A characterization of the existence of solutions for Hamilton-Jacobi equations in ergodic control problems with applications
type: Item
instance of: scholarly article
title: A characterization of the existence of solutions for Hamilton-Jacobi equations in ergodic control problems with applications
zbMATH Open document ID: 0967.35029
published in: Applied Mathematics and Optimization
publication date: 2001-08-30T00:00:00Z
review text: In this paper is considered the first-order Hamilton-Jacobi equation \(H(x,u,Du)= 0\) in \(\Omega\), where \(\Omega\) is an open subset of a real Hilbert space \({\mathcal H}\), \(u\) is the unknown which is a real-valued function on \(\overline\Omega\), \(Du(x)\) denotes the Fréchet derivative at \(x\) which is regarded as an element of \({\mathcal H}\) via Riesz theorem, and \(H\) is a real-valued function on \(\Omega\times\mathbb{R}\times{\mathcal H}\) which is called a Hamiltonian. For this equation, the so-called state-constraint problem is considered and the existence of a viscosity solution is studied. This problem is in the category of ergodic control problems and hence, the Hamiltonian \(H(x,u,p)\) is not assumed to be strictly increasing with respect to \(u\). One of the main results characterizes the existence of a bounded viscosity solution in terms of a sort of value functions of the state-constraint problem associated with the Hamilton-Jacobi equation. One of the applications of the main results is the reexamination of an example and the other is concerned with calculation of effective Hamiltonians in periodic homogenization of Hamilton-Jacobi equations.
Mathematics Subject Classification ID: 35F20
Mathematics Subject Classification ID: 35R15
Mathematics Subject Classification ID: 49J15
Mathematics Subject Classification ID: 49L20
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 49L25
zbMATH DE Number: 1485112
zbMATH Keywords: state-constraint problem
zbMATH Keywords: effective Hamiltonians
zbMATH Keywords: periodic homogenization
author: Mariko Arisawa
author: Hitoshi Ishii
author: P.-L. Lions
reviewed by: Vasile Iftode
MaRDI profile type: Publication
full work available at URL: s002450010001
OpenAlex ID: W2054625457
DOI: 10.1007/S002450010001
description: scientific article; zbMATH DE number 1485112