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

Abstract

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 ¿+¿x(¿) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> or ¿¿+¿x(¿) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> , depending on the choice of the approximation, where ¿ and ¿x are, respectively, the time and space discretization steps. We give numerical examples in dimension 2.