlinear one.We carry out the numerical experiments for all the above algorithms. Numericalexperiments show that the algorithms are e?cient.