Abstract
Complex pursuit-evasion games with complete information under state variable inequality constraints are investigated. By exploitation of Isaacs’ minimax principle, necessary conditions of first and second order are derived for the optimal trajectories. These conditions give rise to multipoint boundary-value problems, which yield open-loop representations of the optimal strategies along the optimal trajectories. The multipoint boundary-value problems are accurately solved by the multiple shooting method. The computed open-loop representations can thereafter be used to synthesize the optimal strategies globally.
As an illustrative example, the evasion of an aircraft from a pursuing missile is investigated. The flight of the aircraft is restricted by various control variable inequality contraints and by a state variable inequality constraint for the dynamic pressure. The optimal trajectories exhibit boundary arcs with regular and singular constrained controls. The influence of various singular surfaces in the state space including a low-dimensional universal surface is discussed.
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References
Başar, T., and Olsder, G. J., Dynamic Noncooperative Game Theory, Academic Press, London, 1982, 2nd ed., 1994.
Breitner, M. H., Construction of the Optimal Feedback Controller for Constrained Optimal Control Problems with Unknown Disturbances, in: R. Bulirsch and D. Kraft (Eds.), Computational Optimal Control, International Series of Numerical Mathematics 115, Birkhauser, Basel, 1994, 147–162.
Breitner, M. H., Real-Time Capable Approximation of Optimal Strategies in Complex Differential Games, in: M. Breton, G. Zaccour (Eds.), Proceedings of the Sixth International Symposium on Dynamic Games and Applications, St-Jovite, Quebec, GERAD, Ecole des Hautes Etudes Commerciales, Montreal, 1994 370–374.
Breitner, M. H., Robust-optimale Riickkopplungssteuerungen gegen unvorhersehbare Einfliisse: Differentialspielansatz, numerische Berech- nung und Echtzeitapproximation, PhD thesis, in preparation.
Breitner, M. H., and Pesch H. J., Reentry Trajectory Optimization under Atmospheric Uncertainty as a Differential Game, in: T. Başar, and A. Haurie (Eds.), Advances in Dynamic Games and Applications, Annals of the International Society of Dynamic Games 1, 1994, Birkhauser, Basel, 70–88.
Breitner, M. H., Grimm, W., and Pesch, H. J., Barrier Trajectories of a Realistic Missile/Target Pursuit-Evasion Game, in: R. P. Hamalainen, and H. K. Ehtamo (Eds.), Differential Games — Developments in Modelling and Computation, Lecture Notes in Control and Information Sciences 156,1991, Springer, Berlin, 48–57.
Breitner, M. H., Pesch, H. J., and Grimm, W., Complex Differential Games of Pursuit-Evasion Type with State Constraints, Part 1: Necessary Conditions for Optimal Open-Loop Strategies, Part 2: Numerical Computation of Optimal Open-Loop Strategies, Journal of Optimization Theory and Applications 78, 1993, 419–441, 443–463.
Bryson, A. E., Denham, W. F., and Dreyfus, S. E., Optimal Programming Problems with Inequality Constraints, I: Necessary Conditions for Extremal Solutions, AIAA Journal 1, 1963, 2544–2550.
bulirsch, r., Die Mehrzielmethode zur numerischen Losung von Randwertproblemen und Aufgaben der optimalen Steuerung, Report of the Carl-Cranz-Gesellschaft, DLR, Oberpfaffenhofen, Germany, 1971; Reprint, Department of Mathematics, Munich University of Technology, Munich, 1985 and 1993.
Fink, G., Numerische Berechnung optimaler Strategien fur ein realitatsnahes Differentialspiel einer Verfolgung im dreidimensionalen Raum, Diploma thesis, Department of Mathematics, Munich University of Technology, Munich, 1994.
Greenwood, N. J. C., Applied Differential Games in the Aerial Scenario, PhD thesis, Department of Mathematics, University of Queensland, Brisbane, 1994.
Grimm, W., Lenkung von Hochleistungsflugzeugen. Vergleich von optimaler Steuerung und fastoptimaler Echtzeitsteuerung, PhD thesis, Department of Mathematics, Munich University of Technology, Munich, 1992.
Guelman, M., Shinar, J., and Green, J., Qualitative Study of a Planar Pursuit Evasion Game in the Atmosphere, in: Proceedings of the AIAA Guidance, Navigation and Control Conference, Minneapolis, 1988.
