LaxHopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part II: Computational methods

Christian G. Claudel; Alexandre M. Bayen

doi:10.1109/TAC.2010.2045439

LaxHopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part II: Computational methods

Christian G. Claudel, Alexandre M. Bayen

Research output: Contribution to journal › Article › peer-review

95 Scopus citations

Abstract

This article presents a new method for explicitly computing solutions to a HamiltonJacobi partial differential equation for which initial, boundary and internal conditions are prescribed as piecewise affine functions. Based on viability theory, a LaxHopf formula is used to construct analytical solutions for the individual contribution of each affine condition to the solution of the problem. The results are assembled into a LaxHopf algorithm which can be used to compute the solution to the partial differential equation at any arbitrary time at no other cost than evaluating a semi-analytical expression numerically. The method being semi-analytical, it performs at machine accuracy (compared to the discretization error inherent to finite difference schemes). The performance of the method is assessed with benchmark analytical examples. The running time of the algorithm is compared with the running time of a Godunov scheme.

Original language	English (US)
Article number	5461008
Pages (from-to)	1158-1174
Number of pages	17
Journal	IEEE Transactions on Automatic Control
Volume	55
Issue number	5
DOIs	https://doi.org/10.1109/TAC.2010.2045439
State	Published - May 2010
Externally published	Yes

Keywords

Hamilton-Jacobi (HJ)
Initial conditions (ICs)
Lax-Hopf formula
Partial differential equation (PDE)
Piecewise affine (PWA)
Terminal conditions (TCs)

ASJC Scopus subject areas

Control and Systems Engineering
Computer Science Applications
Electrical and Electronic Engineering

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10.1109/TAC.2010.2045439

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title = "LaxHopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part II: Computational methods",

abstract = "This article presents a new method for explicitly computing solutions to a HamiltonJacobi partial differential equation for which initial, boundary and internal conditions are prescribed as piecewise affine functions. Based on viability theory, a LaxHopf formula is used to construct analytical solutions for the individual contribution of each affine condition to the solution of the problem. The results are assembled into a LaxHopf algorithm which can be used to compute the solution to the partial differential equation at any arbitrary time at no other cost than evaluating a semi-analytical expression numerically. The method being semi-analytical, it performs at machine accuracy (compared to the discretization error inherent to finite difference schemes). The performance of the method is assessed with benchmark analytical examples. The running time of the algorithm is compared with the running time of a Godunov scheme.",

keywords = "Hamilton-Jacobi (HJ), Initial conditions (ICs), Lax-Hopf formula, Partial differential equation (PDE), Piecewise affine (PWA), Terminal conditions (TCs)",

author = "Claudel, {Christian G.} and Bayen, {Alexandre M.}",

note = "Funding Information: Manuscript received June 17, 2008; revised April 12, 2009. Current version published May 12, 2010. This work was supported in part by VIMADES. Recommended by Associate Editor D. Dochain.",

year = "2010",

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AU - Claudel, Christian G.

AU - Bayen, Alexandre M.

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N2 - This article presents a new method for explicitly computing solutions to a HamiltonJacobi partial differential equation for which initial, boundary and internal conditions are prescribed as piecewise affine functions. Based on viability theory, a LaxHopf formula is used to construct analytical solutions for the individual contribution of each affine condition to the solution of the problem. The results are assembled into a LaxHopf algorithm which can be used to compute the solution to the partial differential equation at any arbitrary time at no other cost than evaluating a semi-analytical expression numerically. The method being semi-analytical, it performs at machine accuracy (compared to the discretization error inherent to finite difference schemes). The performance of the method is assessed with benchmark analytical examples. The running time of the algorithm is compared with the running time of a Godunov scheme.

AB - This article presents a new method for explicitly computing solutions to a HamiltonJacobi partial differential equation for which initial, boundary and internal conditions are prescribed as piecewise affine functions. Based on viability theory, a LaxHopf formula is used to construct analytical solutions for the individual contribution of each affine condition to the solution of the problem. The results are assembled into a LaxHopf algorithm which can be used to compute the solution to the partial differential equation at any arbitrary time at no other cost than evaluating a semi-analytical expression numerically. The method being semi-analytical, it performs at machine accuracy (compared to the discretization error inherent to finite difference schemes). The performance of the method is assessed with benchmark analytical examples. The running time of the algorithm is compared with the running time of a Godunov scheme.

KW - Hamilton-Jacobi (HJ)

KW - Initial conditions (ICs)

KW - Lax-Hopf formula

KW - Partial differential equation (PDE)

KW - Piecewise affine (PWA)

KW - Terminal conditions (TCs)

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JF - IEEE Transactions on Automatic Control

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ER -

LaxHopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part II: Computational methods

Abstract

Keywords

ASJC Scopus subject areas

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