Abstract
A methodology to obtain an approximate solution of a singularly perturbed nonlinear differential game is presented. The outcome of the game with approximate strategies, defined as extended value, is related to the saddle-point value of the game. In an example of a simple pursuit-evasion game, it is shown that the proposed methodology leads to an easily implementable feedback form solution with fairly accurate results. This approach seems to be attractive for analyzing realistic air-combat models without solving a two-point boundary-value problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Othling, W. L., Jr.,Application of Differential Game Theory to Pursuit-Evasion Problems of Two Aircrafts, Air Force Institute of Technology, Wright-Patterson Air Force Base, PhD Dissertation, 1970.
Lynch, U. H. D.,Games Barriers and Their Application in Air-to-Air Combat, Air Force Institute of Technology, Wright-Patterson Air Force Base, PhD Dissertation, 1973.
Jenkins, P. G.,A General Payoff for Differential Game Role Determination in Two-Aircraft Combat, Air Force Institute of Technology, Wright-Patterson Air Force Base, MS Thesis, 1974.
Miller, E. L.,Application of Differential Games to Air-to-Air Combat Problems, Wright-Patterson Air Force Base, Report No. AFFDL-TR-7547, 1975.
Merz, A. W.,Application of Differential Game Theory to Role Determination in Aerial Combat, National Aeronautics and Space Administration, Report No. CR-137713, 1975.
Merz, A. W., andHague, D. S.,Coplanar Tail-Chase Aerial Combat as a Differential Game, AIAA Journal, Vol. 15, No. 10, 1977.
Williamson Noble, S. M. D.,Toward a Differential Game Solution to a Practical Two-Aircraft Pursuit-Evasion Problem in Three Dimensional Space, Air Force Institute of Technology, Wright-Patterson Air Force Base, MS Thesis, 1971.
Roberts, D. A., andMontgomery, R. C.,Development and Application of a Gradient Method for Solving Differential Games, National Aeronautics and Space Administration, Technical Note No. TN-D-6502, 1971.
Anderson, G. M.,A Near-Optimal Closed-Loop Solution Method for Nonsignular Zero-Sum Differential Games, Journal of Optimization Theory and Applications, Vol. 13, No. 3, 1974.
Leatham, A. L., andLynch, U. H. D.,Two Numerical Methods to Solve Realistic Air-to-Air Combat Differential Games, American Institute of Aeronautics and Astronautics, Paper No. 74–22, 1974.
Jarmark, B.,Near-Optimal Closed-Loop Strategy for Aerial Combat Games, The Royal Institute of Technology, Sweden, Report No. TRITA-REG-7602, 1976.
Jarmark, B.,Convergence Control in Differential Dynamic Programming Applied to Air-to-Air Combat, AIAA Journal, Vol. 14, No. 1, 1976.
Miller, L. E.,Differential Game Computer Program, Wright-Patterson Air Force Base, Report No. AFFDL-TM-75-66-FXG, 1976.
Tabak, D.,Numerical Solutions of Differential Game Problems, International Journal of Systems Science, Vol. 6, No. 6, 1975.
Kelley, H. J.,Aircraft Maneuver Optimization by Reduced-Order Approximation, Control and Dynamic Systems, Vol. 10, Edited by T. C. Leondes, Academic Press, New York, New York, 1973.
Ardema, M. D.,Singular Perturbations in Flight Mechanics, National Aeronautics and Space Administration, Report No. TM-X-62-380, August, 1974.
Calise, A. J.,Singular Perturbations Methods for Variational Problems in Aircraft Flight, IEEE Transactions on Automatic Control, Vol. AC-21, No. 3, 1976.
Calise, A. J.,Extended Energy Management Methods for Flight Performance Optimization, AIAA Journal, Vol. 15, No. 3, 1977.
Calise, A. J.,Singular Perturbation Analysis Approach for Systems with Highly Coupled Dynamics, Dynamics Research Corporation, 1977.
Gardner, B. F., Jr.,Zero-Sum Nash Strategies for Systems with Fast and Slow Modes, Proceedings of the 15th Allerton Conference on Communications, Computers, and Controls, University of Illinois, Urbana, Illinois, 1977.
O'Malley, R. E., Jr.,Singular Perturbations and Optimal Control, Mathematical Control Theory, Springer-Verlag, New York, New York, 1978.
Merari, A.,Singular Perturbation Technique Applied to Flight Mechanics Problems, Technion-Israel Institute of Technology, MS Thesis (in preparation).
Simakova, E. N.,Differential Game Pursuit Game, Automatika i Telemekhanika, No. 2, 1967.
Isaacs, R.,Differential Games, John Wiley and Sons, New York, New York, 1965.
Kokotović, P. V.,A Control Engineer's Introduction to Singular Perturbations, Joint Automatic Control Conference, Stanford, California, 1972.
Tihonov, A. N.,Systems of Differential Equations Containing Small Parameters Multiplying Some of the Derivatives (in Russian), Maternaticheskii Sbornik, Vol. 73, No. 3, 1952.
Wasow, W. R.,Asymptotic Expansions for Ordinary Differential Equations. John Wiley and Sons (Interscience Publishers), New York, New York, 1965.
Author information
Authors and Affiliations
Additional information
Communicated by J. V. Breakwell
This research was partially supported by AFSC Contract No. F-49620-79-6-0135. The authors are grateful to Prof. J. V. Breakwell for encouraging the approach taken in this research. Thanks are also due to Dr. S. Gutman and Dr. J. Lewin for their useful comments.
Rights and permissions
About this article
Cite this article
Farber, N., Shinar, J. Approximate solution of singularly perturbed nonlinear pursuit-evasion games. J Optim Theory Appl 32, 39–73 (1980). https://doi.org/10.1007/BF00934842
Issue Date:
DOI: https://doi.org/10.1007/BF00934842