Hamilton-Jacobi-Bellman Equations

Numerical Methods and Applications in Optimal Control

Dante Kalise, Karl Kunisch, Zhiping Rao, Dante Kalise, Karl Kunisch, Zhiping Rao

• Publisher: De Gruyter
• Year: 2018
• Print ISBN: 9783110542639
• eBook ISBN: 9783110542714
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Optimal feedback control arises in different areas such as aerospace engineering, chemical processing, resource economics, etc. In this context, the application of dynamic programming techniques leads to the solution of fully nonlinear Hamilton-Jacobi-Bellman equations. This book presents the state of the art in the numerical approximation of Hamilton-Jacobi-Bellman equations, including post-processing of Galerkin methods, high-order methods, boundary treatment in semi-Lagrangian schemes, reduced basis methods, comparison principles for viscosity solutions, max-plus methods, and the numerical approximation of Monge-Ampère equations. This book also features applications in the simulation of adaptive controllers and the control of nonlinear delay differential equations.

Contents
From a monotone probabilistic scheme to a probabilistic max-plus algorithm for solving Hamilton–Jacobi–Bellman equations
Improving policies for Hamilton–Jacobi–Bellman equations by postprocessing
Viability approach to simulation of an adaptive controller
Galerkin approximations for the optimal control of nonlinear delay differential equations
Efficient higher order time discretization schemes for Hamilton–Jacobi–Bellman equations based on diagonally implicit symplectic Runge–Kutta methods
Numerical solution of the simple Monge–Ampere equation with nonconvex Dirichlet data on nonconvex domains
On the notion of boundary conditions in comparison principles for viscosity solutions
Boundary mesh refinement for semi-Lagrangian schemes
A reduced basis method for the Hamilton–Jacobi–Bellman equation within the European Union Emission Trading Scheme

Marianne Akian and Eric Fodjo

1From a monotone probabilistic scheme to a probabilistic max-plus algorithm for solving Hamilton–Jacobi–Bellman equations

Note: The first author was partially supported by the ANR project MALTHY, ANR-13-INSE-0003, by ICODE, and by PGMO, a joint program of EDF and FMJH (Fondation Mathématique Jacques Hadamard).

Abstract: In a previous work (Akian, Fodjo, 2016), we introduced a lower complexity probabilistic max-plus numerical method for solving fully nonlinear Hamilton–Jacobi–Bellman equations associated with diffusion control problems involving a finite set-valued (or switching) control and possibly a continuum-valued control. This method was based on the idempotent expansion properties obtained by McEneaney, Kaise, and Han (2011) and on the numerical probabilistic method proposed by Fahim, Touzi, and Warin (2011) for solving some fully nonlinear parabolic partial differential equations (PDE). A difficulty of the algorithm of Fahim, Touzi, and Warin is in the critical constraints imposed on the Hamiltonian to ensure the monotonicity of the scheme, hence the converg...

Table of contents

• Cover
• Title Page
• Copyright
• Preface
• Contents
• List of contributing authors
• 1 From a monotone probabilistic scheme to a probabilistic max-plus algorithm for solving Hamilton–Jacobi–Bellman equations
• 2 Improving policies for Hamilton–Jacobi–Bellman equations by postprocessing
• 3 Viability approach to simulation of an adaptive controller
• 4 Galerkin approximations for the optimal control of nonlinear delay differential equations
• 5 Efficient higher order time discretization schemes for Hamilton–Jacobi–Bellman equations based on diagonally implicit symplectic Runge–Kutta methods
• 6 Numerical solution of the simple Monge–Ampère equation with nonconvex Dirichlet data on nonconvex domains
• 7 On the notion of boundary conditions in comparison principles for viscosity solutions
• 8 Boundary mesh refinement for semi-Lagrangian schemes
• 9 A reduced basis method for the Hamilton–Jacobi–Bellman equation within the European Union Emission Trading Scheme
• Index
• Radon Series on Computational and Applied Mathematics