Abstract
In Part 1 of this paper (Ref. 1), necessary conditions for optimal open-loop strategies in differential games of pursuit-evasion type have been developed for problems which involve state variable inequality constraints and nonsmooth data. These necessary conditions lead to multipoint boundary-value problems with jump conditions. These problems can be solved very efficiently and accurately by the well-known multiple-shooting method. By this approach, optimal open-loop strategies and their associated saddle-point trajectories can be computed for the entire capture zone of the game. This also includes the computation of optimal open-loop strategies and saddle-point trajectories on the barrier of the pursuit-evasion game. The open-loop strategies provide an open-loop representation of the optimal feedback strategies. Numerical results are obtained for a special air combat scenario between one medium-range air-to-air missile and one high-performance aircraft in a vertical plane. A dynamic pressure limit for the aircraft imposes a state variable inequality constraint of the first order. Special emphasis is laid on realistic approximations of the lift, drag, and thrust of both vehicles and the atmospheric data. In particular, saddle-point trajectories on the barrier are computed and discussed. Submanifolds of the barrier which separate the initial values of the capture zone from those of the escape zone are computed for two representative launch positions of the missible. By this way, the firing range of the pursuing missile is determined and visualized.
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Breitner, M. H., Pesch, H. J., andGrimm, W.,Complex Differential Games of Pursuit-Evasion Type with State Constraints, Part 1: Necessary Conditions for Optimal Open-Loop Strategies, Journal of Optimization Theory and Applications, Vol. 78, pp. 419–441, 1993.
Järmark, B.,Convergence Control in Differential Dynamic Programming Applied to Air-to-Air Combat, AIAA Journal, Vol. 14, pp. 118–121, 1976.
Miele, A., Damoulakis, J. N., Cloutier, J. R., andTietze, J. L.,Sequential Gradient-Restoration Algorithm for Optimal Control Problems with Nondifferential Constraints, Journal of Optimization Theory and Applications, Vol. 13, pp. 218–255, 1974.
Eichberger, H.,Ein sequentieller Gradienten-Restaurations-Algorithmus zur Behandlung einer Klasse von differentiellen Spielen mit vorgegebener Spieldauer, Diploma Thesis, Department of Mathematics, Berlin University of Technology, Berlin, Germany, 1977.
Dolezal, I. A Gradient-Type Algorithm for the Numerical Solution of Two-Player Zero-Sum Differential Game Problems, Kybernetika, Vol. 14, pp. 429–446, 1978.
Quintana, V. H., andDavison, E. J.,Two Numerical Techniques to Solve Differential Game Problems, International Journal of Control, Vol. 16, pp. 465–474, 1972.
Tolwinski, B.,Numerical Solution of n-Person Nonzero-Sum Differential Games, Control and Cybernetics, Vol. 7, pp. 37–50, 1978.
Bulirsch, R.,Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung, Report of the Carl-Cranz Gesellschaft, Carl-Cranz Gesellschaft, Oberpfaffenhofen, Germany, 1971.
Stoer, J., andBulirsch, R.,Introduction to Numerical Analysis, Springer-Verlag, New York, New York, 1993.
Bayen, H.,Numerische Lösung von differentiellen Spielen mit der Mehrzielmethode, Diploma Thesis, Department of Mathematics, University of Cologne, Cologne, Germany, 1974.
Bulirsch, R., Montrone, F., andPesch, H. J.,Abort Landing in the Presence of a Windshear as a Minimax Optimal Control Problem, Part 1; Necessary Conditions, Journal of Optimization Theory and Applications, Vol. 70, pp. 1–23, 1991.
Bulirsch, R., Montrone, F., andPesch, H. J.,Abort Landing in the Presence of a Windshear as a Minimax Optimal Control Problem, Part 2; Multiple Shooting and Homotopy, Journal of Optimization Theory and Applications, Vol. 70, pp. 221–252, 1991.
Oberle, H. J.,Numerische Berechnung optimaler Steuerungen von Heizung und Kühlung für ein realistisches Sonnenhausmodell, Habilitationsschrift, Munich University of Technology, Munich, Germany, 1982.
Shinar, J., andGutman, S.,Three-Dimensional Optimal Pursuit and Evasion with Bounded Control, IEEE Transactions on Automatic Control, Vol. 25, pp. 492–496, 1980.
Gutman, S., andKatz, D.,On Guaranteed-Cost, Closed-Form Guidance via Simple Linear Differential Games, Proceedings of the 27th IEEE Conference on Decision and Control, Austin, Texas, pp. 1421–1424, 1988.
Guelman, M., Shinar, J., andGreen, A.,Qualitative Study of a Planar Pursuit-Evasion Game in the Atmosphere, Paper No. 88-4158-CP, Proceedings of the AIAA Guidance, Navigation, and Control Conference, Minneapolis, Minnesota, 1988.
Shinar, J., andGazit, S.,Optimal No-Escape Envelopes of Guided Missiles, Paper No. 85-1960, Proceedings of the AIAA Guidance, Navigation, and Control Conference, Snowmass, Colorado, 1985.
Oberle, H. J., andGrimm, W.,BNDSCO:A Program for the Numerical Solution of Optimal Control Problems, Internal Report No. 515-89/22, Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen, Germany, 1989.
Deuflhard, P.,Ein Newton-Verfahren bei fastsingulär Funktionalmatrix zur Lösung von nichtlinearen Randwertaufgaben mit der Mehrzielmethode, Dissertation, Department of Mathematics, University of Cologne, Cologne, Germany, 1972.
Breitner, M. H.,Numerische Berechnung der Barriere eines realistischen Differentialspiels, Diploma Thesis, Department of Mathematics, Munich University of Technology, Munich, Germany, 1990.
Katzir, S., Cliff, E. M., andLutze, F. H.,A Comparison of Dynamic Models for Optimal Midcourse Guifance, Proceedings of the AIAA Guidance, Navigation and Control Conference, Boston, Massachusetts, 1989.
Breitner, M. H., Grimm, W., andPesch, H. J.,Barrier Trajectories of a Realistic Missile/Target Pursuit-Evasion Game, Differential Games: Developments in Modelling and Computation, Edited by R. P. Hämäläinen and H. K. Ehtamo, Springer-Verlag, Berlin, Germany, pp. 48–57, 1991.
Isaacs, R.,Differential Games, John Wiley and Sons, New York, New York, 1965; see also Krieger, New York, New York, 3rd Printing, 1975.
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Communicated by R. Bulirsch
This paper is dedicated to the memory of Professor John V. Breakwell.
The authors would like to express their sincere and grateful appreciation to Professors R. Bulirsch and K. H. Well for their encouraging interest in this work.
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Breitner, M.H., Pesch, H.J. & Grimm, W. Complex differential games of pursuit-evasion type with state constraints, part 2: Numerical computation of optimal open-loop strategies. J Optim Theory Appl 78, 443–463 (1993). https://doi.org/10.1007/BF00939877
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DOI: https://doi.org/10.1007/BF00939877