Numerical methods for solving differential games with nonterminal payoff
Keywords:
control theory, differential games, feedback strategies, resource constraints
Abstract
Linear convex positional differential games with geometric constraints on control actions and nonterminal payoff which evaluates a norm of a set of motion deviations at given instants of time from given target points are considered. Cases when the saddle point in a small game is either present or absent together with possible presence of additional integral constraints on control actions are studied. In each of these cases numerical methods for calculating the game value in appropriate classes of strategies and for constructing corresponding optimal control laws are elaborated. Numerical methods are based on backward constructions of upper convex hulls of auxiliary program functions. Domains of these functions are approximated by a pixel method, functions are stored as tables, upper convex hull is computed approximately as a lower envelope of a finite family of supporting hyperplanes to subgraphs of these functions. Details of software implementation for modern computational systems are discussed. Results of simulations in model examples are given.
References
1. Azimov A.Ya. On one way of pursuit in linear differential games with integral constraints, Izvestiya Akademii Nauk SSSR. Tekhnicheskaya Kibernetika, 1974, no. 2, pp. 31-35 (in Russian).
2. Isaacs R. Differential games, New York: John Wiley and Sons, 1965. Translated under the title Differentsial'nye igry, Moscow: Mir, 1967, 479 p.
3. Al'brekht E.G. Constructing the approximate solutions of some quasilinear differential games, Proc. Steklov Inst. Math., 2000, suppl. 1, pp. S24-S34.
4. Balashov M.V. On the P-property of compact convex sets, Mathematical Notes, 2002, vol. 71, issue 3, pp. 295-304. DOI: 10.1023/A:1014864822124
5. Batukhtin V.D. Extremal aiming in a nonlinear convergence game, Soviet Mathematics. Doklady, 1972, vol. 13, pp. 1427-1431.
6. Berdyshev Yu.I. Problem of successive optimization without time decomposition, Cybernetics, 1987, vol. 23, issue 4, pp. 474-479. DOI: 10.1007/BF01078903
7. Berdyshev Yu.I. A problem of the sequential approach to a group of moving points by a third-order non-linear control system, J. Appl. Math. Mech., 2002, vol. 66, issue 5, pp. 709-718. DOI: 10.1016/S0021-8928(02)90001-8
8. Berdyshev Yu.I., Chentsov A.G. Optimization of a weighted criterion function in one control problem, Cybernetics, 1986, vol. 22, issue 1, pp. 67-74. DOI: 10.1007/BF01078629
9. Botkin N.D. Differential pursuit game with mixed control constraints, Automation and Remote Control, 1992, vol. 53, no. 6, part 1, pp. 793-800.
10. Gomoyunov M.I., Lukoyanov N.Yu. On the stability of a procedure for solving a minimax control problem for a positional functional, Proceedings of the Steklov Institute of Mathematics, 2015, vol. 288, suppl. 1, pp. 54-69. DOI: 10.1134/S0081543815020078
11. Gornov A.Yu. Vychislitel'nye tekhnologii resheniya zadach optimal'nogo upravleniya (Computational technologies of solving optimal control problems), Novosibirsk: Nauka, 2009, 279 p.
12. Gornov A.Yu., Tyatyushkin A.I., Finkelstein E.A. Numerical methods for solving applied optimal control problems, Computational Mathematics and Mathematical Physics, 2013, vol. 53, issue 12, pp. 1825-1838. DOI: 10.1134/S0965542513120063
13. Grigorenko N.L. On structure of one class of differential games with common integral constraints, Upravlyaemye Sistemy, 1974, no. 12, pp. 23-31 (in Russian).
14. Grigorenko N.L., Kamzolkin D.V., Luk'yanova L.N. A numerical algorithm for solving a nonstationary problem of optimal control, Proceedings of the Steklov Institute of Mathematics, 2011, vol. 275, suppl. 1, pp. 49-56. DOI: 10.1134/S0081543811090045
15. Grigorenko N.L., Kiselev Yu.N., Lagunova N.V., Silin D.B., Trin'ko N.G. Solution methods for differential games, Computational Mathematics and Modeling, 1996, vol. 7, issue 1, pp. 101-116. DOI: 10.1007/BF01128750
16. Dar'in A.N., Kurzhanskii A.B. Method of dynamical programming in problems of control synthesis under polytypic and double contstraints, Problemy upravleniya i prilozheniya (tekhnika, proizvodstvo, ekonomika): trudy mezhdunarodnoi konferentsii (Control problems and applications (engineering, manufacturing, economics): proceedings of international conference), vol. 2, Minsk, 2005, pp. 51-65 (in Russian).
