On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations
ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 1, pp. 33-54.

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.

DOI : 10.1051/m2an:2002002
Classification : 65N06, 65N15, 41A25, 49L20, 49L25
Mots clés : Hamilton-Jacobi-Bellman equation, viscosity solution, approximation schemes, finite difference methods, convergence rate
@article{M2AN_2002__36_1_33_0,
     author = {Barles, Guy and Jakobsen, Espen Robstad},
     title = {On the convergence rate of approximation schemes for {Hamilton-Jacobi-Bellman} equations},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {33--54},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {1},
     year = {2002},
     doi = {10.1051/m2an:2002002},
     mrnumber = {1916291},
     zbl = {0998.65067},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2002002/}
}
TY  - JOUR
AU  - Barles, Guy
AU  - Jakobsen, Espen Robstad
TI  - On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2002
SP  - 33
EP  - 54
VL  - 36
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2002002/
DO  - 10.1051/m2an:2002002
LA  - en
ID  - M2AN_2002__36_1_33_0
ER  - 
%0 Journal Article
%A Barles, Guy
%A Jakobsen, Espen Robstad
%T On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2002
%P 33-54
%V 36
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2002002/
%R 10.1051/m2an:2002002
%G en
%F M2AN_2002__36_1_33_0
Barles, Guy; Jakobsen, Espen Robstad. On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations. ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 1, pp. 33-54. doi : 10.1051/m2an:2002002. http://www.numdam.org/articles/10.1051/m2an:2002002/

[1] G. Barles and P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second-order equations. Asymptotic Anal. 4 (1991) 271-283. | Zbl

[2] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997). | MR | Zbl

[3] F. Bonnans and H. Zidani, Consistency of generalized finite difference schemes for the stochastic HJB equation. Preprint. | Zbl

[4] F. Camilli and M. Falcone, An approximation scheme for the optimal control of diffusion processes. RAIRO Modél. Math. Anal. Numér. 29 (1995) 97-122. | Numdam | Zbl

[5] I. Capuzzo-Dolcetta, On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming. Appl. Math. Optim. 10 (1983) 367-377. | MR | Zbl

[6] M.G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second-order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1-67. | Zbl

[7] M.G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations. Math. Comp. 43 (1984) 1-19. | Zbl

[8] W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, New York (1993). | MR | Zbl

[9] H. Ishii and P.-L Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differential Equations 83 (1990) 26-78. | Zbl

[10] 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. | Zbl

[11] N.V. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equations. St. Petersbg Math. J. 9 (1997) 639-650. | Zbl

[12] N.V. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients. Probab. Theory Relat. Fields 117 (2000) 1-16. | Zbl

[13] H.J. Kushner, Numerical Methods for Approximations in Stochastic Control Problems in Continuous Time. Springer-Verlag, New York (1992). | Zbl

[14] P.-L. Lions, Existence results for first-order Hamilton-Jacobi equations. Ricerche Mat. 32 (1983) 3-23. | Zbl

[15] P.-L. Lions, 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. | Zbl

[16] P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part II: Viscosity solutions and uniqueness. Comm. Partial Differential Equations 8 (1983) 1229-1276. | Zbl

[17] 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). | Zbl

[18] P.-L. Lions and B. Mercier, Approximation numérique des équations de Hamilton-Jacobi-Bellman. RAIRO Anal. Numér. 14 (1980) 369-393. | Numdam | Zbl

[19] J.L. Menaldi, Some estimates for finite difference approximations. SIAM J. Control Optim. 27 (1989) 579-607. | MR | Zbl

[20] P.E. Souganidis, Approximation schemes for viscosity solutions of Hamilton-Jacobi equations. J. Differential Equations 59 (1985) 1-43. | Zbl

Cité par Sources :