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A max-plus finite element method for solving finite horizon deterministic optimal control problems

Marianne Akian, Stephane Gaubert, Asma Lakhoua

TL;DR

A max-plus analogue of the Petrov-Galerkin finite element method, to solve finite horizon deterministic optimal control problems, and obtains a nonlinear discretized semigroup corresponding to a zero-sum two players game.

Abstract

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 $(Δt)^{1/2}+Δx(Δt)^{-1}$, 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.

A max-plus finite element method for solving finite horizon deterministic optimal control problems

TL;DR

A max-plus analogue of the Petrov-Galerkin finite element method, to solve finite horizon deterministic optimal control problems, and obtains a nonlinear discretized semigroup corresponding to a zero-sum two players game.

Abstract

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.

Paper Structure

This paper contains 11 sections, 7 theorems, 44 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on residuation and projections over semimodules
  3. Residuation, semimodules, and linear maps
  4. Projectors on semimodules
  5. The max-plus finite element method
  6. Max-plus variational formulation
  7. Ideal max-plus finite element method
  8. Effective max-plus finite element method
  9. Comparison with the method of Fleming and McEneaney
  10. Error analysis
  11. Numerical results

Key Result

Theorem 1

Let $B:\mathcal{U}\to\mathcal{X}$ and $C:\mathcal{X}\to\mathcal{Y}$ be two residuated linear operators. Let $\Pi_{B}^{C}=B\circ(C\circ B)^{\sharp}\circ C$. We have $\Pi_{B}^{C}=\Pi_{B}^{}\circ \Pi_{}^{C}$, where $\Pi_{B}^{}=B\circ B^{\sharp}$ and $\Pi_{}^{C}=C^{\sharp}\circ C$. Moreover, $\Pi_{B}^{C

Figures (5)

  • Figure 1: Max-plus approximation of a linear quadratic control problem (Example \ref{['ex-lq']})
  • Figure 2: A bad choice of test functions for the distance problem (Example \ref{['ex-dist']})
  • Figure 3: A good choice of test functions for the distance problem (Example \ref{['ex-dist']})
  • Figure 4: Value function and its max-plus approximation (Example \ref{['ex-falcone1']})
  • Figure 5: Value function and its max-plus approximation (Example \ref{['ex-falcone2']})

Theorems & Definitions (15)

  • Theorem 1: Projection on an image parallel to a kernel
  • Theorem 1: Projection on an image parallel to a kernel
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof : Sketch of proof
  • Lemma 5
  • Definition 6: Lipschitz finite elements
  • Definition 7: Quadratic finite elements
  • ...and 5 more