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On the convergence rate of approximation schemes forHamilton-Jacobi-Bellman Equations

Published online by Cambridge University Press:  15 April 2002

Guy Barles  and
Espen Robstad Jakobsen
Guy Barles
Affiliation:
Laboratoire de Mathématiques et Physique Théorique, University of Tours, Parc de Grandmont, 37200 Tours, France. barles@univ-tours.fr.
Espen Robstad Jakobsen
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway.
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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.

Keywords

Hamilton-Jacobi-Bellman equationviscosity solutionapproximation schemesfinite difference methodsconvergence rate.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 1 , January 2002 , pp. 33 - 54
Copyright
© EDP Sciences, SMAI, 2002

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References

Barles, G. and Souganidis, P.E., Convergence of approximation schemes for fully nonlinear second-order equations. Asymptotic Anal. 4 (1991) 271-283.
M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997).
F. Bonnans and H. Zidani, Consistency of generalized finite difference schemes for the stochastic HJB equation. Preprint.
Camilli, F. and Falcone, M., An approximation scheme for the optimal control of diffusion processes. RAIRO Modél. Math. Anal. Numér. 29 (1995) 97-122. CrossRef
Capuzzo-Dolcetta, I., On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming. Appl. Math. Optim. 10 (1983) 367-377. CrossRef
Crandall, M.G., Ishii, H. and Lions, P.-L., User's guide to viscosity solutions of second-order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1-67. CrossRef
Crandall, M.G. and Lions, P.-L., Two approximations of solutions of Hamilton-Jacobi equations. Math. Comp. 43 (1984) 1-19. CrossRef
W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, New York (1993).
Ishii, H. and Lions, P.-L, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differential Equations 83 (1990) 26-78. CrossRef
E.R. Jakobsen and K.H. Karlsen, Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations. To appear in J. Differential Equations.
Krylov, N.V., On the rate of convergence of finite-difference approximations for Bellman's equations. St. Petersbg Math. J. 9 (1997) 639-650.
Krylov, N.V., On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients. Probab. Theory Relat. Fields 117 (2000) 1-16. CrossRef
H.J. Kushner, Numerical Methods for Approximations in Stochastic Control Problems in Continuous Time. Springer-Verlag, New York (1992).
Lions, P.-L., Existence results for first-order Hamilton-Jacobi equations. Ricerche Mat. 32 (1983) 3-23.
Lions, P.-L., Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part I: The dynamic programming principle and applications. Comm. Partial Differential Equations 8 (1983) 1101-1174. CrossRef
Lions, P.-L., Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part II: Viscosity solutions and uniqueness. Comm. Partial Differential Equations 8 (1983) 1229-1276. CrossRef
P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part III, in Nonlinear Partial Differential Equations and Appl., Séminaire du Collège de France, Vol. V, Pitman, Ed., Boston, London (1985).
Lions, P.-L. and Mercier, B., Approximation numérique des équations de Hamilton-Jacobi-Bellman. RAIRO Anal. Numér. 14 (1980) 369-393.
Menaldi, J.L., Some estimates for finite difference approximations. SIAM J. Control Optim. 27 (1989) 579-607. CrossRef
Souganidis, P.E., Approximation schemes for viscosity solutions of Hamilton-Jacobi equations. J. Differential Equations 59 (1985) 1-43. CrossRef