Differential games of inf-sup type and Isaacs equations
type: Item
instance of: scholarly article
title: Differential games of inf-sup type and Isaacs equations
zbMATH Open document ID: 1111.49026
author: Hidehiro Kaise
author: S. J. Sheu
published in: Applied Mathematics and Optimization
publication date: 2006-02-08T00:00:00Z
review text: In the paper the dynamics of the form \[ \frac{d\,x}{d\,s}=f(s,x,a,b)\tag{1} \] are studied, dependent on a pair of control functions \(a(s)\), \(b(s)\), along with a game payoff function \[ J(t,x,a,b)=\int_t^Tl(s,x(s),a(s),b(s))\,ds+\Phi(x(T)).\tag{2} \] The functions \(a(s)\) and \(b(s)\) are regarded as, respectively, the control of the minimizing and the control of the maximizing player. The maximizing player is interpreted as a disturbance in \(H^{\infty}\) control, therefore the inf-sup game value function \[ V(t,x)=\inf_{\alpha}\sup_b J(t,x,\alpha(b),b) \] is examined, where the supremum is taken over all feasible \(b\), while the infimum is computed with respect to a class of strategies of the minimizing player, i.e. the maps of the function \(b\) into \(a\). The main results of the paper are concerned with establishing a relationship between the Dynamic Programming Principle (DDP) and the infinitesimal generator for the game value function, and with obtaining this generator. It has been shown that, if DDP holds and the infinitesimal generator has been obtained, then the value function of the inf-sup type game is a viscosity solution of the corresponding Isaacs equation. More specifically, in the case of progressive strategies the infinitesimal generator is the lower Hamiltonian related with (1) and (2) (Theorem 3.3), and in the case of strictly progressive strategies this generator is equal to the upper Hamiltonian (Theorem 3.5). Additionally, for strategies situated between strictly progressive and progressive strategies the infinitesimal generator is defined, in an asymptotic sense, by a convex combination of the lower and the upper Hamiltonians (Theorem 4.2).
reviewed by: Krzysztof Tchoń
Mathematics Subject Classification ID: 49N70
Mathematics Subject Classification ID: 49L20
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 49J35
zbMATH DE Number: 5004511
zbMATH Keywords: differential game
zbMATH Keywords: dynamic programming
zbMATH Keywords: viscosity solutions
zbMATH Keywords: infinitesimal generators
zbMATH Keywords: strategy classes
MaRDI profile type: Publication
full work available at URL: s00245-004-0818-6
OpenAlex ID: W2093253246
DOI: 10.1007/S00245-004-0818-6
description: scientific article; zbMATH DE number 5004511