Bahrom T. Samatov     (Namangan State University, 316 Uychi Str., Namangan, 116019, Uzbekistan)
Ulmasjon B. Soyibboev     (Namangan State University, 316 Uychi Str., Namangan, 116019, Uzbekistan)


In this paper, we study the well-known problem of Isaacs called the "Life line" game when movements of players occur by acceleration vectors, that is, by inertia in Euclidean space. To solve this problem, we investigate the dynamics of the attainability domain of an evader through finding solvability conditions of the pursuit-evasion problems in favor of a pursuer or an evader. Here a pursuit problem is solved by a parallel pursuit strategy. To solve an evasion problem, we propose a strategy for the evader and show that the evasion is possible from given initial positions of players. Note that this work develops and continues studies of Isaacs, Petrosjan, Pshenichnii, Azamov, and others performed for the case of players' movements without inertia.


Differential game, Pursuit, Evasion, Acceleration, Strategy, Attainability domain, Lifeline.

Full Text:



  1. 1. Alekseev V.M., Tikhomirov V.M., Fomin S.V. Optimal’noye upravleniye [Optimal Control]. Moscow: Nauka,  1979. 432 p. (in Russian)
  2. Aubin J.-P., Cellina A. Differential Inclusions. Set-Valued Maps and Viability Theory. Berlin-Heidelberg-New York-Tokyo: Springer-Verlag, 1984. 342 p. DOI: 10.1007/978-3-642-69512-4
  3. Azamov A. On the quality problem for simple pursuit games with constraint. Serdica Bulgariacae Math. Publ., 1986. Vol. 12, No. 1, P. 38–43. (in Russian) 
  4. Azamov A.A., Samatov B.T. The Π-strategy: analogies and applications. In: Proc. The Fourth Int. Conf. Game Theory and Management (GTM 2010), June 28-30, 2010, St. Petersburg, Russia, vol. 4. 2010. P. 33–47.
  5. Berkovitz L.D. Differential game of generalized pursuit and evasion. SIAM J. Contr., 1986. Vol. 24, No. 3. P. 361–373. DOI: 10.1137/0324021
  6. Blagodatskikh A.I., Petrov N.N. Konfliktnoe vzaimodejstvie grupp upravlyaemyh ob”ektov [Conflict Interaction Between Groups of Controlled Objects]. Izhevsk: Udmurt State University, 2009. 266 p. (in Russian)
  7. Blagodatskikh V.I. Vvedenie v optimal’noye upravleniye [Introduction to Optimal Control]. Moscow: Vysshaya Shkola, 2001. 239 p. (in Russian)
  8. Chikrii A.A. Conflict–Controlled Processes. Dordrecht: Kluwer Academic Publishers, 1997. 404 p. DOI: 10.1007/978-94-017-1135-7 
  9. Dar’in A.N., Kurzhanskii A.B. Control under indeterminacy and double constraints. Differ. Equ., 2003. Vol. 39, No. 11. P. 1554–1567. DOI: 10.1023/B:DIEQ.0000019347.24930.a3
  10. Elliot R.J., Kalton N.J. The existence of value in differential games of pursuit and evasion. J. Differential Equations, 1972. Vol. 12, No. 3. P. 504–523. DOI: 10.1016/0022-0396(72)90022-8
  11. Fleming W.H. The convergence problem for differential games, II. In: Advances in Game Theory, 2nd ed. Annals of Math., vol. 52. 1964. P. 195–210. DOI: 10.1515/9781400882014-013
  12. Friedman A. Differential Games, 2nd ed. Pure Appl. Math., vol. 25. New York: Wiley Int., 1971. 350 p.
  13. Grigorenko N.L. Matematicheskie metody upravleniya neskol’kimi dinamicheskimi processami [Mathematical Methods of Control for Several Dynamic Processes]. Moscow: Izdat. Gos. Univ., 1990. 198 p. (in Russian)
  14. Hajek O. Pursuit Games: An Introduction to the Theory and Applications of Differential Games of Pursuit and Evasion. Mineola, NY: Dove. Pub. Inc., 2008. 266 p. 
  15. Ho Y., Bryson A., Baron S. Differential games and optimal pursuit-evasion strategies. IEEE Trans. Autom. Control, 1965. Vol. 10, No. 4. P. 385–389. DOI: 10.1109/TAC.1965.1098197 
  16. Ibragimov G.I. A game of optimal pursuit of one object by several. J. Appl. Math. Mech., 1998. Vol. 62, No. 2, P. 187–192. DOI: 10.1016/S0021-8928(98)00024-0 
  17. Ibragimov G.I. Optimal pursuit with countably many pursuers and one evader. Differ. Equ., 2005. Vol. 41, No. 5. P. 627–635. DOI: 10.1007/s10625-005-0198-y
  18. Ibragimov G.I. The optimal pursuit problem reduced to an infinite system of differential equations. J. Appl. Math. Mech., 2013. Vol. 77, No. 5, P. 470–476. DOI: 10.1016/j.jappmathmech.2013.12.002
  19. Ibragimov G.I. Optimal pursuit time for a differential game in the Hilbert space \(l_2\) . Sci. Asia, 2013. Vol. 39S. P. 25–30. DOI: 10.2306/scienceasia1513-1874.2013.39S.025
