A semi-Lagrangian scheme for Hamilton–Jacobi–Bellman equations with oblique derivatives boundary conditions

Elisa Calzola; Elisabetta Carlini; Xavier Dupuis; Francisco J. Silva

doi:10.1007/s00211-022-01336-6

A semi-Lagrangian scheme for Hamilton–Jacobi–Bellman equations with oblique derivatives boundary conditions

Elisa Calzola, Elisabetta Carlini, Xavier Dupuis, Francisco J. Silva

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4 Scopus citations

Abstract

We investigate in this work a fully-discrete semi-Lagrangian approximation of second order possibly degenerate Hamilton–Jacobi–Bellman (HJB) equations on a bounded domain O⊂ RN (N= 1 , 2 , 3) with oblique derivatives boundary conditions. These equations appear naturally in the study of optimal control of diffusion processes with oblique reflection at the boundary of the domain. The proposed scheme is shown to satisfy a consistency type property, it is monotone and stable. Our main result is the convergence of the numerical solution towards the unique viscosity solution of the HJB equation. The convergence result holds under the same asymptotic relation between the time and space discretization steps as in the classical setting for semi-Lagrangian schemes on O= RN. We present some numerical results, in dimensions N=1,2, on unstructured meshes, that confirm the numerical convergence of the scheme.

Original language	English (US)
Pages (from-to)	49-84
Number of pages	36
Journal	Numerische Mathematik
Volume	153
Issue number	1
DOIs	https://doi.org/10.1007/s00211-022-01336-6
State	Published - Dec 24 2022
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2023-03-03
Acknowledged KAUST grant number(s): OSR-2017-CRG6-3452.04
Acknowledgements: The first two authors would like to thank the Italian Ministry of Instruction, University and Research (MIUR) for supporting this research with funds coming from the PRIN Project 2017 (2017KKJP4X entitled “Innovative numerical methods for evolutionary partial differential equations and applications”). Xavier Dupuis thanks the support by the EIPHI Graduate School (Contract ANR-17-EURE-0002). Elisa Calzola, Elisabetta Carlini and Francisco J. Silva were partially supported by KAUST through the subaward agreement OSR-2017-CRG6-3452.04.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1007/s00211-022-01336-6

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title = "A semi-Lagrangian scheme for Hamilton–Jacobi–Bellman equations with oblique derivatives boundary conditions",

abstract = "We investigate in this work a fully-discrete semi-Lagrangian approximation of second order possibly degenerate Hamilton–Jacobi–Bellman (HJB) equations on a bounded domain O⊂ RN (N= 1 , 2 , 3) with oblique derivatives boundary conditions. These equations appear naturally in the study of optimal control of diffusion processes with oblique reflection at the boundary of the domain. The proposed scheme is shown to satisfy a consistency type property, it is monotone and stable. Our main result is the convergence of the numerical solution towards the unique viscosity solution of the HJB equation. The convergence result holds under the same asymptotic relation between the time and space discretization steps as in the classical setting for semi-Lagrangian schemes on O= RN. We present some numerical results, in dimensions N=1,2, on unstructured meshes, that confirm the numerical convergence of the scheme.",

author = "Elisa Calzola and Elisabetta Carlini and Xavier Dupuis and Silva, {Francisco J.}",

note = "KAUST Repository Item: Exported on 2023-03-03 Acknowledged KAUST grant number(s): OSR-2017-CRG6-3452.04 Acknowledgements: The first two authors would like to thank the Italian Ministry of Instruction, University and Research (MIUR) for supporting this research with funds coming from the PRIN Project 2017 (2017KKJP4X entitled “Innovative numerical methods for evolutionary partial differential equations and applications”). Xavier Dupuis thanks the support by the EIPHI Graduate School (Contract ANR-17-EURE-0002). Elisa Calzola, Elisabetta Carlini and Francisco J. Silva were partially supported by KAUST through the subaward agreement OSR-2017-CRG6-3452.04. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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T1 - A semi-Lagrangian scheme for Hamilton–Jacobi–Bellman equations with oblique derivatives boundary conditions

AU - Calzola, Elisa

AU - Carlini, Elisabetta

AU - Dupuis, Xavier

AU - Silva, Francisco J.

N1 - KAUST Repository Item: Exported on 2023-03-03 Acknowledged KAUST grant number(s): OSR-2017-CRG6-3452.04 Acknowledgements: The first two authors would like to thank the Italian Ministry of Instruction, University and Research (MIUR) for supporting this research with funds coming from the PRIN Project 2017 (2017KKJP4X entitled “Innovative numerical methods for evolutionary partial differential equations and applications”). Xavier Dupuis thanks the support by the EIPHI Graduate School (Contract ANR-17-EURE-0002). Elisa Calzola, Elisabetta Carlini and Francisco J. Silva were partially supported by KAUST through the subaward agreement OSR-2017-CRG6-3452.04. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2022/12/24

Y1 - 2022/12/24

N2 - We investigate in this work a fully-discrete semi-Lagrangian approximation of second order possibly degenerate Hamilton–Jacobi–Bellman (HJB) equations on a bounded domain O⊂ RN (N= 1 , 2 , 3) with oblique derivatives boundary conditions. These equations appear naturally in the study of optimal control of diffusion processes with oblique reflection at the boundary of the domain. The proposed scheme is shown to satisfy a consistency type property, it is monotone and stable. Our main result is the convergence of the numerical solution towards the unique viscosity solution of the HJB equation. The convergence result holds under the same asymptotic relation between the time and space discretization steps as in the classical setting for semi-Lagrangian schemes on O= RN. We present some numerical results, in dimensions N=1,2, on unstructured meshes, that confirm the numerical convergence of the scheme.

AB - We investigate in this work a fully-discrete semi-Lagrangian approximation of second order possibly degenerate Hamilton–Jacobi–Bellman (HJB) equations on a bounded domain O⊂ RN (N= 1 , 2 , 3) with oblique derivatives boundary conditions. These equations appear naturally in the study of optimal control of diffusion processes with oblique reflection at the boundary of the domain. The proposed scheme is shown to satisfy a consistency type property, it is monotone and stable. Our main result is the convergence of the numerical solution towards the unique viscosity solution of the HJB equation. The convergence result holds under the same asymptotic relation between the time and space discretization steps as in the classical setting for semi-Lagrangian schemes on O= RN. We present some numerical results, in dimensions N=1,2, on unstructured meshes, that confirm the numerical convergence of the scheme.

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A semi-Lagrangian scheme for Hamilton–Jacobi–Bellman equations with oblique derivatives boundary conditions

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