Uniqueness to elliptic and parabolic Hamilton–Jacobi–Bellman equations with non-smooth boundary

Sébastien Chaumont

doi:10.1016/j.crma.2004.08.009

Partial Differential Equations

Uniqueness to elliptic and parabolic Hamilton–Jacobi–Bellman equations with non-smooth boundary
[Unicité aux équations d'Hamilton–Jacobi–Bellman elliptiques et paraboliques avec frontière irrégulière.]

Présenté par : Pierre-Louis Lions

Sébastien Chaumont ¹

¹ Institut Élie Cartan, université Henri Poincaré Nancy I, B.P. 239, 54506 Vandœuvre-lès-Nancy cedex, France

Comptes Rendus. Mathématique, Volume 339 (2004) no. 8, pp. 555-560.

Résumé
Abstract

Dans le cadre de la théorie des solutions de viscosité, on donne une extension du principe de comparaison fort pour l'équation d'Hamilton–Jacobi–Bellman (HJB) avec condition au bord de type Dirichlet au cas de certains domaines irréguliers. En particulier, ce résultat est applicable aux problèmes paraboliques posés dans des domaines cylindriques.

In the framework of viscosity solutions, we give an extension of the strong comparison result for Hamilton–Jacobi–Bellman (HJB) equations with Dirichlet boundary conditions to the case of some non-smooth domains. In particular, it may be applied to parabolic problems on cylindrical domains.

Reçu le : 2004-04-24
Accepté le : 2004-08-28
Publié le : 2004-10-01

DOI : 10.1016/j.crma.2004.08.009

Affiliations des auteurs :

Sébastien Chaumont ¹

¹ Institut Élie Cartan, université Henri Poincaré Nancy I, B.P. 239, 54506 Vandœuvre-lès-Nancy cedex, France

@article{CRMATH_2004__339_8_555_0,
     author = {S\'ebastien Chaumont},
     title = {Uniqueness to elliptic and parabolic {Hamilton{\textendash}Jacobi{\textendash}Bellman} equations with non-smooth boundary},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {555--560},
     publisher = {Elsevier},
     volume = {339},
     number = {8},
     year = {2004},
     doi = {10.1016/j.crma.2004.08.009},
     language = {en},
}

TY  - JOUR
AU  - Sébastien Chaumont
TI  - Uniqueness to elliptic and parabolic Hamilton–Jacobi–Bellman equations with non-smooth boundary
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 555
EP  - 560
VL  - 339
IS  - 8
PB  - Elsevier
DO  - 10.1016/j.crma.2004.08.009
LA  - en
ID  - CRMATH_2004__339_8_555_0
ER  -

%0 Journal Article
%A Sébastien Chaumont
%T Uniqueness to elliptic and parabolic Hamilton–Jacobi–Bellman equations with non-smooth boundary
%J Comptes Rendus. Mathématique
%D 2004
%P 555-560
%V 339
%N 8
%I Elsevier
%R 10.1016/j.crma.2004.08.009
%G en
%F CRMATH_2004__339_8_555_0

Sébastien Chaumont. Uniqueness to elliptic and parabolic Hamilton–Jacobi–Bellman equations with non-smooth boundary. Comptes Rendus. Mathématique, Volume 339 (2004) no. 8, pp. 555-560. doi : 10.1016/j.crma.2004.08.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.08.009/

Bibliographie
Cité par

[1] G. Barles Solutions de Viscosité des Equations de Hamilton–Jacobi, Math. Appl., vol. 17, Springer-Verlag, 1994

[2] G. Barles Nonlinear Neumann boundary conditions for quasilinear elliptic equations and applications, J. Differential Equations, Volume 154 (1999) no. 1, pp. 191-224

[3] G. Barles; R. Buckdahn; E. Pardoux Backward stochastic differential equations and integral-partial differential equations, Stochastics Stochastics Rep., Volume 60 (1997), pp. 57-83

[4] G. Barles; J. Burdeau The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit time control problems, Comm. Partial Differential Equations, Volume 20 (1995) no. 1/2, pp. 129-178

[5] G. Barles; E. Rouy A strong comparison result for the Bellman equation arising in stochastic exit time control problems and its applications, Comm. Partial Differential Equations, Volume 23 (1998) no. 11/12, pp. 1945-2033

[6] G. Barles; P.E. Souganidis Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., vol. 4, Elsevier, North-Holland, 1991, pp. 273-283

[7] M.G. Crandall; H. Ishii; P.-L. Lions User's guide to viscosity solutions of 2nd order PDE, Bull. Amer. Math. Soc., Volume 27 (1992) no. 1, pp. 1-67

[8] M.A. Katsoulakis Viscosity solutions of second order fully nonlinear elliptic equations with state constraints, Indiana Univ. Math. J., Volume 43 (1994) no. 2, pp. 493-517

[9] N.V. Krylov Controlled Diffusion Processes, Springer-Verlag, Berlin, 1980

[10] P.-L. Lions Optimal control of diffusion processes and HJB equations part II: viscosity solutions and uniqueness, Comm. Partial Differential Equations, Volume 8 (1983), pp. 1229-1276

Christoph Reisinger; Yufei Zhang Regularity and Stability of Feedback Relaxed Controls, SIAM Journal on Control and Optimization, Volume 59 (2021) no. 5, p. 3118 | DOI:10.1137/20m1312435
Athena Picarelli; Christoph Reisinger; Julen Rotaetxe Arto Some regularity and convergence results for parabolic Hamilton-Jacobi-Bellman equations in bounded domains, Journal of Differential Equations, Volume 268 (2020) no. 12, p. 7843 | DOI:10.1016/j.jde.2019.11.081
Dmitry B. Rokhlin; Georgii Mironenko Regular finite fuel stochastic control problems with exit time, Mathematical Methods of Operations Research, Volume 84 (2016) no. 1, p. 105 | DOI:10.1007/s00186-016-0536-2
Dmitry B. Rokhlin Verification by Stochastic Perron's Method in stochastic exit time control problems, Journal of Mathematical Analysis and Applications, Volume 419 (2014) no. 1, p. 433 | DOI:10.1016/j.jmaa.2014.04.062
J. H. Witte; C. Reisinger A Penalty Method for the Numerical Solution of Hamilton–Jacobi–Bellman (HJB) Equations in Finance, SIAM Journal on Numerical Analysis, Volume 49 (2011) no. 1, p. 213 | DOI:10.1137/100797606
Espen R. Jakobsen Monotone Schemes, Encyclopedia of Quantitative Finance (2010) | DOI:10.1002/9780470061602.eqf12010

Cité par 6 documents. Sources : Crossref

Commentaires - Politique