Journal
IEEE TRANSACTIONS ON AUTOMATIC CONTROL
Volume 55, Issue 5, Pages 1158-1174Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TAC.2010.2045439
Keywords
Hamilton-Jacobi (HJ); initial conditions (ICs); Lax-Hopf formula; partial differential equation (PDE); piecewise affine (PWA); terminal conditions (TCs)
Funding
- VIMADES
- Direct For Computer & Info Scie & Enginr [0845076, GRANTS:13642843] Funding Source: National Science Foundation
- Division Of Computer and Network Systems [GRANTS:13642843, 0845076] Funding Source: National Science Foundation
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This article presents a new method for explicitly computing solutions to a Hamilton-Jacobi partial differential equation for which initial, boundary and internal conditions are prescribed as piecewise affine functions. Based on viability theory, a Lax-Hopf 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 Lax-Hopf 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.
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