17. Dar'in A.N., Kurzhanskii A.B. Control under indeterminacy and double constraints, Differential Equations, 2003, vol. 39, issue 11, pp. 1554-1567. DOI: 10.1023/B:DIEQ.0000019347.24930.a3
18. Dvurechenskii P.E., Ivanov G.E. Algorithms for computing Minkowski operators and their application in differential games, Computational Mathematics and Mathematical Physics, 2014, vol. 54, issue 2, pp. 235-264. DOI: 10.1134/S0965542514020055
19. Dykhta V.A., Samsonyuk O.N. Optimal'noe impul'snoe upravlenie s prilozheniyami (Optimal impulse control and its applications), Moscow: Fizmatlit, 2003, 256 p.
20. Zhukovskii V.I., Chikrii A.A. Lineino-kvadratichnye differentsial'nye igry (Linear quadratic differential games), Kiev: Naukova dumka, 1994, 241 p.
21. Zavalishchin S.T., Sesekin A.N. Impul'snye protsessy. Modeli i prilozheniya (Impulse procesess. Models and applications), Moscow: Nauka, 1991, 256 p.
22. Ivanov G.E., Kazeev V.A. Minimax algorithm for constructing an optimal control strategy in differential games with a Lipschitz payoff, Computational Mathematics and Mathematical Physics, 2011, vol. 51, issue 4, pp. 550-574. DOI: 10.1134/S0965542511040075
23. Ivanov G.E., Polovinkin E.S. On strongly convex linear differential games, Differentsial'nye uravneniya, 1995, vol. 31, no. 10, pp. 1641-1648 (in Russian).
24. Ignatenko A.P. On one approach problem with integral-geometry constraints, Teoriya optimal'nykh reshenii, 2007, no. 6, pp. 74-79 (in Russian).
25. Ioffe A.D., Tikhomirov V.M. Teoriya ekstremal'nykh zadach (Theory of extremal problems), Moscow: Nauka, 1974, 479 p.
26. Isakova E.A., Logunova G.V., Patsko V.S. Computation of stable bridges for linear differential games with fixed time of termination, Algoritmy i programmy resheniya lineinykh differentsial'nykh igr, Sverdlovsk: Ural Scientific Center, 1984, pp. 127-158 (in Russian).
27. Kleimenov A.F. Neantagonisticheskie pozitsionnye differentsial'nye igry (Nonantagonistic positional differential games), Yekaterinburg: Nauka, 1993, 185 p.
28. Kovrizhnykh A.Yu. On the problem of conflict control with a quasipositional functional, Proceedings of the Steklov Institute of Mathematics, 2000, suppl. 2, pp. S79-S93.
29. Cormen T.H., Leiserson C.E., Rivest R.L., Stein C. Introduction to Algorithms, MIT press, 2001, 1202 p. Translated under the title Algoritmy: postroenie i analiz, Moscow: Vil'yams, 2005, 1296 p.
30. Krasovskii A.A., Krasovskii A.N. Nonlinear positional differential game in the class of mixed strategies, Proceedings of the Steklov Institute of Mathematics, 2012, vol. 277, issue 1, pp. 137-143. DOI: 10.1134/S0081543812040098
31. Krasovskii A.N. On minimax position control, Journal of Applied Mathematics and Mechanics, 1980, vol. 44, issue 4, pp. 425-430. DOI: 10.1016/0021-8928(80)90028-3
32. Krasovskii A.N. Construction of mixed strategies on the basis of stochastic programs, Journal of Applied Mathematics and Mechanics, 1987, vol. 51, issue 2, pp. 144-149. DOI: 10.1016/0021-8928(87)90055-4
33. Krasovskii A.N. Sintez smeshannykh strategii upravleniya (Synthesis of mixed control strategies), Sverdlovsk: Ural State University, 1988, 151 p.
34. Krasovskii A.N., Reshetova T.N. Upravlenie pri defitsite informatsii: Uchebnoe posobie (Control under lack of information: textbook), Sverdlovsk: Ural State University, 1990, 104 p.
