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The maxplus finite element method for solving deterministic optimal control problems: basic properties and convergence analysis, in "SIAM J. Control and Opt.", to appear
, 2007
"... Abstract. We introduce a maxplus analogue of the PetrovGalerkin finite element method to solve finite horizon deterministic optimal control problems. The method relies on a maxplus variational formulation. We show that the error in the sup norm can be bounded from the difference between the value ..."
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Cited by 25 (4 self)
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Abstract. We introduce a maxplus analogue of the PetrovGalerkin finite element method to solve finite horizon deterministic optimal control problems. The method relies on a maxplus 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 maxplus and minplus semimodules, when the maxplus analogue of the stiffness matrix is exactly known. In general, the stiffness matrix must be approximated: this requires approximating the operation of the LaxOleinik 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 δ + ∆x(δ) −1 or √ δ + ∆x(δ) −1, depending on the choice of the approximation, where δ and ∆x are respectively the time and space discretization steps. We compare our method with another maxplus based discretization method previously introduced by Fleming and McEneaney. We give numerical examples in dimension 1 and 2.
A CurseofDimensionalityFree Numerical Method for a Class of HJB PDEs
 Proc. 16th IFAC World Congress
, 2005
"... Abstract: Maxplus methods have been explored for solution of firstorder, nonlinear HamiltonJacobiBellman partial differential equations (HJB PDEs) and corresponding nonlinear control problems. These methods exploit the maxplus linearity of the associated semigroups. Although these methods pro ..."
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Cited by 17 (6 self)
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Abstract: Maxplus methods have been explored for solution of firstorder, nonlinear HamiltonJacobiBellman partial differential equations (HJB PDEs) and corresponding nonlinear control problems. These methods exploit the maxplus linearity of the associated semigroups. Although these methods provide advantages, they still suffer from the curseofdimensionality. Here we consider HJB PDEs where the Hamiltonian takes the form of a (pointwise) maximum of linear/quadratic forms. We obtain a numerical method not subject to the curseofdimensionality. The method is based on construction of the dualspace semigroup corresponding to the HJB PDE. This dualspace semigroup is constructed from the dualspace semigroups corresponding to the constituent Hamiltonians. Copyright 2005 IFAC.
CurseofComplexity Attenuation in the CurseofDimensionalityFree Method for HJB PDEs
, 2008
"... Recently, a curseofdimensionalityfree method was developed for solution of HamiltonJacobiBellman partial differential equations (HJB PDEs) for nonlinear control problems, using semiconvex duality and maxplus analysis. The curseofdimensionalityfree method may be applied to HJB PDEs where t ..."
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Cited by 15 (6 self)
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Recently, a curseofdimensionalityfree method was developed for solution of HamiltonJacobiBellman partial differential equations (HJB PDEs) for nonlinear control problems, using semiconvex duality and maxplus analysis. The curseofdimensionalityfree method may be applied to HJB PDEs where the Hamiltonian is given as (or wellapproximated by) a pointwise maximum of quadratic forms. Such HJB PDEs also arise in certain switched linear systems. The method constructs the correct solution of an HJB PDE from a maxplus linear combination of quadratics. The method completely avoids the curseofdimensionality, and is subject to cubic computational growth as a function of space dimension. However, it is subject to a curseofcomplexity. In particular, the number of quadratics in the approximation grows exponentially with the number of iterations. Efficacy of such a method depends on the pruning of quadratics to keep the complexity growth at a reasonable level. Here we apply a pruning algorithm based on semidefinite programming. Computational speeds are exceptional, with an example HJB PDE in sixdimensional Euclidean space solved to the indicated quality in approximately 30 minutes on a typical desktop machine.
