Abstract
In a linear pursuit-evasion game with elliptical vectograms, singular surfaces appear in the plane of symmetry containing the ellipses minor axes and, as a consequence, locally nonsmooth isocost tubes are generated. Both dispersal and attractive focal surfaces are encountered. The dispersal surface is a zone of initial conditions for trajectories leaving the plane of symmetry. Optimal trajectories attracted by the focal surface merge tangentially to the plane of symmetry and remain there until the boundary of the dispersal zone is reached. Determination of the saddle-point strategies in the focal surface leads to constructing the isocost surfaces in the entire game space.
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References
Ho, Y. C., Bryson, A. E., Jr., andBaron, S.,Differential Games and Optimal Pursuit-Evasion Strategies, IEEE Transactions on Automatic Control, Vol. AC-10, No. 10, 1965.
Mishchenko, E. F., andPontryagin, L. S.,Linear Differential Games, Soviet Mathematics Doklady, Vol. 8, No. 3, 1967.
Borgest, W., andVaraiya, P.,Target Function Approach to Linear Pursuit Problems, IEEE Transactions on Automatic Control, Vol. AC-16, No. 5, 1971.
Aumasson, C.,Approche du Problem de l'Interception Tridimensionnelle Optimale par la Theorie des Jeux Differentiels Lineaires Quadratiques, Recherche Aerospaciale, Vol. 176, No. 5, 1976.
Bernhard, P.,Linear-Quadratic, Two-Person, Zero-Sum Differential Games: Necessary and Sufficient Conditions, Journal of Optimization Theory and Applications, Vol. 27, No. 1, 1979.
Gutman, S., andLeitmann, G.,Optimal Strategies in the Neighborhood of a Collision Course, AIAA Journal, Vol. 14, No. 9, 1976.
Gutman, S.,An Optimal Guidance for Homing Missiles, Journal of Guidance and Control, Vol. 2, No. 4, 1979.
Gutman, S.,Remarks on Capture-Avoidance Games, Journal of Optimization Theory and Applications, Vol. 32, No. 3, 1980.
Shinar, J., andGutman, S.,Three-Dimensional Optimal Pursuit and Evasion with Bounded Controls, IEEE Transactions on Automatic Control, Vol. AC-25, No. 3, 1980.
Anderson, G. M.,Comparison of Optimal Control and Differential Game Intercept Missile Guidance Laws, Journal of Guidance and Control, Vol. 4, No. 2, 1981.
Shinar, J.,Solution Techniques for Realistic Pursuit-Evasion Games, Control and Dynamic Systems, Vol. 17, Edited by C. T. Leondes, Academic Press, New York, New York, 1981.
Isaacs, R.,Differential Games, John Wiley and Sons, New York, New York, 1967.
Breakwell, J. V.,Some Differential Games with Interesting Discontinuities, Stanford University, Department of Aeronautics and Astronautics, 1973.
Bernhard, P.,Singular Surfaces in Differential Games: An Introduction, Proceedings of the Workshop on Differential Games and Applications, Enschede, Holland, 1977; Edited by P. Hagedornet al., Springer-Verlag, Berlin, Germany, 1977.
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Communicated by J. V. Breakwell
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Shinar, J., Medinah, M. & Biton, M. Singular surfaces in a linear pursuit-evasion game with elliptical vectograms. J Optim Theory Appl 43, 431–456 (1984). https://doi.org/10.1007/BF00934465
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DOI: https://doi.org/10.1007/BF00934465