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A max-plus finite element method for solving finite horizon deterministic optimal control problems

Marianne Akian·Stephane Gaubert·Asma Lakhoua
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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, and exploits the properties of projectors on max-plus semimodules. We obtain a nonlinear discretized semigroup, corresponding to a zero-sum two players game. We give an error estimate of order $(\Delta t)^{1/2}+\Delta x(\Delta t)^{-1}$, for a subclass of problems in dimension 1. We compare our method with a max-plus based discretization method previously introduced by Fleming and McEneaney.

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Marianne Akian, Stephane Gaubert, Asma Lakhoua. 2004-04-08. A max-plus finite element method for solving finite horizon deterministic optimal control problems. https://arxiv.org/abs/math/0404184

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