Differential Games

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Differential Games

 

a branch of the mathematical theory of control, dealing with the control of an object in conflict situations. In differential games the possibilities of the players are described by differential equations containing control vectors, manipulated by the players. In choosing his control, each player can use only current information on the behavior of other players. There are two categories of differential games: those for two players and those for many players. The most thoroughly studied games are the differential games of pursuit, in which there are two players—the pursuer and the pursued. The goal of the pursuer is to guide a vector z(t) to a given set M in the shortest possible time. The goal of the pursued is to delay the moment of arrival of the vector z(t) in M. Fundamental results in the study of differen-tail games were obtained during the 1960’s in the USSR by L. S. Pontriagin, N. N. Krasovskii, E. F. Mishchenko, and B. N. Pshenichnyi and in the USA by R. Isaacs, L. Berkovitz, and W. Fleming.

M. S. NIKOL’SKII

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
The largest class of games that can be used in the game control of dynamic transport processes and among them controlling the movement of ships, planes, and cars represent differential games, described by state and output equations, and state and control constraints [20-24].
Petrosyan, "Stable solutions of differential games with many participants," vol.
Petrosyan, Cooperative Stochastic Differential Games, Springer, NewYork, NY, USA, 2006.
This type of inequalities also arises in zero sum stochastic differential games of mixed type where each player uses both continuous control and stopping times.
[26] proved the existence and uniqueness of solutions of fully coupled FBSDEL and then obtained the existence of an open-loop Nash equilibrium point for nonzero sum stochastic differential games by using this result.
The author covers stochastic differential equations, backward stochastic differential equations, stochastic control, continuous time stochastic optimization and control, probabilistic approaches to stochastic control, stochastic differential games, and a wide variety of other related subjects.
Isaacs, Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, Wiley, New York Press, 1965.
Differential games of capitalism were first explored by Lancaster (1973), (3) who adopted a two player non-cooperative differential game where the workers control the share of their consumption in total output while the capitalists control the share of investment in the surplus.
Other areas covered include dynamic programming, optimal control for polynomial systems, multi-variable frequency-domain techniques, and differential games. Background in the state-variable representation of systems is assumed; an appendix offers a review of matrix algebra.
Chapters cover general optimal control problems, finite-dimensional optimization, optimization of dynamic systems with general performance criteria, terminal equality constraints, the linear quadratic control problem, and linear quadratic differential games. Detailed proofs and chapter problems are included.
The first is based on the mathematical theory of differential games. The second one, called "the elastic bands", was developed by Quinlan and Khatib and relies on repulsive potential field generation.

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