ESAIM: Mathematical Modelling and Numerical Analysis

Research Article

On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman Equations

Barles, Guya1 and Jakobsen, Espen Robstada2

a1 Laboratoire de Mathématiques et Physique Théorique, University of Tours, Parc de Grandmont, 37200 Tours, France. barles@univ-tours.fr.

a2 Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway.

Abstract

Using systematically a tricky idea of N.V. Krylov, we obtain general results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-Jacobi-Bellman equations with variable coefficients. This result applies in particular to control schemes based on the dynamic programming principle and to finite difference schemes despite, here, we are not able to treat the most general case. General results have been obtained earlier by Krylov for finite difference schemes in the stationary case with constant coefficients and in the time-dependent case with variable coefficients by using control theory and probabilistic methods. In this paper we are able to handle variable coefficients by a purely analytical method. In our opinion this way is far simpler and, for the cases we can treat, it yields a better rate of convergence than Krylov obtains in the variable coefficients case.

(Received September 11 2001)

(Online publication April 15 2002)

Key Words:

  • Hamilton-Jacobi-Bellman equation;
  • viscosity solution;
  • approximation schemes;
  • finite difference methods;
  • convergence rate.

Mathematics Subject Classification:

  • 65N06;
  • 65N15;
  • 41A25;
  • 49L20;
  • 49L25
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