Article Outline
Glossary
Definition of the Subject
Introduction
Qualitative and Quantitative Differential Games
Existence of a Value for Zero Sum Differential Games
Nonantagonist Differential Games
Miscellaneous
Future Directions
Bibliography
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsAbbreviations
- Dynamics:
-
This is the law which governs the evolution of the system: for differential games it is a differential equation.
- Strategies:
-
This is the way a player chooses his control asa function of the state of the system and of the action of his opponents.
- Information:
-
This is the set of parameters known by the player in order to build his strategy.
Bibliography
Primary Literature
Aubin JP (1991)Viability Theory. Birkhäuser, Basel
Aumann RJ, Maschler M (1995)Repeated games with incomplete information.MIT Press, Cambridge
Bardi M, Capuzzo Dolcetta I (1997)Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations.Systems and Control: Foundations and Applications, vol xvii.Birkhäuser, Boston, pp 570
Basar T, Bernhard P (1995)\( { H\sp \infty } \)-optimal control and related minimax design problems.A dynamic game approach, 2nd edn.Systems and Control: Foundations and Applications.Birkhäuser, Basel
Berkovitz LD (1994)A theory of differential games. In: Basar T et al (eds) Advances indynamic games and applications, Birkhäuser, Basel; Ann Int Soc Dyn Games 1:3–22
Bernhard P (1988)Differential games in Systems and control Encyclopedia. In: Singh MG (ed) Theory Technology Application.Pergamon Press, Oxford
Bettiol P, Cardaliaguet P, Quincampoix M (2006)Zero-sum state constrained differential games: Existence of value for Bolza problem.Int J Game Theor 34(4):495–527
Breakwell JV (1977)Zero-sum differential games with terminal payoff.In: Hagedorn P, Knobloch HW, Olsder GH (eds) Differential Game and Applications.Lecture Notes in Control and Information Sciences, vol 3.Springer, Berlin
Buckdahn R, Cardaliaguet P, Rainer C (2004)Nash equilibrium payoffs for nonzero-sum stochastic differential games.SIAM J Control Optim 43(2):624–642
Cardaliaguet P (1996)A differential game with two players and one target.SIAM J Control Optim 34(4):1441–1460
Cardaliaguet P (1997)Nonsmooth semi-permeable barriers, Isaacs equation, and application to a differential gamewith one target and two players.Appl Math Optim 36:125–146
Cardaliaguet P, Plasckacz S (2003)Existence and uniqueness of a Nash equilibrium feedback for a simple non zero-sum differential game.Int J Game Theor 32(4):561–593
Cardaliaguet P, Quincampoix M, Saint-Pierre P (1999) Numerical methods for optimal control and numerical games. In: Bardi M, Parthasarathy T, Raghavan TES (eds) Annals of International Society of Dynamical Games. Birkhäuser, Basel, pp 177–249
Cardaliaguet P, Quincampoix M, Saint-Pierre P (2001)Pursuit differential games with state constraints.SIAM J Control Optim 39(5):1615–1632
Cardaliaguet P, Quincampoix M (2008)Determinist Differential games under probability knowledge of initial condition.Int Game Theor Rev 6(1):1–16
Elliot N, Kalton N (1972)The existence of value in differential games.Mem Am Math Soc 126:67
Evans LC, Souganidis PE (1984)Differential games and representation formulas for solutions of Hamilton–Jacobi equations.Indiana Univ Math J 282:487–502
Fleming W, Souganidis P (1989)On the existence of value functions of two-player, zero-sum stochastic differential games.Indiana Univ Math J 38(2):293–314
Flynn J (1973)Lion and Man: The Boundary Constraints.SIAM J Control 11:397
Klejmenov AF (1993)Nonantagonistic positional differential games.Nauka, Ekaterinburg (in Russian)
Krassovski NN, Subbotin AI (1988)Game-Theorical Control Problems.Springer, New York
Isaacs R (1965)Differential Games.Wiley, New York
Jorgensen S, Quincampoix M, Vincent T (2007)Advances in Dynamic Games Theory.Annals of International Society of Dynamic Games.Birkhäuser, Basel
Melikyan AA (1998)Generalized characteristics of first order PDEs.Applications in optimal control and differential games.Birkhäuser, Boston
Petrosjan LA (2004)Cooperation in games with incomplete information.Nonlinear analysis and convex analysis.Yokohama Publ, pp 469–479
Pontryagin N (1968)Linear Differential Games, I, II.Sov Math Doklady 8(3–4):769–771; 910–913
Plaskacz S, Quincampoix M (2000)Discontinuous Mayer control problem under state-constraints.Topol Method Nonlinear Analysis 15:91–100
Quincampoix M (1992)Differential inclusions and target problems.SIAM J Control Optim 30(2):324–335
Quincampoix M, Veliov V (2005)Optimal control of uncertain systems with incomplete information for the disturbance.Siam J Control Optim 43(4):1373–1399
Rainer C (2007)On two different approaches to nonzero-sum stochastic differential games.Appl Math Optim 56:131–144
Roxin E (1969)The axiomatic approach in differential games.J Optim Theor Applt 3:153–163
Subbotin AI (1995)Generalized solutions of first-order PDEs, The dynamical optimization perspective.Translated from the Russian. Systems and Control: Foundations and Applications.Birkhäuser, Boston
Varaiya P (1967)The existence of solution to a differential game.SIAM J Control Optim 5:153–162
Varaiya P, Lin J (1967)Existence of saddle points in differential game.SIAM J Control Optim 7(1):141–157
Von Neumann J, Morgenstern O (1946)Theory of Games and Economic Behaviour.Princeton University Press, Princeton
Books and Reviews
Bardi M, Raghavan TES, Parthasarath T (1999)Stochastic and differential games.Theory and numerical methods, Dedicated to Prof. Subbotin AI.Annals of the International Society of Dynamic Games, vol 4.Birkhäuser, Boston
Basar T, Olsder GJ (1999)Dynamic noncooperative game theory, 2nd edn.Classics in Applied Mathematics, vol 23.SIAM, Society for Industrial and Applied Mathematics, Philadelphia
Blaquiére A, Gérard F, Leitman G (1969)Quantitative and Qualitative Games.Academic Press, New York
Dockner E, Jrgensen S, Van Long N, Sorger G (2000)Differential games in economics and management science.Cambridge University Press, New York
Hajek O (1975)Pursuit games.Academic Press, New York
Patsko VS, Turova VL (2000)Numerical study of differential games with the homicidal chauffeur dynamics.Russian Academy of Sciences, Institute of Mathematics and Mechanics, Ekaterinburg
Petrosyan LA (1993)Differential games of pursuit.Series on Optimization, vol 2.World Scientific, Singapore
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag
About this entry
Cite this entry
Quincampoix, M. (2012). Differential Games. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_55
Download citation
DOI: https://doi.org/10.1007/978-1-4614-1800-9_55
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-1799-6
Online ISBN: 978-1-4614-1800-9
eBook Packages: Computer ScienceReference Module Computer Science and Engineering