35. Krasovskii N.N. Igrovye zadachi o vstreche dvizhenii (Game-theoretical problems of meeting of motions), Moscow: Nauka, 1970, 420 p.
36. Krasovskii N.N. On the problem of damping of a linear system under minimum control intensity, Journal of Applied Mathematics and Mechanics, 1965, vol. 29, issue 2, pp. 245-253. DOI: 10.1016/0021-8928(65)90028-6
37. Krasovskii N.N. Upravlenie dinamicheskoi sistemoi (Control of a dynamic system), Moscow: Nauka, 1985, 516 p.
38. Krasovskii N.N., Lukoyanov N.Yu. Problem of conflict control with hereditary information, Journal of Applied Mathematics and Mechanics, 1996, vol. 60, no. 6, pp. 869-882. DOI: 10.1016/S0021-8928(96)00109-8
39. Krasovskii N.N., Reshetova T.N. On the program synthesis of a guaranteed control, Problems of Control and Information Theory, 1988, vol. 17, no. 6, pp. 1-11.
40. Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial'nye igry (Positional differential games), Moscow: Nauka, 1974, 456 p.
41. Krasovskii N.N., Tret'yakov V.E. To a pursuit problem under constraints on impulses of control forces, Differentsial'nye Uravneniya, 1966, vol. 2, no. 5, pp. 587-599 (in Russian).
42. Krasovskii A.N., Krasovskii N.N., Tret'yakov V.E. Stochastic programmed design for a deterministic positional differential game, Journal of Applied Mathematics and Mechanics, 1981, vol. 45, issue 4, pp. 425-430. DOI: 10.1016/0021-8928(81)90081-2
43. Kryazhimskii A.V. On the theory of positional differential games of approach-evasion, Doklady Akad. Nauk SSSR, 1978, vol. 239, no. 4, pp. 779-782 (in Russian).
44. Kurzhanskii A.B. Upravlenie i nablyudenie v usloviyakh neopredelennosti (Control and observation under conditions of uncertainty), Moscow: Nauka, 1977, 392 p.
45. Ledyaev Yu.S. Regular differential games with mixed constraints on the controls, Proceedings of the Steklov Institute of Mathematics, 1986, vol. 167, pp. 233-242.
46. Lokshin M.D. Differential games with integral restrictions on the controlling actions, Differential Equations, 1992, vol. 28, no. 11, pp. 1602-1610.
47. Lukoyanov N.Yu. The problem of computing the value of a differential game for a positional functional, Journal of Applied Mathematics and Mechanics, 1998, vol. 62, issue 2, pp. 177-186. DOI: 10.1016/S0021-8928(98)00023-9
48. Lukoyanov N.Yu. The problem of conflicting control with mixed constraints, Journal of Applied Mathematics and Mechanics, 1995, vol. 59, no. 6, pp. 911-919. DOI: 10.1016/0021-8928(95)00124-7
49. Lukoyanov N.Yu. A conflict control problem under mixed restrictions on control actions, Differential Equations, 1995, vol. 31, no. 9, pp. 1427-1436.
50. Lukoyanov N.Yu. How to compute the value of a positional differential game, Differential Equations, 2001, vol. 37, issue 1, pp. 17-26. DOI: 10.1023/A:1019260029465
51. Lukoyanov N.Yu. A differential game with nonterminal cost, Journal of Computer and Systems Sciences International, 1997, vol. 36, no. 1, pp. 77-82.
52. Maksimov V.I. On the existence of a saddle point in a difference-differential encounter-evasion game, Journal of Applied Mathematics and Mechanics, 1978, vol. 42, no. 1, pp. 13-20. DOI: 10.1016/0021-8928(78)90083-7
53. Mamadaliev N. On a pursuit problem with integral constraints on the players’ controls, Siberian Mathematical Journal, 2015, vol. 56, issue 1, pp. 107-124. DOI: 10.1134/S0037446615010115
54. Mezentsev A.V. Differentsial'nye igry s integral'nymi ogranicheniyami na upravleniya (Differential games with integral control constraints), Moscow: Moscow State University, 1988, 135 p.
