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# The max-plus finite element method for solving deterministic optimal control problems: basic properties and convergence analysis

Abstract : We introduce a max-plus analogue of the Petrov-Galerkin finite element method to solve finite horizon deterministic optimal control problems. The method relies on a max-plus variational formulation. We show that the error in the sup norm can be bounded from the difference between the value function and its projections on max-plus and min-plus semimodules, when the max-plus analogue of the stiffness matrix is exactly known. In general, the stiffness matrix must be approximated: this requires approximating the operation of the Lax-Oleinik semigroup on finite elements. We consider two approximations relying on the Hamiltonian. We derive a convergence result, in arbitrary dimension, showing that for a class of problems, the error estimate is of order $\delta+\Dtax(\delta)^-1$ or $\sqrt\delta+\Dtax(\delta)^-1$, depending on the choice of the approximation, where $\delta$ and $\Dtax$ are respectively the time and space discretization steps. We compare our method with another max-plus based discretization method previously introduced by Fleming and McEneaney. We give numerical examples in dimension 1 and 2.
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https://hal.inria.fr/inria-00071395
Contributor : Rapport de Recherche Inria <>
Submitted on : Tuesday, May 23, 2006 - 5:10:58 PM
Last modification on : Friday, May 25, 2018 - 12:02:03 PM
Document(s) archivé(s) le : Sunday, April 4, 2010 - 10:10:15 PM

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• HAL Id : inria-00071395, version 1

### Citation

Marianne Akian, Stéphane Gaubert, Asma Lakhoua. The max-plus finite element method for solving deterministic optimal control problems: basic properties and convergence analysis. [Research Report] RR-5874, INRIA. 2006. ⟨inria-00071395⟩

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