Gutman, S., and Katz, D., On Guaranteed-Cost, Closed-Form Guidance via Simple Linear Differential Games, in: Proceedings of the 27th IEEE Conference on Decision and Control, Austin, 1988.
Hiltmann, P., Chudej, K., and Breitner, M. H., Eine modifizierte Mehrzielmethode zur Losung von Mehrpunkt-Randwertproblemen — Benutzeranleitung, Report No. 14, Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 255 “Transatmospharische Flugsysteme”, Department of Mathematics, Munich University of Technology, Munich, Germany, 1993.
Isaacs, R., Games of Pursuit, Paper No. P-257, RAND Corporation, Santa Monica, California, 1951.
Isaacs, R., Differential Games, Wiley, New York, New York, 1965; Krieger, New York, New York, 1975.
Jacobson, D. H., Lele, M. M., and Speyer, J. L., New Necessary Conditions of Optimality for Control Problems with State Variable Inequality Constraints, Journal of Mathematical Analysis and Applications 35, 255–284, 1971.
Lachner, R., Realitatsnahe Modellierung eines Differentialspieles mit Zustandsbeschrankungen und numerische Berechnung optimaler Trajek- torien, Diploma thesis, Department of Mathematics, Munich University of Technology, Munich, Germany, 1992.
Lachner, R., Breitner, M. H., and Pesch, H. J., Optimal Strategies of a Complex Pursuit-Evasion Game, in: R. Fritsch, and M. Toepell (Eds.), Proceedings of the 2nd Gauss Symposium, Munich, Germany, 1993; to appear in Journal of Computing and Information, 1995.
Lachner, R., Breitner, M. H., and Pesch, H. J., Efficient Numerical Solution of Differential Games with Application to Air Combat, Report No. 466, Deutsche Forschungsgemeinschaft, Schwerpunkt “Anwendungsbezogene Optimierung und Steuerung”, Department of Mathematics, Munich University of Technology, Munich, 1993.
Leitmann, G., Einfilhrung in die Theorie optimaler Steuerung und der Differentialspiele, R. Oldenburg, Munich, 1974.
Maurer, H., Optimale Steuerprozesse mit Zustandsbeschrankungen, Habilitationsschrift, University of Wurzburg, Wurzburg, 1976.
Miele, A., Flight Mechanics I, Theory of Flight Paths, Addison Wesley, Reading, 1962.
Pesch, H. J., Offline and Online Computation of Optimal Trajectories in the Aerospace Field, in: A. Miele and A. Salvetti (Eds.), Applied Mathematics in Aerospace Science and Engineering, Plenum, New York, 1994, 165–219.
Pesch, H. J., Solving Optimal Control and Pursuit-Evasion Game Problems of High Complexity, in: R. Bulirsch, and D. Kraft (Eds.), Computational Optimal Control, International Series of Numerical Mathematics 115, Birkhauser, Basel, 1994, 43–64.
Pesch, H. J., Gabler, I., Miesbach, S., and Breitner, M. H., Synthesis of Optimal Strategies for Differential Games by Neural Networks, in: G. J. Olsder (Ed.), Advances in Dynamic Games and Applications, Annals of the International Society of Dynamic Games, this volume, Birkhauser, Boston, 1995.
Seywald, H., Cliff, E. M., and Well, K. H., Range Optimal Trajectories for an Aircraft Flying in the Vertical Plane, Journal of Guidance, Control, and Dynamics 17, 1994, 389–398.
Stoer, J., Bulirsch, R., Introduction to Numerical Analysis, Springer, New York, 2nd ed., 1993.
von Stryk, O., Numerische Losung optimaler Steuerungsprob- leme: Diskretisierung, Parameteroptimierung und Berechnung der adjungierten Variablen, Fortschritt-Berichte VDI, Reihe 8, Nr. 441, VDI- Verlag, Diisseldorf, 1995
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Lachner, R., Breitner, M.H., Pesch, H.J. (1995). Three-Dimensional Air Combat: Numerical Solution of Complex Differential Games. In: Olsder, G.J. (eds) New Trends in Dynamic Games and Applications. Annals of the International Society of Dynamic Games, vol 3. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4274-1_8
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DOI: https://doi.org/10.1007/978-1-4612-4274-1_8
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