Stochastic and Differential Games: Theory and Numerical Methods / Edition 1

by Martino Bardi, T.E.S. Raghavan, T. Parthasarathy

Overview

The theory of two-person, zero-sum differential games started at the be- ginning of the 1960s with the works of R. Isaacs in the United States and L. S. Pontryagin and his school in the former Soviet Union. Isaacs based his work on the Dynamic Programming method. He analyzed many special cases of the partial differential equation now called Hamilton- Jacobi-Isaacs-briefiy HJI-trying to solve them explicitly and synthe- sizing optimal feedbacks from the solution. He began a study of singular surfaces that was continued mainly by J. Breakwell and P. Bernhard and led to the explicit solution of some low-dimensional but highly nontriv- ial games; a recent survey of this theory can be found in the book by J. Lewin entitled Differential Games (Springer, 1994). Since the early stages of the theory, several authors worked on making the notion of value of a differential game precise and providing a rigorous derivation of the HJI equation, which does not have a classical solution in most cases; we mention here the works of W. Fleming, A. Friedman (see his book, Differential Games, Wiley, 1971), P. P. Varaiya, E. Roxin, R. J. Elliott and N. J. Kalton, N. N. Krasovskii, and A. I. Subbotin (see their book Po- sitional Differential Games, Nauka, 1974, and Springer, 1988), and L. D. Berkovitz. A major breakthrough was the introduction in the 1980s of two new notions of generalized solution for Hamilton-Jacobi equations, namely, viscosity solutions, by M. G. Crandall and P. -L.

Product Details

ISBN-13:	9780817640293
Publisher:	Birkhäuser Boston
Publication date:	06/01/1999
Series:	Annals of the International Society of Dynamic Games , #4
Edition description:	1999
Pages:	381
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

I. Zero-Sum Differential Games and Numerical Methods.- 1 Constructive Theory of Positional Differential Games and Generalized Solutions to Hamilton—Jacobi Equations.- 2 Two-Player, Zero-Sum Differential Games and Viscosity Solutions.- 3 Numerical Methods for Pursuit-Evasion Games via Viscosity Solutions.- 4 Set-Valued Numerical Analysis for Optimal Control and Differential Games.- II. Shastic and Nonzero-Sum Games and Applications.- 5 An Introduction to Gambling Theory and Its Applications to Shastic Games.- 6 Discounted Shastic Games: A Complex Analytic Perspective.- 7 Nonzero-Sum Shastic Games.- 8 The Power of Threats in Shastic Games.- 9 A Markov Game Approach for Optimal Routing Into a Queuing Network.- 10 On Linear Complementarity and A Discounted Polyshastic Game.