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

Guy Barles; Espen Robstad Jakobsen

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 36, Issue: 1, page 33-54
- ISSN: 0764-583X

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topBarles, Guy, and Jakobsen, Espen Robstad. "On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman Equations." ESAIM: Mathematical Modelling and Numerical Analysis 36.1 (2010): 33-54. <http://eudml.org/doc/194095>.

@article{Barles2010,

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.
},

author = {Barles, Guy, Jakobsen, Espen Robstad},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Hamilton-Jacobi-Bellman equation; viscosity solution;
approximation schemes; finite difference methods; convergence rate.; approximation; finite difference method; convergence; monotone approximation schemes; dynamic programming},

language = {eng},

month = {3},

number = {1},

pages = {33-54},

publisher = {EDP Sciences},

title = {On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman Equations},

url = {http://eudml.org/doc/194095},

volume = {36},

year = {2010},

}

TY - JOUR

AU - Barles, Guy

AU - Jakobsen, Espen Robstad

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 36

IS - 1

SP - 33

EP - 54

AB -
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.

LA - eng

KW - Hamilton-Jacobi-Bellman equation; viscosity solution;
approximation schemes; finite difference methods; convergence rate.; approximation; finite difference method; convergence; monotone approximation schemes; dynamic programming

UR - http://eudml.org/doc/194095

ER -

## References

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- H.J. Kushner, Numerical Methods for Approximations in Stochastic Control Problems in Continuous Time. Springer-Verlag, New York (1992).
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- P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part I: The dynamic programming principle and applications. Comm. Partial Differential Equations8 (1983) 1101-1174.
- P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part II: Viscosity solutions and uniqueness. Comm. Partial Differential Equations8 (1983) 1229-1276.
- 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).
- P.-L. Lions and B. Mercier, Approximation numérique des équations de Hamilton-Jacobi-Bellman. RAIRO Anal. Numér.14 (1980) 369-393.
- J.L. Menaldi, Some estimates for finite difference approximations. SIAM J. Control Optim.27 (1989) 579-607.
- P.E. Souganidis, Approximation schemes for viscosity solutions of Hamilton-Jacobi equations. J. Differential Equations59 (1985) 1-43.

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