Skip to main content
SpringerLink
Log in

A Numerical Method for Solving Time-Optimal Differential Games with a Lifeline

  • mathematical game theory and applications
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

Time-optimal differential games with a lifeline are considered. In such games, there are two sets of interest: the first player tries to guide the system into a target set as soon as possible, while the second player counteracts him and wins if the system reaches another set (called the lifeline). A numerical method for solving time-optimal games with a lifeline is suggested. With this method, the value function is designed as a viscosity solution to the corresponding boundary-value problem for the Hamilton–Jacobi equation. The convergence of the method is established.

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.

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

Similar content being viewed by others

References

  1. Isaacs, R.Differential Games. A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, New York: Wiley, 1965. Translated under the title Differentsial’nye igry, Moscow: Mir, 1967.

  2. Dutkevich, Yu. G. & Petrosjan, L. A. Games with a Life-line. The Case of l-capture. SIAM J. Control 10(no. 1), 40–47 (1972).

    Article  MathSciNet  Google Scholar 

  3. Krasovskii, N. N. & Subbotin, A. I. Pozitsionnye differentsialanye igry (Positional Differential Games). (Nauka, Moscow, 1974).

    Google Scholar 

  4. Munts, N.V. and Kumkov, S.S., On the Coincidence of the Minimax Solution and the Value Function in a Time-Optimal Game with a Lifeline, Proc. Steklov Inst. Math., 2019, vol. 305, suppl. 1, S125-S139.

  5. Petrosjan, L. A. Dispersion Surfaces in One Family of Pursuit Games. Dokl. Akad. Nauk Arm. SSR 43(no. 4), 193–197 (1966).

    Google Scholar 

  6. Petrosjan, L. A. Differential Games of Pursuit. (World Scientific, Singapore, 1993).

    Book  Google Scholar 

  7. Subbotin, A. I. Generalized Solutions of First Order PDEs: The Dynamical Optimization Perspective. (Birkhauser, Boston, 1995).

    Book  Google Scholar 

  8. Bardi, M. & Capuzzo-Dolcetta, I. Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. (Birkhäuser, Boston, 1997).

    Book  Google Scholar 

  9. Bardi, M., Falcone, M., and Soravia, P.Numerical Methods for Pursuit-Evasion Games via Viscosity Solutions, in Stochastic and Differential Games: Theory and Numerical Methods: Annals of Int. Society on Dynamic Games, Boston: Birkhäuser, 1999, vol. 4, pp. 105–175.

  10. Barles, G. & Perthame, B. Discontinuous Solutions of Deterministic Optimal Stopping Time Problems. RAIRO Model. Math. Anal. no. 21, 557–579 (1987).

    Article  MathSciNet  Google Scholar 

  11. Barles, G. Solutions de viscosite des equations de Hamilton-Jacobi, in Mathematiques et Applications 17 (Springer, Paris, 1994).

    MATH  Google Scholar 

  12. Cardaliaguet, P., Quincampoix, M. & Saint-Pierre, P. Some Algorithms for Differential Games with Two Players and One Target. RAIRO-Modélis. Math. Anal. Numér. 28(no. 4), 441–461 (1994).

    Article  MathSciNet  Google Scholar 

  13. Cardaliaguet, P., Quincampoix, M. & Saint-Pierre, P. Set-Valued Numerical Analysis for Optimal Control and Differential Games. Ann. Int. Soc. Dynamic Games 4, 177–247 (1999).

    MathSciNet  MATH  Google Scholar 

  14. Cardaliaguet, P., Quincampoix, M. & Saint-Pierre, P. Pursuit Differential Games with State Constraints. SIAM J. Control Optimiz. 39(no. 5), 1615–1632 (2001).

    Article  MathSciNet  Google Scholar 

  15. Cardaliaguet, P., Quincampoix, M. & Saint-Pierre, P. Differential Games Through Viability Theory: Old and Recent Results. Ann. Int. Soc. Dynamic Games 9, 3–35 (2007).

    Article  MathSciNet  Google Scholar 

  16. Crandall, M. G., Evans, L. C. & Lions, P.-L. Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. Trans. Am. Math. Soc. 282(no. 2), 487–502 (1984).

    Article  MathSciNet  Google Scholar 

  17. Crandall, M. G., Ishii, H. & Lions, P. L. Useras Guide to Viscosity Solutions of Second Order Partial Differential Equations. Bull. Am. Math. Soc. no. 27, 1–67 (1992).

    Article  Google Scholar 

  18. Isaacs, R. Differential Games. (Wiley, New York, 1965).

    MATH  Google Scholar 

  19. Krasovskii, N. N. & Subbotin, A. I. Game-Theoretical Control Problems. (Springer-Verlag, New York, 1988).

    Book  Google Scholar 

  20. Munts, N.V. and Kumkov, S.S.The Existence of the Value Function in a Time-Optimal Game with Lifeline, Proc. 47th Int. Youth School-Conf. "Modern Problems in Mathematics and Its Applications,” Yekaterinburg, 2016, pp. 94–99.

  21. Patsko, V. S. & Turova, V. L. Level Sets of the Value Function in Differential Games with the Homicidal Chauffeur Dynamics. Int. Game Theory Rev. 3(no. 1), 67–112 (2001).

    Article  MathSciNet  Google Scholar 

  22. Patsko, V.S. and Turova, V.L.Numerical Study of Differential Games with the Homicidal Chauffeur Dynamics, Yekaterinburg: Inst. Mat. Mekh., 2000. http://sector3.imm.uran.ru/preprint2000_01/ preprint2000.pdf.

  23. Petrosjan, L. A. A Family of Differential Survival Games in the Space \({{\mathbb{R}}}^{n}\). Soviet Math. Dokl. no. 6, 377–380 (1965).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Saint Petersburg, Russia

    N.V. Munts & S.S. Kumkov

Authors
  1. N.V. Munts
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S.S. Kumkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Munts, N., Kumkov, S. A Numerical Method for Solving Time-Optimal Differential Games with a Lifeline. Autom Remote Control 81, 1545–1561 (2020). https://doi.org/10.1134/S0005117920080159

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117920080159

Keywords

Search

Navigation