6 research outputs found
The max-plus finite element method for optimal control problems: further approximation results
We develop the max-plus finite element method to solve finite horizon
deterministic optimal control problems. This method, that we introduced in a
previous work, relies on a max-plus variational formulation, and exploits the
properties of projectors on max-plus semimodules. We prove here a convergence
result, in arbitrary dimension, showing that for a subclass of problems, the
error estimate is of order , where and
are the time and space steps respectively. We also show how the
max-plus analogues of the mass and stiffness matrices can be computed by convex
optimization, even when the global problem is non convex. We illustrate the
method by numerical examples in dimension 2.Comment: 13 pages, 2 figure
A max-plus finite element method for solving finite horizon deterministic optimal control problems
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 , 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.Comment: 13 pages, 5 figure
The max-plus finite element method for solving deterministic optimal control problems: basic properties and convergence analysis
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 or , depending on the
choice of the approximation, where and 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.Comment: 31 pages, 11 figure
Méthode des éléments finis max-plus pour la résolution numérique de problèmes de commande optimale déterministe
PARIS-BIUSJ-Thèses (751052125) / SudocPARIS-BIUSJ-Physique recherche (751052113) / SudocSudocFranceF
© 2010 Academic Journals Full Length Research Paper
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