55. Mikhalev D.K., Ushakov V.N. Two algorithms for approximate construction of the set of positional absorption in the game problem of pursuit, Automation and Remote Control, 2007, vol. 68, issue 11, pp. 2056-2070. DOI: 10.1134/S0005117907110136
56. Mishchenko E.F. Pursuit and evasion problems in differential games theory, Izvestiya Akademii Nauk SSSR. Tekhnicheskaya Kibernetika, 1971, no. 5, pp. 3-9 (in Russian).
57. Nikol'skii M.S. Linear differential pursuit games with integral constraints, Differential Equations, 1992, vol. 28, no. 2, pp. 190-193.
58. Nikol'skii M.S. Pervyi pryamoi metod L.S. Pontryagina v differentsial'nykh igrakh (The first direct method of L.S. Pontryagin in differential games), Moscow: Moscow State University, 1984, 65 p.
59. Nikol'skii M.S. A direct method in linear differential games with integral constraints, Upravlyaemye Sistemy, 1969, no. 2, pp. 49-58 (in Russian).
60. Nikol'skii M.S. A direct method in linear differential games with general integral constraints, Differentsial'nye Uravneniya, 1972, vol. 8, no. 6, pp. 964-971 (in Russian).
61. Osipov Yu.S. Differential games of systems with aftereffect, Doklady Akad. Nauk SSSR, 1971, vol. 196, no. 4, pp. 779-782 (in Russian).
62. Osipov Yu.S., Pimenov V.G. On positional control under aftereffect in the controlling forces, Journal of Applied Mathematics and Mechanics, 1981, vol. 45, issue 2, pp. 159-163. DOI: 10.1016/0021-8928(81)90028-9
63. Patsko V.S., Turova V.L. Chislennoe reshenie differentsial'nykh igr na ploskosti (Numerical solving of differential games on a plain), Yekaterinburg: Ural Branch of the Russian Academy of Sciences, 1995, 77 p.
64. Petrov N.N. About existence of the value of a pursuit game, Doklady Akad. Nauk SSSR, 1970, vol. 190, no. 6, pp. 621-624 (in Russian).
65. Petrov N.N. On a group pursuit problem, Automation and Remote Control, 1996, vol. 57, no. 6, pp. 808-813.
66. Petrosyan L.A. Differentsial'nye igry presledovaniya (Pursuit differential games), Leningrad: Leningrad State University, 1977, 222 p.
67. Polovinkin E.S., Ivanov G.E., Balashov M.V., Konstantinov R.V., Khorev A.V. An algorithm for the numerical solution of linear differential games, Sbornik: Mathematics, 2001, vol. 192, no. 10, pp. 1515-1542. DOI: 10.1070/SM2001v192n10ABEH000604
68. Pontryagin L.S. About linear differential games. I, Doklady Akad. Nauk SSSR, 1967, vol. 174, no. 6, pp. 1278-1280 (in Russian).
69. Pontryagin L.S. About linear differential games. II, Doklady Akad. Nauk SSSR, 1967, vol. 175, no. 4, pp. 764-766 (in Russian).
70. Pshenichnyi B.N. Structure of differential games, Doklady Akad. Nauk SSSR, 1969, vol. 184, no. 2, pp. 285-187 (in Russian).
71. Pshenichnyi B.N., Onopchuk Yu.N. Linear differential games with integral constraints, Izvestiya Akademii Nauk SSSR. Tekhnicheskaya Kibernetika, 1968, no. 1, pp. 13-22 (in Russian).
72. Pshenichnyi B.N., Sagaidak M.I. Differential games of prescribed duration, Cybernetics, 1970, vol. 6, issue 2, pp. 72-83. DOI: 10.1007/BF01070503
73. Rokafellar R. Convex analysis, Princeton: Princeton University Press, 1970. Translated under the title Vypuklyi analiz, Moscow: Mir, 1973, 469 p.
74. Samatov B.T. The pursuit-evasion problem under integral-geometric constraints on pursuer controls, Automation and Remote Control, 2013, vol. 74, issue 7, pp. 1072-1081. DOI: 10.1134/S0005117913070023
75. Solomatin A.M., Ushakov V.N. Construction of a positional absorption set in a linear game with integral constraints, Upravlenie i Otsenivanie v Dinamicheskikh Sistemakh, 1982, pp. 74-89 (in Russian).