  20. Isaacs R. Differential Games. NY: John Wiley and Sons, 1965. 385 p. 
  21. Ivanov R.P., Ledyayev Yu.S. Time optimality for the pursuit of several objects with simple motion in a differential game. Proc. Steklov Inst. Math., 1983. Vol. 158, P. 93–103.
  22. Kornev D.V., Lukoyanov N.Yu. On a minimax control problem for a positional functional under geometric and integral constraints on control actions. Proc. Steklov Inst. Math., 2016. Vol. 293, No. Supl. 1. P. S85–S100. DOI: 10.1134/S0081543816050096
  23. Krasovskii N.N., Subbotin A.I. Game-Theoretical Control Problems. NY: Springer, 2011. 517 p. 
  24. Munts N.V., Kumkov S.S. Numerical method for solving time-optimal differential games with lifeline. Mat. Teor. Igr. Pril., 2018. Vol. 10, No. 3. P. 48–75. (in Russian)
  25. Munts N.V., 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, No. Supl. 1. P. S125–S139. DOI: 10.1134/S0081543819040138
  26. Pang J.-S., Stewart D.E. Differential variational inequalities. Math. Program., 2008. Vol. 113, No. 2, Ser. A. P. 345–424. DOI: 10.1007/s10107-006-0052-x
  27. Pashkov A.G., Terekhov S.D. A differential game of approach with two pursuers and one evader. J. Optim. Theory Appl., 1987. Vol. 55, No. 2. P. 303–311. DOI: 10.1007/BF00939087
  28. Petrosjan L.A. Differential Games of Pursuit. Ser. Optim, vol. 2. Singapore: World Scientific Publ., 1993. 340 p. DOI: 10.1142/1670 
  29. Pontryagin L.S. Izbrannye trudy [Selected Works]. Moscow: MAKS Press, 2004. 551 p. (in Russian) 
  30. Pshenichnyi B.N. Simple pursuit by several objects. Cybernet. Systems Anal., 1976. Vol. 12, No. 5. P. 484–485.
  31. Pshenichnyi B.N., Chikrii A.A., Rappoport I.S. An efficient method for solving differential games with many pursuers. Dokl. Akad. Nauk SSSR, 1981. Vol. 256, No. 3. P. 530–535. (in Russian)
  32. Samatov B.T. On a pursuit-evasion problem under a linear change of the pursuer resource. Siberian Adv. Math., 2013. Vol. 23, No. 10. P. 294–302. DOI: 10.3103/S1055134413040056
  33. Samatov B.T. The pursuit-evasion problem under integral-geometric constraints on pursuer controls. Autom. Remote Control, 2013. Vol. 74, No. 7. P. 1072–1081. DOI: 10.1134/S0005117913070023
  34. Samatov B.T. The Π-strategy in a differential game with linear control constraints. J. Appl. Math. Mech., 2014. Vol. 78, No. 3. P. 258–263. DOI: 10.1016/j.jappmathmech.2014.09.008
  35. Samatov B.T. Problems of group pursuit with integral constraints on controls of the players. I. Cybernet. Systems Anal., 2013. Vol. 49, No. 5. P. 756–767. DOI: 10.1007/s10559-013-9563-7
  36. Samatov B.T. Problems of group pursuit with integral constraints on controls of the players. II. Cybernet. Systems Anal., 2013. Vol. 49, No. 6. 907–921. DOI: 10.1007/s10559-013-9581-5
  37. Samatov B.T., Ibragimov G., Khodjibayeva I.V. Pursuit-evasion differential games with Grönwall-type constraints on controls. Ural Math. J., 2020. Vol. 6, No. 2. P. 95–107. DOI: 10.15826/umj.2020.2.010
  38.  Satimov N.Yu., Rikhsiev B.B., Khamdamov A.A. On a pursuit problem for n-person linear differential and discrete games with integral constraints. Math. USSR Sb., 1983. Vol. 46, No. 4. P. 459–471. DOI: 10.1070/SM1983v046n04ABEH002946
  39. Shiyuan J., Zhihua Q. Pursuit-evasion games with multi-pursuer vs. One fast evader. In: Proc. 8th World Congress on Intelligent Control and Automation, July 7–9, 2010, Jinan, China. IEEE Xplore, 2010. P. 3184–3189. DOI: 10.1109/WCICA.2010.5553770 
  40. Subbotin A.I. Generalization of the main equation of differential game theory. J. Optim. Theory Appl., 1984. Vol. 43, No. 1. P. 103–133. DOI: 10.1007/BF00934749
  41. Subbotin A.I., Chentsov A.G. Optimizaciya garantii v zadachah upravleniya [Optimization of Guaranteed Result in Control Problems]. M: Nauka, 1981. 288 p. (in Russian)
  42. Ushakov V.N. Extremal strategies in differential games with integral constraints. J. Appl. Math. Mech., 1972. Vol. 36, No. 1. P. 12–19. DOI: 10.1016/0021-8928(72)90076-7
  43. Ushakov V.N., Ershov A.A., Ushakov A.V., Kuvshinov O.A. Control system depending on a parameter. Ural Math. J., 2021. Vol. 7, No. 1. P. 120–159. DOI: 10.15826/umj.2021.1.011

DOI: http://dx.doi.org/10.15826/umj.2021.2.007

Article Metrics

Metrics Loading ...


  • There are currently no refbacks.