A.G. Chentsov, D.M. Khachai. Relaxation of the pursuit-evasion differential game and iterative methods ... P. 246-268

A variant of the program iteration method called stability iterations is used for a differential game of pursuit-evasion. The successful solvability set of one of the problems generating the game is found as a limit of the iterative procedure in the space of sets whose elements are positions of the game. The game is defined by a pair of closed sets, one of the which is the objective set in the pursuit problem (the first player's problem) and the other specifies the state constraints in this problem. For the positions not belonging to the solvability set of the pursuit problem, it is interesting to determine the smallest ''size'' of a neighborhood of the two mentioned sets for which the first player can implement the guidance to the neighborhood of the objective set corresponding to this ''size'' within the similar neighborhood of the second set, i.e., the set specifying the state constraints. Similar constructions are considered for the sets realized at each stage of the iterative procedure. We use the connection of these constructions with the mentioned smallest ''size'' of neighborhoods of the sets that are parameters of the differential game in the sense of guaranteed realizability of guidance under the replacement of the original sets by these neighborhoods.

Keywords: differential game of pursuit–evasion, program iteration method, guaranteed guidance

Received September 24, 2018

Revised November 08, 2018

Accepted November 12, 2018

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 16-01-00505).

Alexander Georgievich Chentsov, Dr. Phys.-Math. Sci, RAS Corresponding Member, Prof., Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia; Ural Federal University, Yekaterinburg, 620002 Russia,
e-mail: chentsov@imm.uran.ru

Daniil Mikhailovich Khachai, graduate student, Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, 620990 Russia; Ural Federal University, Ekaterinburg, 620002 Russia, e-mail: dmx@imm.uran.ru


1.   Krasovskii N.N., Subbotin A.I. An alternative for the game problem of convergence. J. Appl. Math. Mech., 1970, vol. 34, no. 6, pp. 948–965. doi: 10.1016/0021-8928(70)90158-9

2.   Krasovskii N.N., Subbotin A.I. Game-theoretical control problems. New York, Springer, 1988, 517 p. ISBN: 978-1-4612-8318-8 . Original Russian text published in Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial’nye igry. Moscow, Nauka Publ., 1974, 456 p.

3.   Isaacs R. Differential games. N Y, John Wiley and Sons, 1965, 384 p. ISBN: 0471428604 .

4.   Berkovitz L.D. Differential games of generalized pursuit and evasion. Appl. Math. Optim., 1988, vol. 17, no. 1, pp. 177–183. doi: 10.1007/BF01448365

5.   Elliott R.J., Kalton N.J. Values in differential games. Bull. Amer. Math. Soc., 1972, vol. 78, no. 3, pp. 427–431. doi: 10.1090/S0002-9904-1972-12929-X

6.   Chentsov A.G. On a game problem of guidance with information memory. Sov. Math., Dokl., 1976, vol. 17, pp. 411–414.

7.   Chentsov A.G. On a game problem of converging at a given instant of time. Math. USSR-Sb., 1976, vol. 28, no. 3, pp. 353–376. doi: 10.1070/SM1976v028n03ABEH001657

8.   Subbotin A.I., Chentsov A.G. Optimizatsiya garantii v zadachakh upravleniya [Optimization of guarantee in control problems]. Moscow, Nauka Publ., 1981, 288 p.

9.   Ushakov V.N., Ershov A.A. On the solution of control problems with fixed terminal time. Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki, 2016, vol. 26, no. 4, pp. 543–564 (in Russian). doi: 10.20537/vm160409

10.   Ushakov V.N., Matviychuk A.R. To solution of control problems of nonlinear systems on a finite time interval. Izv. IMI UdGU, 2015, no. 2 (46), pp. 202–215 (in Russian).

11.   Ushakov V.N., Ukhobotov V.I., Ushakov A.V., Parshikov G.V. On solving approach problems for control systems. Proc. Steklov Inst. Math., 2015, vol. 291, no. 1, pp. 263–278. doi: 10.1134/S0081543815080210

12.   Krasovskii N.N. A differential game of approach and evasion. I. Engrg. Cybernetics, 1973, vol. 11, no. 2, pp. 189–203.

