Skip to main content
SpringerLink
Log in

Computation of Solvability Set for Differential Games in the Plane with Simple Motion and Non-convex Terminal Set

  • Published:
Dynamic Games and Applications Aims and scope Submit manuscript

Abstract

The paper suggests an algorithm for an exact construction of solvability set in a differential game with simple motion in the plane, with a fixed terminal time and a polygonal (in the general case, non-convex) terminal set. Some examples of solvability sets are computed.

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
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Aubin JP (1991) Viability theory. Birkhäuser, Boston

    MATH  Google Scholar 

  2. Bardi M, Capuzzo-Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Birkhäuser, Boston

    Book  MATH  Google Scholar 

  3. Bardi M, Evans LC (1984) On Hopf’s formulas for solutions of HJ equations. Nonlinear Anal 8:1373–1381

    Article  MathSciNet  MATH  Google Scholar 

  4. Botkin ND, Ryazantseva EA (1992) An algorithm of constructing the solvability set for linear differential games of high dimension. Trudy Inst Mat i Mekh UrO RAN 2:128–134 (in Russian)

    MATH  Google Scholar 

  5. Bryson A, Ho YC (1969) Applied optimal control: optimization, estimation, and control. Blaisdell Publishing Company, Waltham

    Google Scholar 

  6. Cardaliaguet P, Quincampoix M, Saint-Pierre P (1999) Set-valued numerical analysis for optimal control and differential games. In: Bardi M, Raghavan TES, Parthasarathy T (eds) Stochastic and differential games: theory and numerical methods, Annals of the International Society of Dynamic Games, vol 4. Birkhäuser, Boston, pp 177–247

    Chapter  Google Scholar 

  7. Chen M, Tomlin CJ (2018) Hamilton–Jacobi reachability: some recent theoretical advances and applications in unmanned airspace management. Annu Rev Control Robot Auton Syst 1:333–358

    Article  Google Scholar 

  8. Ganebny SA, Kumkov SS, Le Menec S, Patsko VS (2012) Study of linear game with two pursuers and one evader: different strength of pursuers. In: Cardaliaguet P, Cressman R (eds) Advances in dynamic games. Annals of the International Society of Dynamic Games, vol 12. Birkhäuser, Boston, pp 269–292

    Google Scholar 

  9. Hadwiger H (1957) Vorlesungen uber Inhalt, Oberflache und Isoperimetrie. Springer, Berlin

    Book  MATH  Google Scholar 

  10. Hopf E (1965) Generalized solutions of nonlinear equations of first order. J Math Mech 14(6):951–973

    MathSciNet  MATH  Google Scholar 

  11. Isaacs R (1965) Differential games. Wiley, New York

    MATH  Google Scholar 

  12. Isakova E, Logunova G, Patsko V (1984) Construction of stable bridges in a linear differential game with fixed terminal time. In: Subbotin AI, Patsko VS (eds) Algorithms and programs of solution of linear differential games: collection of scientific works. Inst. Mat. Mekh, Sverdlovsk, pp 127–158 (in Russian)

    Google Scholar 

  13. Kamneva LV, Patsko VS (2016) Maximal stable bridge in game with simple motions in the plane. In: Thuijsman F, Wagener F (eds) Advances in dynamic and evolutionary games: theory, applications, and numerical methods, Annals of the International Society of Dynamic Games, vol 14. Birkhäuser, Basel, pp 139–163

    Chapter  Google Scholar 

  14. Krasovskii NN (1970) Game problems on the encounter of motions. Nauka, Moscow (in Russian)

    MATH  Google Scholar 

  15. Krasovskii NN, Subbotin AI (1974) Positional differential games. Nauka, Moscow (in Russian)

    MATH  Google Scholar 

  16. Krasovskii NN, Subbotin AI (1988) Game-theoretical control problems. Springer, New York

    Book  Google Scholar 

  17. Kumkov SS, Le Menec S, Patsko VS (2017) Zero-sum pursuit-evasion differential games with many objects: survey of publications. Dyn Games Appl 7(4):609–633

    Article  MathSciNet  MATH  Google Scholar 

  18. Margellos K, Lygeros J (2011) Hamilton–Jacobi formulation for reach-avoid differential games. IEEE Trans Autom Control 56(8):1849–1861

    Article  MathSciNet  MATH  Google Scholar 

  19. Patsko VS (1996) Special aspects of convex hull constructing in linear differential games of small dimension. IFAC Proc Vol 29(8):19–24

    Article  Google Scholar 

  20. Pontryagin LS (1967) Linear differential games, II. Soviet Math Dokl 8:910–912

    MATH  Google Scholar 

  21. Pshenichnyy BN, Sagaydak MI (1971) Differential games with fixed time. J Cybern 1(1):117–135

    Article  MathSciNet  Google Scholar 

  22. Subbotin AI (1988) A piecewise-linear cost function for a differential game with simple moves. Optimal control and differential games, Trudy Mat. Inst. Steklov., vol 185. Nauka, Moscow, pp 242–251 (Russian); Proceedings of the Steklov Inst Math 185(2): 269–278 (1990)

  23. Subbotin AI (1995) Generalized solutions of first-order PDEs: the dynamical optimization perspective. Birkhäuser, Boston

    Book  Google Scholar 

  24. Zarkh MA, Patsko VS (1987) Numerical solution of a third-order directed game. Izv Akad Nauk SSSR Tekhn Kibernet 6: 162–169 (Russian); Soviet J Comput Syst Sci 26(4): 92–99 (1988)

  25. Zarkh MA, Ivanov AG (1992) Construction of the value function in the linear differential game with the fixed terminal time. Trudy Inst Mat i Mekh UrO RAN 2:140–155 (in Russian)

    MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the Russian Foundation for Basic Research under Project No. 18-01-00410.

Author information

Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics, 16 S.Kovalevskaya str., Yekaterinburg, Russia, 620108

    Liudmila Kamneva

Authors
  1. Liudmila Kamneva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liudmila Kamneva.

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

Kamneva, L. Computation of Solvability Set for Differential Games in the Plane with Simple Motion and Non-convex Terminal Set. Dyn Games Appl 9, 724–750 (2019). https://doi.org/10.1007/s13235-018-00292-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13235-018-00292-x

Keywords

Search

Navigation