Advertisement

Cooperative 2-On-1 Bounded-Control Linear Differential Games

  • Shmuel Y. HayounEmail author
  • Tal Shima

Abstract

A linearized 2-on-1 engagement is considered, in which the players’ controls are bounded and have first order dynamics and equal terminal instants. A capturability analysis is performed, presenting necessary and sufficient conditions for the feasibility of exact capture against any target maneuver and for arbitrary control dynamics. Wishing to formulate the engagement as a zero-sum differential game, a suitable cost function is proposed and validated, and the resulting optimization problem and its solution are presented. Construction and analysis of the game space for the case of strong pursuers is shown, and the players’ closed form optimal controls are derived.

Keywords

Differential Game Capture Zone Game Space Target Maneuver Proposed Cost Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ben-Asher, J., Yaesh, I.: Advances in Missile Guidance Theory (Progress in Astronautics and Aeronautics), vol. 180, pp. 89–126. AIAA, Washington D.C (1998)Google Scholar
  2. 2.
    Borowko, P., Rzymowski, W.: On the Game of Two Cars. Journal of Optimization Theory and Applications 44(3), 381–396 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bryson, A.E., Ho, Y.C.: Applied Optimal Control. Ann. Mat. Pura. Appl. Blaisdell Publishing Company, Waltham, Massachusetts (1969)Google Scholar
  4. 4.
    Cockayne, E.: Plane Pursuit with Curvature Constraints. SIAM Journal on Applied Mathematics 15(6), 1511–1516 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Foley, M.H., Schmitendorf, W.E.: A Class of Differential Games with Two Pursuers Versus One Evader. IEEE Transactions on Automatic Control 19(3), 239–243 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Ganebny, S.A., Kumkov, S.S., Le Mènec, S., Patsko, V.S.: Model Problem in a Line with Two pursuers and One Evader. Dynamic Games and Applications 2(2), 228–257 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Gutman, S.: On Optimal Guidance for Homing Missiles. Journal of Guidance and Control 3(4), 296–300 (1979)CrossRefGoogle Scholar
  8. 8.
    Gutman, S., Leitmann, G.: Optimal Strategies in the Neighborhood of a Collision Course. AIAA Journal 14(9), 1210–1212 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Ho, Y.C., Bryson Jr, A.E., Baron, S.: Differential Games and Optimal Pursuit-Evasion Strategies. IEEE Transactions on Automatic Control AC-10(4), 385–389 (1965)Google Scholar
  10. 10.
    Isaacs, R.: Differential Games. John Wiley, New York (1965)zbMATHGoogle Scholar
  11. 11.
    Le Mènec, S.: Linear Differential Game with Two Pursuers and One Evader. Advances in Dynamic Games 25(6), 209–226 (2011)CrossRefGoogle Scholar
  12. 12.
    Leatham, A.L.: Some Theoretical Aspects of Nonzero Sum Differential Games and Applications to Combat Problems. Wright-Patterson AFB OH School of Engineering 10(AFIT/DS/MC/71-3) (1971)Google Scholar
  13. 13.
    Lewin, J.: Differential games: Theory and Methods for Solving Game Problems with Singular Surfaces. Springer, New York (1994)CrossRefGoogle Scholar
  14. 14.
    Liu, Y., Qi, N., Tang, Z.: Linear Quadratic Differential Game Strategies with Two-Pursuit Versus Single-Evader. Chinese Journal of Aeronautics 25(6), 896–905 (2012)CrossRefGoogle Scholar
  15. 15.
    Perelman, A., Shima, T., Rusnak, I.: Cooperative Differential Games Strategies for Active Aircraft Protection from a Homing Missile. AIAA Journal of Guidance, Control, and Dynamics 34(3), 761–773 (2011)CrossRefGoogle Scholar
  16. 16.
    Rublein, G.T.: On Pursuit with Curvature Constraints. SIAM Journal on Control 10(1), 37–39 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Rzymowski, W.: On The Game of 1+n Cars. Journal of Mathematical Analysis and Applications 99(1), 109–122 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Shima, T.: Capture Conditions in a Pursuit-Evasion Game Between Players with Biproper Dynamics. Journal of Optimization Theory and Applications 126(3), 503–528 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Shima, T., Shinar, J.: Time-varying linear pursuit-evasion game models with bounded controls. AIAA Journal of Guidance, Control, and Dynamics 25(3), 425–432 (2002)CrossRefGoogle Scholar
  20. 20.
    Shinar, J., Shinar, J.: Solution Techniques for Realistic Pursuit-Evasion Games. Advances in Control and Dynamic Systems 17, 63–124 (1981)CrossRefzbMATHGoogle Scholar
  21. 21.
    Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Aerospace EngineeringTechnion-Israel Institute of TechnologyHaifaIsrael

Personalised recommendations