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A max-plus finite element method for solving finite horizon deterministic optimal control problems
TLDR
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 $\sqrt{\Dta t}+\Dta x(\Dta t)^{-1}$, 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.read more
Citations
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Journal ArticleDOI
A Curse-of-Dimensionality-Free Numerical Method for Solution of Certain HJB PDEs
TL;DR: This work considers HJB PDEs in which the Hamiltonian takes the form of a (pointwise) maximum of linear/quadratic forms, and obtains a numerical method not subject to the curse of dimensionality.
Journal ArticleDOI
The Max-Plus Finite Element Method for Solving Deterministic Optimal Control Problems: Basic Properties and Convergence Analysis
TL;DR: A max-plus analogue of the Petrov-Galerkin finite element method to solve finite horizon deterministic optimal control problems and derives 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.
Proceedings ArticleDOI
Curse-of-complexity attenuation in the curse-of-dimensionality-free method for HJB PDEs
TL;DR: A pruning algorithm based on semidefinite programming is applied to solution of Hamilton-Jacobi-Bellman partial differential equations (HJB PDEs) for nonlinear control problems, using semiconvex duality and max-plus analysis to avoid the curse-of-dimensionality.
Journal ArticleDOI
A new fundamental solution for differential Riccati equations arising in control
TL;DR: In this article, the authors considered the matrix differential Riccati equation (DRE) as a finite-dimensional solution to a deterministic linear/quadratic control problem and proposed a semiconvex dual of the associated semigroup.
Journal ArticleDOI
Convergence Rate for a Curse-of-Dimensionality-Free Method for a Class of HJB PDEs
TL;DR: This work obtains specific error bounds for a previously obtained numerical method not subject to the curse-of-dimensionality of HJB PDEs where the Hamiltonian takes the form of a (pointwise) maximum of linear/quadratic forms.
References
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Book
Controlled Markov processes and viscosity solutions
Wendell H. Fleming,H. Mete Soner +1 more
TL;DR: In this paper, an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions is given, as well as a concise introduction to two-controller, zero-sum differential games.
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Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations
TL;DR: In this paper, the main ideas on a model problem with continuous viscosity solutions of Hamilton-Jacobi equations are discussed. But the main idea of the main solutions is not discussed.
Journal ArticleDOI
Synchronization and Linearity: An Algebra for Discrete Event Systems
TL;DR: This book proposes a unified mathematical treatment of a class of 'linear' discrete event systems, which contains important subclasses of Petri nets and queuing networks with synchronization constraints, which is shown to parallel the classical linear system theory in several ways.
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Solutions de viscosité des équations de Hamilton-Jacobi
TL;DR: In this paper, the authors present a panorama assez complet for solving the equations of Hamilton-Jacobi du premier ordre and their applications in controle optimal deterministe and perturbations singulieres, en particulier du type Grandes deviations.
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Idempotent Analysis and Its Applications
TL;DR: In this article, a generalized solution of Bellman's Differential Equation and multiplicative additive asymptotics is presented, which is based on the Maslov Optimziation Theory.