The issues related to the relaxation of a game problem of approach on a finite time interval are considered. In the original problem, it is assumed that the following sets are given: a target set closed in the position space and a set that determines state constraints and whose sections corresponding to fixed times are closed in the state space. The game termination conditions are relaxed by replacing these sets with their neighborhoods defined in different topologies of the position space; the “sizes” of the neighborhoods are related by a proportionality coefficient in the form of a priority parameter. For each value of this parameter and a fixed position, we find the value of the relaxed problem, which coincides with the minimax in the class of quasi-strategies for a special quality functional. It is established that the resulting position function depends on the parameter continuously as a mapping of the positive semiaxis to the Tikhonov power of the real line with the position space as the index set. Regions of uniform continuity are specified for the corresponding calculation functions (for a fixed position).
Keywords: differential game, quasi-strategy, program iteration method
Received January 18, 2021
Revised January 29, 2021
Accepted February 1, 2021
Alexander Georgievich Chentsov, Dr. Phys.-Math. Sci, RAS Corresponding Member, Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: chentsov@imm.uran.ru
REFERENCES
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. N Y: 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. Pontryagin L.S. Linear differential games. I, II. Dokl. Akad. Nauk SSSR, 1967, vol. 174, no. 6, pp. 1278–1280; vol. 175, no. 4, pp. 764–766 (in Russian).
4. Pontryagin L.S. The mathematical theory of optimal processes and differential games. Trudy Mat. Inst. Steklov, 1985, vol. 169, pp. 119–158 (in Russian).
5. Krasovskii N.N. Teoriya upravleniya dvizheniem [Theory of motion control]. Moscow: Nauka Publ., 1968, 476 p.
6. Krasovskii N.N. Igrovye zadachi o vstreche dvizhenii [Game problems on the encounter of motions]. Moscow: Nauka Publ., 1970, 420 p.
7. Krasovskii N.N. Upravlenie dinamicheskoi sistemoi [Control of a dynamical system]. Moscow: Nauka Publ., 1985, 516 p.
8. Kurzhanskii A.B. Upravlenie i nablyudenie v usloviyakh neopredelennosti [Control and observation under the conditions of uncertainty]. Moscow: Nauka Publ., 1977, 392 p.
9. Osipov Yu.S. An alternative in a differential-difference game. Dokl. Akad. Nauk SSSR, 1971, vol. 197, no. 5, pp. 1022–1025 (in Russian).
10. Osipov Yu.S. On the theory of differential games in distributed parameter systems. Dokl. Akad. Nauk SSSR, 1975, vol. 223, no. 6, pp. 1314–1317 (in Russian).
11. Pshenichnyi B.N. The structure of differential games. Dokl. Akad. Nauk SSSR, 1969, vol. 184, no. 2, pp. 285–287 (in Russian).
12. Subbotin A.I. A generalization of the basic equation of the theory of differential games. Dokl. Akad. Nauk SSSR, 1980, vol. 254, no. 2, pp. 293–297 (in Russian).
13. Subbotin A.I. Generalized solutions of first-order PDEs. The dynamical optimization perspective. Basel: Birkh$\ddot{\mathrm{a}}$user, 1995, 314 p. doi: 10.1007/978-1-4612-0847-1 . Translated to Russian under the title Obobshchennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka: Perspektivy dinamicheskoi optimizatsii. Moscow; Izhevsk: Inst. Komp’yuter. Issled. Publ., 2003, 336 p.
14. Subbotin A.I., Chentsov A.G. Optimizatsiya garantii v zadachakh upravleniya [Guarantee optimization in control problems]. Moscow: Nauka Publ., 1981, 288 p.
15. Kryazhimskii A.V. On the theory of positional differential games of approach-evasion. Soviet Math. Dokl., 1978, vol. 19, no. 2, pp. 408–412.
16. Isaacs R. Differential games. N Y: John Wiley and Sons, 1965, 384 p. ISBN: 0471428604 . Translated to Russian under the title Differentsial’nye igry. Moscow: Mir Publ., 1967, 480 p.
17. Krasovskii N.N. A differential game of approach and evasion. I. Engineering Cybernetics, 1973, vol. 11, no. 2, pp. 189–203; 1973, vol. 11, no. 3, pp. 376–394.
18. Chentsov A.G. The structure of a certain game-theoretic approach problem. Soviet Math. Dokl., 1975, vol. 16, no. 5, pp. 1404–1408.
19. Chentsov A. On a game problem of guidance. Soviet. Math., Dokl., 1976, vol. 17, pp. 73–77.
20. 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
21. Chistyakov S.V. On solutions for game problems of pursuit. Prikl. Mat. Mekh., 1977, vol. 41, no. 5, pp. 825–832 (in Russian).
22. 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
23. Chentsov A.G. The program iteration method in a game problem of guidance. Proc. Steklov Inst. Math., 2017, vol. 297, no. 1, pp. 43–61. doi: 10.1134/S0081543817050066
24. Chentsov A.G. Some questions of differential game theory with phase constraints. Izv. IMI UdGU, 2020, vol. 56, pp. 138–184 (in Russian). doi: 10.35634/2226-3594-2020-56-10
25. Kuratowski K., Mostowski A. Set theory. Warszawa: PWN - Polish Scientific Publishers, 1968, 417 p. ISBN: 9780444534170 . Translated to Russian under the title Teoriya mnozhestv. Moscow: Mir Publ., 1970, 416 p.
26. Dieudonn$\acute{\mathrm{e}}$ J. Foundations of modern analysis. N Y: Acad. Press, 1960, 361 p. Translated to Russian under the title Osnovy sovremennogo analiza. Moscow: Mir Publ., 1964, 430 p.
27. Chentsov A.G. Elementy konechno-additivnoi teorii mery, I [Elements of finitely additive measure theory, I]. Ekaterinburg: USTU-UPI Publ., 2008, 388 p.
28. 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.
29. Dunford N., Schwartz J.T. Linear Operators. I. General Theory. New York: Interscience Publishers, 1958, 858 p. ISBN: 0470226056 . Translated to Russian under the title Lineinye operatory. Obshchaya teoriya. Moscow: Inostr. Lit. Publ., 1962, 896 p.
30. 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.
31. 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
32. 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
33. Chentsov A.G. The programmed iterations method in a game problem of guidance. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2016, vol. 26, no. 2, pp. 271–282 (in Russian). doi: 10.20537/vm160213
34. Chentsov A.G. Relaxation of the game problem of guidance connected with alternative in guidance-evasion differential game. Russian Universities Reports. Mathematics, 2020, vol. 25, no. 130, pp. 196–244 (in Russian). doi: 10.20310/2686-9667-2020-25-130-196-244
Cite this article as: A.G. Chentsov. On the relaxation of a game problem of approach with priority elements, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 2, pp. 281–297; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 317, Suppl. 1, pp. S55–S70.