76. Straustrup B. The C++ programming language: special edition, Addison Wesley, 2000, 1029 p. Translated under the title Yazyk programmirovaniya C++. Spetsial'noe izdanie, Moscow: Binom, 2011, 1136 p.
77. Subbotin A.I., Ushakov V.N. Alternative for an encounter-evasion differential game with integral constraints on the players'controls, Journal of Applied Mathematics and Mechanics, 1975, vol. 39, issue 3, pp. 367-375. DOI: 10.1016/0021-8928(75)90001-5
78. Subbotin A.I., Chentsov A.G. Optimizatsiya garantii v zadachakh upravleniya (Optimization of guarantee in control problems), Moscow: Nauka, 1981, 288 p.
79. Subbotin A.I. Minimaksnye neravenstva i uravneniya Gamil'tona-Yakobi (Minimax inequalities and Hamilton-Jacobi equations), Moscow: Nauka, 1991, 216 p.
80. Subbotina N.N., Subbotin A.I. Alternative for the encounter-evasion differential game with constraints on the momenta of the players' controls, Journal of Applied Mathematics and Mechanics, 1975, vol. 39, issue 3, pp. 376-385. DOI: 10.1016/0021-8928(75)90002-7
81. Subbotina N.N. The method of characteristics for Hamilton-Jacobi equations and applications to dynamical optimization, Journal of Mathematical Sciences, 2004, vol. 135, issue 3, pp. 2955-3091. DOI: 10.1007/s10958-006-0146-2
82. Taras'ev A.M. Approximation schemes for constructing minimax solutions of Hamilton-Jacobi equations, Journal of Applied Mathematics and Mechanics, 1994, vol. 58, issue 2, pp. 207-221. DOI: 10.1016/0021-8928(94)90049-3
83. Taras'ev A.M., Uspenskii A.A., Ushakov V.N. Approximation schemes and finite-difference operators for constructing generalized solutions of Hamilton-Jacobi equations, Izv. Ross. Akad. Nauk, Tekh. Kibern., 1994, no. 3, pp. 173-185 (in Russian).
84. Taras'ev A.M., Uspenskii A.A., Ushakov V.N. Finite-difference method of constructing functions of optimal guaranteed result, Gagarinskie nauchnye chteniya po kosmonavtike i aviatsii. 1991 (Gagarin Readings on space and aviation), Moscow: Nauka, 1992, pp. 166-172.
85. Ukhobotov V.I. On a class of differential games with an integral constraint, Journal of Applied Mathematics and Mechanics, 1977, vol. 41, issue 5, pp. 838-844. DOI: 10.1016/0021-8928(77)90166-6
86. Ukhobotov V.I. A type of linear game with mixed constraints on the controls, Journal of Applied Mathematics and Mechanics, 1987, vol. 51, issue 2, pp. 139-144. DOI: 10.1016/0021-8928(87)90054-2
87. Ukhobotov V.I., Gushchin D.V. About one class of similar differential game with mixed limitations, Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki, 2010, issue 3, pp. 81-86 (in Russian). DOI: 10.20537/vm100310
88. Ushakov V.N. Extremal strategies in differential games with integral constraints, Journal of Applied Mathematics and Mechanics, 1972, vol. 36, issue 1, pp. 12-19. DOI: 10.1016/0021-8928(72)90076-7
89. Ushakov V.N., Khripunov A.P. Approximate construction of solutions in game-theoretic control problems, Journal of Applied Mathematics and Mechanics, 1997, vol. 61, issue 3, pp. 401-408. DOI: 10.1016/S0021-8928(97)00051-8
90. Fan Ky. Minimax theorems, Proceedings of the National Academy of Sciences of the United States of America, 1953, vol. 39, no. 1, pp. 42-47.
91. Chentsov A.G. On a game problem of converging at a given instant of time, Mathematics of the USSR-Sbornik, 1976, vol. 28, no. 3, pp. 353-376. DOI: 10.1070/SM1976v028n03ABEH001657
92. Chernous'ko F.L. Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem. Metod ellipsoidov (Phase state estimation in dynamical systems. Method of ellipsoids), Moscow: Nauka, 1988, 319 p.
93. Chernous'ko F.L., Melikyan A.A. Igrovye zadachi upravleniya i poiska (Game problems of control and search), Moscow: Nauka, 1978, 270 p.
