Abstract
This paper presents a new computational approach for a class of pursuit-evasion games of degree. The saddle-point problem is decomposed into two subproblems that are solved by turns iteratively. The subproblems are ordinary optimal control problems that can be solved efficiently using discretization and nonlinear programming techniques. Hence it is not necessary to supply an initial guess of the adjoint variables or a hypothesis on the switching structure of the solution. Furthermore, in the presented approach the numerical differentiation of the payoff is avoided. We test the algorithm with different numerical examples and compare the results with solutions obtained by an indirect method. In the test examples the method converges rapidly from a rough initial guess, and the results coincide well with the reference solutions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bardi, M., S. Bottacin, and M. Falcone. “Convergence of Discrete Schemes for Discontinuous Value Functions of Pursuit-Evasion Games”, In: G. J. Olsder, ed. New Trends in Dynamic Games and Applications, pp. 273–304. Birkhäuser, Boston, 1996.
Basar, T. and G. Olsder. Dynamic Noncooperative Game Theory, 2nd ed. Academic Press, London, 1995.
Bazaraa, M., H. Sherali, and C. Shetty. Nonlinear Programming, Theory and Algorithms. 2nd ed., Wiley, New York, 1993.
Breitner, M., H. Pesch, and W. Grimm. “Complex Differential Games of Pursuit-Evasion Type with State Constraints, Part 2: Multiple Shooting and Homotopy”, Journal of Optimization Theory and Applications, 78, No. 3,442–464, 1993.
Ehtamo, H. and T. Raivio. A Feasible Direction Method for Saddle Point Problems. Proceedings of the 8th International Symposium on Dynamic Games and Applications, Maastricht, The Netherlands, pp. 184–188, 1998.
Fiacco, A. Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic Press, New York, 1983.
Hargraves, C. and S. Paris. Direct Trajectory Optimization Using Nonlinear Programming and Collocation. Journal of Guidance, Control and Dynamics, 10, No. 4, 338–342, 1987.
Isaacs, R. Differential Games. Reprint. Krieger, New York, 1975.
Krasovskii, N. and A. Subbotin. Game Theoretical Control Problems. Springer-Verlag, New York, 1988.
Lachner, R., M. Breitner, and H. Pesch. Three-Dimensional Air Combat: Numerical Solution of Complex Differential Games. In: G. Olsder, ed. New Trends in Dynamic Games and Applications, pp. 165–190. Birkhäuser, Boston, 1996.
Lachner, R., M. Breitner, and H. Pesch. Real-Time Collision Avoidance Against Wrong Drivers: Differential Game Approach, Numerical Solution and Synthesis of Strategies with Neural Networks. Proceedings of the 7th International Symposium on Dynamic Games and Applications, Kanagawa, Japan, 564–572, 1996.
Moritz, K., R. Polis, and K. Well. Pursuit-Evasion in Medium-Range Air-Combat Scenarios. Computers and Mathematics with Applications, 13, No. 1–3, 167–180, 1987.
Oberle, H. and W. Grimm. BNDSCO—A Program for the Numerical Solution of Optimal Control Problems. Program manual, 1989.
Raivio, T, H. Ehtamo, and R. Hämäläinen. Simulating the Worst Possible Prediction Error of System Trajectories. Proceedings of the European Simulation Multiconference, Budapest, Hungary, pp. 299–303, 1996.
Raivio, T., H. Ehtamo, and R. Hämäläinen. A Decomposition Method for a Class of Pursuit-Evasion Games. Proceedings of the 7th International Symposium on Dynamic Games and Applications, Kanagawa, Japan, pp. 784–795,1996.
Seywald, H. Trajectory Optimization Based on Differential Inclusion. Journal of Guidance, Control and Dynamics, 17, No. 3, 480–487, 1994.
Shinar, J., M. Guelman, and A. Green. An Optimal Guidance Law for a Planar Pursuit-Evasion Game of Kind, Computers and Mathematics with Applications, 18, No. 1–3, 35–44, 1989.
Stryk von, O. Numerische Lösung optimaler Steuerungsprobleme: Diskretisierung, Parameteroptimierung und Berechnung der adjungierten Variabein. Doctoral Dissertation, Munich University of Technology, 1994 (In German).
Virtanen, K., H. Ehtamo, T. Raivio, and R. Hämäläinen. VIATO—Visual interactive Aircraft Trajectory Optimization. IEEE Transactions on Systems, Man and Cybernetics, Part C, Applications and Reviews, 29, 3,409–421, 1999.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media New York
About this paper
Cite this paper
Raivio, T., Ehtamo, H. (2000). On the Numerical Solution of a Class of Pursuit-Evasion Games. In: Filar, J.A., Gaitsgory, V., Mizukami, K. (eds) Advances in Dynamic Games and Applications. Annals of the International Society of Dynamic Games, vol 5. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1336-9_9
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1336-9_9
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7100-0
Online ISBN: 978-1-4612-1336-9
eBook Packages: Springer Book Archive