Convergence analysis of the Max-Plus Finite Element Method for Solving Deterministic Optimal Control Problems

We consider the Max-Plus Finite Element Method for Solving Deterministic Optimal Control Problems, which is a max-plus analogue of the Petrov-Galerkin finite element method. This method, that we introduced in a previous work, relies on a max-plus variational formulation. The error in the sup-norm can be bounded from the difference between the value function and its projections on max-plus and minplus semimodules when the max-plus analogue of the stiffness matrix is exactly known. We derive here a convergence result in arbitrary dimension for approximations of the stiffness matrix relying on the Hamiltonian, and for arbitrary discretization grid. We show that for a class of problems, the error estimate is of order δ+Delta;x(δ)^-1 or √δ+Δx(δ)^-1, depending on the choice of the approximation, where δ and Δx are, respectively, the time and space discretization steps. We give numerical examples in dimension 2.