Complexity Reduction, Cornices and Pruning
 Proc. of the International Conference on Tropical and Idempotent Mathematics, G.L. Litvinov and S.N. Sergeev (Eds.), AMS
"... Abstract. In maxplus based algorithms for curseofdimensionalityfree solution of HamiltonJacobiBellman partial differential equations, and in sensor tasking algorithms, one is faced with a certain computationalcomplexity growth that must be attenuated. At each step of these algorithms, the s ..."
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Cited by 4 (3 self)
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Abstract. In maxplus based algorithms for curseofdimensionalityfree solution of HamiltonJacobiBellman partial differential equations, and in sensor tasking algorithms, one is faced with a certain computationalcomplexity growth that must be attenuated. At each step of these algorithms, the solutions are represented as maxplus (or minplus) sums of simple functions. Our problem is: Given an approximate solution representation as a maxplus sum of M functions, find the best approximation as a maxplus sum of N functions (with N < M). The main result of the paper is that for certain classes of problems, the optimal reducedcomplexity representation is comprised of a subset of the original set of functions. 1.
Distributed Dynamic Programming for DiscreteTime Stochastic Control, and Idempotent Algorithms
"... Previously, idempotent methods have been found to be extremely fast for solution of dynamic programming equations associated with deterministic control problems. The original methods exploited the idempotent (e.g., maxplus) linearity of the associated semigroup operator. However, it is now known t ..."
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Cited by 3 (2 self)
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Previously, idempotent methods have been found to be extremely fast for solution of dynamic programming equations associated with deterministic control problems. The original methods exploited the idempotent (e.g., maxplus) linearity of the associated semigroup operator. However, it is now known that the curseofdimensionalityfree idempotent methods do not require this linearity. Instead, it is sufficient that certain solution forms are retained through application of the associated semigroup operator. Here, we see that idempotent methods may be used to solve some classes of stochastic control problems. The key is the use of the idempotent distributive property. This allows one to apply the curseofdimensionalityfree idempotent approach. We demonstrate this approach for a class of nonlinear, discretetime stochastic control problems. 1
Idempotent Method for Dynamic Games and Complexity Reduction in MinMax Expansions
"... Abstract — In recent years, idempotent methods (specifically, maxplus methods) have been developed for solution of nonlinear control problems. It was thought that idempotent linearity of the associated semigroup was required for application of these techniques. It is now known that application of ..."
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Cited by 1 (1 self)
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Abstract — In recent years, idempotent methods (specifically, maxplus methods) have been developed for solution of nonlinear control problems. It was thought that idempotent linearity of the associated semigroup was required for application of these techniques. It is now known that application of the maxplus distributive property allows one to apply the maxplus curseofdimensionalityfree approach to stochastic control problems. Here, we see that a similar, albeit more abstract, approach can be applied to deterministic game problems. The main difficulty is a curseofcomplexity growth in the computational cost. Attenuation of this effect requires finding reducedcomplexity approximations to minmax sums of maxplus affine functions. We demonstrate that that problem can be reduced to a pruning problem. I.
Nonlinear PerronFrobenius theory and discrete event systems
"... ABSTRACT. We show how methods from nonlinear spectral theory can be used to analyse the time behaviour of dynamical discrete event systems. RÉSUMÉ. Nous montrons comment analyser le comportement temporel des systèmes à événements discrets à l’aide de résultats de théorie spectrale nonlinéaire. ..."
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ABSTRACT. We show how methods from nonlinear spectral theory can be used to analyse the time behaviour of dynamical discrete event systems. RÉSUMÉ. Nous montrons comment analyser le comportement temporel des systèmes à événements discrets à l’aide de résultats de théorie spectrale nonlinéaire.
unknown title
, 2006
"... Algèbres maxplus et mathématiques de la décision/Maxplus algebras and mathematics of decision ..."
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Algèbres maxplus et mathématiques de la décision/Maxplus algebras and mathematics of decision
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"... Algèbres maxplus et mathématiques de la décision/Maxplus algebras and mathematics of decision ..."
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Algèbres maxplus et mathématiques de la décision/Maxplus algebras and mathematics of decision