13.   Krasovskii N.N. A differential game of approach and evasion. II. Engrg. Cybernetics, 1973, vol. 11, no. 3, pp. 376–394.

14.   Chentsov A.G. The structure of a certain game-theoretic approach problem. Soviet Math. Dokl., 1975, vol. 16, no. 5, pp. 1404–1408.

15.   Chistyakov S.V. On solutions for game problems of pursuit. Prikl. Mat. Mekh., 1977, vol. 41, no. 5, pp. 825–832 (in Russian).

16.   Ukhobotov V.I. Construction of a stable bridge for a class of linear games. J. Appl. Math. Mech., 1977, vol. 41, no. 2, pp. 350–354. doi: 10.1016/0021-8928(77)90021-1

17.   Chentsov A.G. Metod programmnykh iteratsii dlya differentsial’noi igry sblizheniya-ukloneniya [The method of program iterations for a differential approach-evasion game]. Sverdlovsk, 1979. Available from VINITI, no. 1933-79, 102 p.

18.   Chentsov A.G. Stability iterations and an evasion problem with a constraint on the number of switchings. Trudy Inst. Mat. i Mekh. UrO RAN, 2017, vol. 23, no. 2, pp. 285–302 (in Russian). doi: 10.21538/0134-4889-2017-23-2-285-302

19.   Chentsov A.G. O zadache upravleniya s ogranichennym chislom pereklyuchenii [On a control problem with a bounded number of switchings]. Sverdlovsk, 1987. Available from VINITI, no. 4942-B87, 44 p.

20.   Chentsov A.G. O differentsial’nykh igrakh s ogranicheniem na chislo korrektsii, 2 [On differential games with restriction on the number of corrections,   2]. Sverdlovsk, 1980. Available from VINITI, no. 5406-80, 55 p.

21.   Neveu J. Mathematical foundations of the calculus of probability. San Francisco: Holden-Day, 1965, 223 p. Translated to Russian under the title Matematicheskie osnovy teorii veroyatnostei. Moscow: Mir Publ., 1969, 309 p.

22.   Billingsley P. Convergence of probability measures. N Y: Wiley, 1968, 253 p. ISBN: 0471072427. Translated to Russian under the title Skhodimost’ veroyatnostnykh mer. Moscow: Nauka Publ., 1977, 352 p.

23.   Dieudonn$\acute{\mathrm{e}}$ J. Foundations of modern analysis. N Y: Acad. Press Inc., 1960, 361 p. Translated to Russian under the title Osnovy sovremennogo analiza. Moscow: Mir Publ., 1964, 430 p.

24.   Kryazhimskii A.V. On the theory of positional differential games of approach-evasion. Soviet Math. Dokl., 1978, vol. 19, no. 2, pp. 408–412.

25.   Chentsov A.G. The program iteration method in a game problem of guidance. Proc. Steklov Inst. Math., 2017, vol. 297, suppl. 1, pp. 43–61. doi: 10.1134/S0081543817050066

26.   Chentsov A.G. On the game problem of convergence at a given moment of time. Math. USSR-Izv., 1978, vol. 12, no. 2, pp. 426–437. doi: 10.1070/IM1978v012n02ABEH001985

27.   Chistyakov S.V., Nikitin F.F. On zero-sum two-person differential games of infinite duration. Vestn. St. Petersbg. Univ., Math., 2004, vol. 37, no. 3, pp. 28–32.

28.   Chistyakov S.V. Programmed iterations and universal ε-optimal stratagies in a positional differential game. Soviet Math. Dokl., 1992, vol. 44, no. 1, pp. 354–357.

29.   Chistiakov S.V. On functional equations in games of encounter at a prescribed instant. J. Appl. Math. Mech., 1982, vol. 46, no. 5, pp. 704–706. doi: 10.1016/0021-8928(82)90023-5

30.   Chentsov A.G. Iterations of stability and the evasion problem with a constraint on the number of switchings of the formed control. Izv. IMI UdGU, 2017, vol. 49, pp. 17–54 (in Russian). doi: 10.20537/2226-3594-2017-49-02