Skip to main content
SpringerLink
Log in

Abnormal and Singular Solutions in the Target Guarding Problem with Dynamics

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The topic of this paper is a two-player zero-sum differential game known as the target guarding problem. After a brief review of Isaacs’ original problem and solution, a problem with second-order dynamics and acceleration control is considered. It is shown that there are four solution classes satisfying the necessary conditions. The four classes are (i) abnormal and non-singular, (ii) normal and non-singular, (iii) normal and pursuer singular, (iv) normal and evader singular. The normal and totally singular case is ruled out. Closed-form solutions are provided for cases ii–iv. The order of singularity in all cases is infinite. Thus, the problem exhibits many interesting properties: normality, abnormality, non-singularity, infinite-order singularity, and non-uniqueness. A practical example of each class is provided.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Isaacs, R.: Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit. Wiley, New York (1965)

    MATH  Google Scholar 

  2. Leitmann, G., Rocklin, S.: The effect of playability in differential games on the relation between best guaranteed costs and saddlepoint values. J. Optim. Theory Appl. 15, 509–516 (1975)

    Article  MathSciNet  Google Scholar 

  3. Venkatesan, R.H., Sinha, N.K.: The target guarding problem revisited: some interesting revelations. In: Proceedings of the 19th World Congress, The International Federation of Automatic Control, pp. 1556–1561 (2014)

    Article  Google Scholar 

  4. Rusnak, I., Weiss, H., Hexner, G.: Guidance laws in target–missile–defender scenario with an aggressive defender. In: Proceedings of the 18th World Congress, The International Federation of Automatic Control, pp. 9349–9354 (2011)

    Article  Google Scholar 

  5. Rusnak, I.: The lady, the bandits, and the body guards—a two team dynamic game. In: Proceedings of the 16th World Congress, pp. 441–446 (2005)

  6. Lau, G., Liu, H.H.: Real-time path planning algorithm for autonomous border patrol: design, simulation, and experimentation. J. Intell. Robot. Syst. 75(3–4), 517–539 (2014)

    Article  Google Scholar 

  7. Li, D., Cruz, J.B.: A two-player stochastic pursuit-evasion differential game. In: Proceedings of the 46th IEEE Conference on Decision and Control, pp. 4057–4062 (2007)

  8. Sun, H., Li, M., Zhang, W.: Infinite time nonzero-sum linear quadratic stochastic differential games. In: Proceedings of the 29th Chinese Control Conference, pp. 1081–1084 (2010)

  9. Shima, T., Golan, O.: Bounded differential games guidance law for dual-controlled missiles. IEEE Trans. Control Syst. Technol. 14, 719–724 (2006)

    Article  Google Scholar 

  10. Li, D., Cruz, J.B., Chen, G., Kwan, C., Chang, M.: A hierarchical approach to multi-player pursuit-evasion differential games. In: Proceedings of the 44th IEEE Conference on Decision and Control, pp. 5674–5679 (2005)

  11. Ge, J., Tang, L., Reimann, J., Vachtesvanos, G.: Hierarchical decomposition approach for pursuit-evasion differential game with multiple players. In: 2006 IEEE Aerospace Conference (2006)

  12. Horie, K., Conway, B.: Genetic algorithm preprocessing for numerical solution of differential games problems. AIAA J. Guid. Control Dyn. 27(6), 1075–1078 (2004)

    Article  Google Scholar 

  13. Horie, K., Conway, B.: Optimal fighter pursuit-evasion maneuvers found via two-sided optimization. AIAA J. Guid. Control Dyn. 29(1), 105–112 (2006)

    Article  Google Scholar 

  14. Carr, R., Cobb, R., Pachter, M., Pierce, S.: Solution of a pursuit-evasion game using a near optimal strategy. AIAA J. Guid. Control Dyn. 41(4), 841–850 (2018)

    Article  Google Scholar 

  15. Raivio, T., Ehtamo, H.: Visual aircraft identification as a pursuit-evasion game. AIAA J. Guid. Control Dyn. 23(4), 701–708 (2000)

    Article  Google Scholar 

  16. Ehtamo, H., Raivio, T.: On applied nonlinear and bilevel programming for pursuit-evasion games. J. Optim. Theory Appl. 108(1), 65–96 (2001)

    Article  MathSciNet  Google Scholar 

  17. Johnson, P.A.: Numerical solution methods for differential game problems. Master’s Thesis, Massachusetts Institute of Technology (2009)

  18. Harris, M., Acikmese, B.: Lossless convexification of non-convex optimal control problems for state constrained linear systems. Automatica 50, 2304–2311 (2014)

    Article  MathSciNet  Google Scholar 

  19. Harris, M., Acikmese, B.: Maximum divert for planetary landing using convex optimization. J. Optim. Theory Appl. 162(3), 975–995 (2014)

    Article  MathSciNet  Google Scholar 

  20. Berkovitz, L.: A variational approach to differential games. Technical Report, The RAND Corporation, p. 2205 (1960)

  21. Berkovitz, L.: Necessary conditions for optimal strategies in a class of differential games and control problems. SIAM J. Control 5(1), 1–24 (1967)

    Article  MathSciNet  Google Scholar 

  22. Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., Mischenko, E.: The Mathematical Theory of Optimal Processes. Interscience Publishers, Geneva (1962)

    Google Scholar 

  23. Liberzon, D.: Calculus of Variations and Optimal Control Theory. Princeton University Press, Princeton (2012)

    Book  Google Scholar 

  24. McShane, E.: On multipliers for lagrange problems. Am. J. Math. 61, 809–819 (1939)

    Article  MathSciNet  Google Scholar 

  25. Kelley, H.: A second variation test for singular extremals. AIAA J. 2, 1380–1382 (1964)

    Article  MathSciNet  Google Scholar 

  26. Kopp, R., Moyer, H.: Necessary conditions for singular extremals. AIAA J. 3, 1439–1444 (1965)

    Article  Google Scholar 

  27. Goh, B.: Necessary conditions for singular extremals involving multiple control variables. SIAM J. Control 4(4), 716–731 (1966)

    Article  MathSciNet  Google Scholar 

  28. Robbins, H.: A generalized Legendre–Clebsch condition for singular cases of optimal control. IBM J. 11, 361–372 (1967)

    Article  Google Scholar 

  29. Krener, A.: The high order maximal principle and its application to singular extremals. SIAM J. Control Optim. 15, 256–293 (1977)

    Article  MathSciNet  Google Scholar 

  30. Bortins, R., Boustany, N., Powers, W.: The infinite-order singular problem. Optim. Control Appl. Methods 1(3), 279–302 (1980)

    Article  MathSciNet  Google Scholar 

  31. Powers, W.: On the order of singular optimal control problems. J. Optim. Theory Appl. 32(4), 479–489 (1980)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

On the occasion of his retirement as editor of the Journal of Optimization Theory and Applications, I thank Dr. David G. Hull for his service to the journal and The University of Texas. More importantly, I thank him for his personal mentorship and friendship, and I wish him the best.

Author information

Authors and Affiliations

  1. Mechanical and Aerospace Engineering, Utah State University, Logan, USA

    Matthew W. Harris

Authors
  1. Matthew W. Harris
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthew W. Harris.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Harris, M.W. Abnormal and Singular Solutions in the Target Guarding Problem with Dynamics. J Optim Theory Appl 184, 627–643 (2020). https://doi.org/10.1007/s10957-019-01597-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-019-01597-6

Keywords

Search

Navigation