Geodesic
An algorithm for the numerical solution of linear differential games
Sbornik. Mathematics, Tome 192 (2001) no. 10, pp. 1515-1542 
E. S. Polovinkin; G. E. Ivanov; M. V. Balashov; R. V. Konstantinov; A. V. Khorev. An algorithm for the numerical solution of linear differential games. Sbornik. Mathematics, Tome 192 (2001) no. 10, pp. 1515-1542. http://geodesic.mathdoc.fr/item/SM_2001_192_10_a5/
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2001_192_10_a5/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

A numerical algorithm for the construction of stable Krasovskii bridges, Pontryagin alternating sets, and also of piecewise program strategies solving two-person linear differential (pursuit or evasion) games on a fixed time interval is developed on the basis of a general theory. The aim of the first player (the pursuer) is to hit a prescribed target (terminal) set by the phase vector of the control system at the prescribed time. The aim of the second player (the evader) is the opposite. A description of numerical algorithms used in the solution of differential games of the type under consideration is presented and estimates of the errors resulting from the approximation of the game sets by polyhedra are presented.

[1] Pontryagin L. S., “Lineinye differentsialnye igry presledovaniya”, Matem. sb., 112:3 (1980), 307–330 | MR | Zbl

[2] Krasovskii N. N., Subbotin A. I., Pozitsionnye differentsialnye igry, Nauka, M., 1974 | MR

[3] Krasovskii N. N., Upravlenie dinamicheskoi sistemoi. Zadacha o minimume garantirovannogo rezultata, Nauka, M., 1985 | MR

[4] Polovinkin E. S., Elementy teorii mnogoznachnykh otobrazhenii, Izd-vo MFTI, M., 1982

[5] Algoritmy i programmy resheniya lineinykh differentsialnykh igr, UNTs AN SSSR, Sverdlovsk, 1984

[6] Upravlenie s garantirovannym rezultatom, Sbornik nauchnykh trudov, UNTs AN SSSR, Sverdlovsk, 1987

[7] Pozitsionnoe upravlenie s garantirovannym rezultatom, Sbornik nauchnykh trudov, UrO AN SSSR, Sverdlovsk, 1988

[8] Ivanov G. E., Polovinkin E. S., “O silno vypuklykh lineinykh differentsialnykh igrakh”, Differents. uravneniya, 31:9 (1995), 1641–1648 | MR | Zbl

[9] Kurzhanski A. B., Valyi I., Ellipsoidal calculus for estimation and control, Birkhäuser, Boston, 1997 | MR

[10] Rokafellar R. T., Vypuklyi analiz, Mir, M., 1973

[11] Polovinkin E. S., “Silno vypuklyi analiz”, Matem. sb., 187:2 (1996), 103–130 | MR | Zbl

[12] Silin D. B., Trinko N. G., “Modifikatsiya algoritma Grekhema dlya ovypukleniya polozhitelno-odnorodnoi funktsii”, ZhVM i MF, 34:4 (1994), 631–636 | MR | Zbl

[13] Sukharev A. G., Timokhov A. V., Fedorov V. V., Kurs metodov optimizatsii, Nauka, M., 1986 | MR | Zbl

[14] Balashov M. V., “O maksimizatsii vypukloi funktsii na kompakte”, Nekotorye problemy fundamentalnoi i prikladnoi matematiki, MFTI, M., 1997, 17–25

[15] Orlova G. B., Silin D. B.,, “Priblizhennoe vychislenie vypukloi obolochki polozhitelno-odnorodnoi funktsii”, Vestn. MGU. Ser. 15. Vychisl. matem. i kibernetika, 1997, no. 2, 32–35 | MR | Zbl

[16] Dudov S. I., “Vnutrennyaya otsenka vypuklogo mnozhestva telom normy”, ZhVM i MF, 36:5 (1996), 153–159 | MR | Zbl

[17] Leikhtveis K., Vypuklye mnozhestva, Nauka, M., 1985 | MR

[18] Vasilev F. P., Chislennye metody resheniya ekstremalnykh zadach, Nauka, M., 1988 | MR