The concept of value in differential games of survival and viscosity solutions of Hamilton-Jacobi equations
type: Item
instance of: scholarly article
title: The concept of value in differential games of survival and viscosity solutions of Hamilton-Jacobi equations
zbMATH Open document ID: 0758.90093
author: Pierpaolo Soravia
published in: Differential and Integral Equations
publication date: 1993-01-16T00:00:00Z
review text: The paper is concerned with a differential game with two players governed by the state equation \(y'=f\bigl(y,a(t),b(t)\bigr)\), \(y(0)=x\in{\mathbf R}^ N\), with target \({\mathcal T}\subset{\mathbf R}^ N\) and payoff \(\int_ 0^{\tau_ x}h\bigl(y(t),a(t),b(t)\bigr) dt+g\bigl(y(\tau_ x)\bigr)\), where \(\tau_ x\) is the first hitting time. The problem is studied without any restriction on \(\tau_ x\), set to \(+\infty\) if \(y\) never reachs the target. It is proved that the upper and lower value functions obtained by the Fleming and the Friedman approximation schemes converge, under suitable assumptions on \(f, h, g\) and \({\mathcal T}\) to a function satisfying the Hamilton-Jacobi-Isaacs equation.
Mathematics Subject Classification ID: 91A23
Mathematics Subject Classification ID: 49L20
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 49L99
Mathematics Subject Classification ID: 49J20
zbMATH DE Number: 90522
zbMATH Keywords: Friedman approximation scheme
zbMATH Keywords: Fleming approximation scheme
zbMATH Keywords: upper and lower value functions
zbMATH Keywords: Hamilton-Jacobi-Isaacs equation
reviewed by: Luigi Ambrosio
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 90522