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## Archive of Issues

Russia Yekaterinburg
Year
2020
Volume
30
Issue
1
Pages
64-91
 Section Mathematics Title Relaxation of pursuit-evasion differential game and program absorption operator Author(-s) Chentsov A.G.ab, Khachai D.M.a Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa, Ural Federal Universityb Abstract We consider some natural relaxation of pursuit-evasion differential game. For two closed sets, which are parameters, similar guidance problem for $\varepsilon$-neighborhoods is being solved. We are interested in finding a minimal size of such neighborhoods, which allows player I successfully solve his guidance problem in the class of generalized non-anticipating strategies. To resolve above-mentioned differential game, a modification of Program Iterations Method is implemented. Size of the neighborhoods is found as a position function and it's defined by application of special iterative procedure further below. As a corollary, it is shown that desired function is a fixed point of the open-loop operator, which defines the procedure. Keywords pursuit-evasion differential game, program iterations method, guaranteed result UDC 519.6 MSC 05A05, 97N70, 97N80 DOI 10.35634/vm200106 Received 2 January 2020 Language Russian Citation Chentsov A.G., Khachai D.M. Relaxation of pursuit-evasion differential game and program absorption operator, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 1, pp. 64-91. References Isaacs R. Differential games, New York: Wiley, 1965. Pontryagin L.S. Linear differential games of pursuit, Mat. Sb. (N.S.), 1980, vol. 112 (154), no. 3 (7), pp. 307-330. Krasovskii N.N. Igrovye zadachi o vstreche dvizhenii (Game problems on the encounter of motions), M.: Nauka, 1970. Pshenichnii B.N. The structure of differential games, Soviet Math. Dokl., 1969, vol. 184, pp. 285-287. Krasovskii N.N., Subbotin A.I. An alternative for the game problem of convergence, Journal of Applied Mathematics and Mechanics, 1970, vol. 34, no. 6, pp. 948-965. Krasovskii N.N., Subbotin A.I. Game-theoretical control problems, Berlin-Heidelberg: Springer-Verlag, 1988. Roxin E. Axiomatic approach in differential games, Journal of Optimization Theory and Applications, 1969, vol. 3, pp. 153. https://doi.org/10.1007/BF00929440 Elliott R.J., Kalton N.J. Values in differential games, Bull. Amer. Math. Soc., 1972, vol. 78, no. 3, pp. 427-431. Chentsov A.G. On a game problem of guidance, Soviet Math., Doklady, 1976, vol. 17, pp. 73-77. https://zbmath.org/?q=an:0395.90105 Kryazimskii A.V. On the theory of positional differential games of convergence-evasion, Sov. Math., Dokl., 1978, vol. 19, no. 2, pp. 408-412. https://zbmath.org/?q=an:0399.90118 Chistiakov S.V. On solving pursuit game problems, Journal of Applied Mathematics and Mechanics, 1977, vol. 41, no. 5, pp. 845-852. https://doi.org/10.1016/0021-8928(77)90167-8 Ukhobotov V.I. Construction of a stable bridge for a class of linear games, Journal of Applied Mathematics and Mechanics, 1977, vol. 41, no. 2, pp. 350-354. https://doi.org/10.1016/0021-8928(77)90021-1 Chentsov A.G., Khachai D.M. Relaxation of a differential game of approach-evasion and iterative methods, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2018, vol. 24, no. 4, pp. 246-269 (in Russian). https://doi.org/10.21538/0134-4889-2018-24-4-246-269 Subbotin A.I., Chentsov A.G. Optimizatsiya garantii v zadachakh upravleniya (Optimization of guarantee in control problems), M.: Nauka, 1981. Chentsov A.G. The program iteration method in a game problem of guidance, Proceedings of the Steklov Institute of Mathematics, 2017, vol. 297, suppl. 1, pp. 43-61. https://doi.org/10.1134/S0081543817050066 Dunford N., Schwartz J.T. Linear operators. Part I: General theory, New York-London: Interscience, 1958. 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. Chentsov A.G. Stability iterations and an evasion problem with a constraint on the number of switchings, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2017, vol. 23, no. 2, pp. 285-302 (in Russian). https://doi.org/10.21538/0134-4889-2017-23-2-285-302 Full text