Pursuit Problem of Low-Maneuverable Objects with a Ring-Shape Terminal Set


In this paper, we consider a pursuit problem for two moving objects, a pursuer and an evader. The objects move in the same plane under the influence of controlled forces directed always perpendicular to their velocities. The laws of variation of these forces are determined by first-order controllers. The capture is determined by the condition that the relative distance between the objects belongs to a segment with positive endpoints. For the problem considered, we construct a control of the pursuer that guarantees the capture.

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


  1. 1.

    G. M. Fikhtengol’tz, A Course of Differential and Integral Calculus [in Russian], Vol. 1, Fizmatlit, Moscow (2001).

  2. 2.

    S. A. Ganebny, S. S. Kumkov, S. Le Menec, and V. S. Patsko, “Model problem in a line with two pursuers and one evader,” Dyn. Games Appl., 2, No. 2, 228–257 (2012).

  3. 3.

    N. N. Krasovsky and A. I. Subbotin, Positional Differential Games [in Russian], Nauka, Moscow (1974).

  4. 4.

    S. S. Kumkov, S. LeMenec, and V. S. Patsko, “Solvability sets in a pursuit game with two pursuers and one evader,” Tr. Inst. Mat. Mekh. Ural. Otdel. Ross. Akad. Nauk, 20, No. 3, 148–165 (2014).

    Google Scholar 

  5. 5.

    T. Shima and J. Shinar, “Time-varying linear pursuit-evasion game models with bounded controls,” J. Guid. Control Dynam., 25, No. 3, 425–432 (2002).

    Article  Google Scholar 

  6. 6.

    J. Shinar, “Solution techniques for realistic pursuit-evasion games,” Adv. Control Dynam. Syst., 17, 63–124 (1981).

    Article  Google Scholar 

  7. 7.

    J. Shinar, M. Medinah, and M. Biton, “Singular surface in a linear pursuit-evasion game with elliptical vectograms,” J. Optim. Theory Appl., 43, No. 3, 431–456 (1984).

    MathSciNet  Article  Google Scholar 

  8. 8.

    J. Shinar and M. Zarkh, “Pursuit of a faster evader—a linear game with elliptical vectograms,” in: Proc. 7th Int. Symp. on Dynamic Games, Yokosuka, Japan (1996), pp. 855–868.

  9. 9.

    V. Turetsky and V. Y. Glizer, “Continuous feedback control strategy with maximal capture zone in a class of pursuit games,” Int. Game Theory Rev., 7, No. 1, 1–24 (2005).

    MathSciNet  Article  Google Scholar 

  10. 10.

    V. I. Ukhobotov and I. V. Izmest’ev, “Differential games with a ring-shape terminal set,” in: Proc. Int. Conf. “System Dynamics and Control Processes,” September 15–20, 2014 [in Russian], Ekaterinburg (2015), pp. 325–332.

Download references

Author information



Corresponding author

Correspondence to I. V. Izmest’ev.

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 148, Proceedings of the International Conference “Geometric Methods in Control Theory and Mathematical Physics: Differential Equations, Integrability, and Qualitative Theory” (Ryazan, September 15–18, 2016), 2018.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Izmest’ev, I.V., Ukhobotov, V.I. Pursuit Problem of Low-Maneuverable Objects with a Ring-Shape Terminal Set. J Math Sci 248, 397–403 (2020). https://doi.org/10.1007/s10958-020-04880-4

Download citation

Keywords and phrases

  • pursuit problem
  • control
  • terminal set

AMS Subject Classification

  • 49N75
  • 91A23