Viscosity solutions for the dynamic programming equations
type: Item
instance of: scholarly article
title: Viscosity solutions for the dynamic programming equations
zbMATH Open document ID: 0760.49017
DOI: 10.1007/BF01182476
published in: Applied Mathematics and Optimization
publication date: 1992-07-22T00:00:00Z
review text: This paper extends previous work on Hamilton-Jacobi equations of the form \(u+H(x,u,Du)=0\), \(u_ t+H(t,x,u,Du)=0\), by adding an unbounded nonlinear term \(\langle Ax,Du\rangle\), so that the above equations become \(u+\langle Ax,Du\rangle+H(x,u,Du)=0\) and \(u_ t+\langle Ax,Du\rangle+H(t,x,u,Du)=0\). Then it is shown that the minimum time problem for an evolution equation of the form \(\dot x+Ax=Bu\), \(x(0)=y\), \(x(\tau)=0\), is a viscosity solution (with respect to \(y\)) if it is continuous in \(y\). Moreover, with a special unbounded nonlinear term, it is shown that the Neumann problem in convex domains may be interpreted as a Hamilton-Jacobi equation with this nonlinear term.
Mathematics Subject Classification ID: 49L20
Mathematics Subject Classification ID: 49L99
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 49J15
Mathematics Subject Classification ID: 35F20
zbMATH DE Number: 39509
zbMATH Keywords: Hamilton-Jacobi equations
zbMATH Keywords: minimum time problem
zbMATH Keywords: evolution equation
zbMATH Keywords: Neumann problem in convex domains
author: Daniel Tataru
reviewed by: S. P. Banks
MaRDI profile type: Publication
cites work: Q4094029
cites work: The Dynamic Programming Equation for the Time-Optimal Control Problem in Infinite Dimensions
cites work: Some results on non-linear optimal control problems and Hamilton-Jacobi equations in infinite dimensions
cites work: Viscosity Solutions of Hamilton-Jacobi Equations
cites work: Hamilton-Jacobi equations in infinite dimensions. I: Uniqueness of viscosity solutions
cites work: Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. IV: Hamiltonians with unbounded linear terms
cites work: Neumann type boundary conditions for Hamilton-Jacobi equations
cites work: Q3138385
description: scientific article; zbMATH DE number 39509