Mathematics – Optimization and Control
Scientific paper
2006-03-27
SIAM J. Control Optim. Volume 47, Issue 2, pp. 817-848 (2008)
Mathematics
Optimization and Control
31 pages, 11 figures
Scientific paper
10.1137/060655286
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 $\delta+\Delta x(\delta)^{-1}$ or $\sqrt{\delta}+\Delta x(\delta)^{-1}$, depending on the choice of the approximation, where $\delta$ and $\Delta x$ 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.
Akian Marianne
Gaubert Stephane
Lakhoua Asma
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