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We introduce a max-plus analogue of the Petrov-Galerkin finite element methodto solve finite horizon deterministic optimal control problems. The methodrelies on a max-plus variational formulation. We show that the error in the supnorm can be bounded from the difference between the value function and itsprojections on max-plus and min-plus semimodules, when the max-plus analogue ofthe stiffness matrix is exactly known. In general, the stiffness matrix must beapproximated: this requires approximating the operation of the Lax-Oleiniksemigroup on finite elements. We consider two approximations relying on theHamiltonian. We derive a convergence result, in arbitrary dimension, showingthat for a class of problems, the error estimate is of order $\delta+\Deltax(\delta)^{-1}$ or $\sqrt{\delta}+\Delta x(\delta)^{-1}$, depending on thechoice of the approximation, where $\delta$ and $\Delta x$ are respectively thetime and space discretization steps. We compare our method with anothermax-plus based discretization method previously introduced by Fleming andMcEneaney. We give numerical examples in dimension 1 and 2.Comment: 31 pages, 11 figure

The Max-Plus Finite Element Method for Solving Deterministic Optimal Control Problems: Basic Properties and Convergence Analysis