Discrete time high-order schemes for viscosity solutions of Hamilton- Jacobi-Bellman equations. (English) Zbl 0791.65046

Falcone, Marizio; Ferretti, Roberto

Numer. Math. 67, No. 3, 315-344 (1994).

The article introduces a class of high-order approximation schemes for first order Hamilton-Jacobi-Bellman equations in \(\mathbb{R}^ n\) (which apply, in particular, to the stationary first order linear equation). The method used to obtain the above schemes is a discrete version of dynamic programming. A general convergence result applies to the schemes in that class: they converge to the viscosity solution whenever the coefficients in the equation are Lipschitz continuous. An estimate in \(L^ \infty\) of the order of convergence and of the local truncation error is proved under more restrictive assumptions.
Several examples of these schemes corresponding to the orders of convergence 1, 2 and 4 are presented throughout the paper. The last section contains their detailed analysis and comparison in terms of CPU time and numerical errors on some tests which can have smooth or non smooth solutions.

Reviewer: Marizio Falcone (Roma)

MSC:

65K10	Numerical optimization and variational techniques
49L20	Dynamic programming in optimal control and differential games
49L99	Hamilton-Jacobi theories

Keywords:

optimal control; numerical examples; high-order approximation schemes; first order Hamilton-Jacobi-Bellman equations; dynamic programming; convergence; viscosity solution