94. Chikrii A.A. Conflict controlled processes, Springer Netherlands, 1997, XX+404 p. DOI: 10.1007/978-94-017-1135-7
95. Chikrii A.A, Belousov A.A. On linear differential games with convex integral constraints, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2013, vol. 19, no. 4, pp. 308-319 (in Russian).
96. Chistyakov S.V. On solving pursuit game problems, Journal of Applied Mathematics and Mechanics, 1977, vol. 41, issue 5, pp. 845-852. DOI: 10.1016/0021-8928(77)90167-8
97. Alton K., Mitchell I.M. Fast marching methods for stationary Hamilton-Jacobi equations with axis-aligned anisotropy, SIAM J. Numer. Anal., 2008, vol. 47, no. 1, pp. 363-385.
98. Bardi M., Falcone M., Soravia P. Numerical methods for pursuit-evasion games via viscosity solutions, Stochastic and differential games, Annals of the International Society of Dynamic Games, vol. 4, Boston: Birkhäuser, 1999, pp. 105-175. DOI: 10.1007/978-1-4612-1592-9_3
99. Barron E.N. Differential games with maximum cost, Nonlinear Analysis: Theory, Methods and Applications, 1990, vol. 14, issue 11, pp. 971-989. DOI: 10.1016/0362-546X(90)90113-U
100. Basar T., Bernhard P. H-infinity optimal control and related minimax design problems: a dynamic game approach, Boston: Birkhäuser, 1995, 428 p.
101. Berkovitz L.D. Characterization of the values of differential games, Applied Mathematics and Optimization, 1988, vol. 17, issue 1, pp. 177-183. DOI: 10.1007/BF01448365
102. Blaquiere A., Gerard F., Leitmann G. Quantitative and qualitative games, New York etc.: Academic Press, 1969. 172 p.
103. Botkin N.D., Hoffmann K.-H., Turova V.L. Stable numerical schemes for solving Hamilton-Jacobi-Bellman-Isaacs equations, SIAM J. Sci. Comput., 2011, vol. 33, issue 2, pp. 992-1007. DOI: 10.1137/100801068
104. Cardaliaguet P., Quincampoix M., Saint-Pierre P. Numerical methods for optimal control and differential games, Ceremade CNRS URA 749, University of Paris Dauphine, 1995.
105. Cardaliaguet P., Quincampoix M., Saint-Pierre P. Set-valued numerical analysis for optimal control and differential games, Stochastic and differential games, Annals of the International Society of Dynamic Games, vol. 4, Boston: Birkhäuser, 1999, pp. 177-247. DOI: 10.1007/978-1-4612-1592-9_4
106. Chernousko F.L., Ananievski I.M., Reshmin S.A. Control of nonlinear dynamical systems. Methods and applications, Berlin-Heidelberg: Springer, 2008, 396 p. DOI: 10.1007/978-3-540-70784-4
107. Elliot R.J., Kalton N.J. The existence of value for differential games, Memoirs of the American Mathematical Society, 1972, no. 126, 67 p. DOI: 10.1090/memo/0126
108. Evans L.C., Ishii H. Differential games and nonlinear first order PDE on bounded domains, Manuscripta Mathematica, 1984, vol. 49, issue 2, pp. 109-139. DOI: 10.1007/BF01168747
109. Falcone M. Recent results in the approximation of nonlinear optimal control problems, Large-scale scientific computing, Lecture Notes in Computer Science, vol. 8353, Berlin-Heidelberg: Springer, 2014, pp. 15-32. DOI: 10.1007/978-3-662-43880-0_2
110. Fleming W.H. A note on differential games of prescribed duration, Contributions to the Theory of Games, Annals of Mathematical Studies, 1957, vol. 3, pp. 407-416.
111. Fleming W.H. The convergence problem for differential games, Journal of Mathematical Analysis and Applications, 1961, vol. 3, issue 1, pp. 102-116. DOI: 10.1016/0022-247X(61)90009-9
112. Fleming W.H. The convergence problem for differential games, Advances in Game Theory. (AM-52) (Annals of Mathematics Studies), Princeton: Princeton University Press, 1964, pp. 195-210.
113. Friedman A. Differential games, New York: Wiley Interscience, 1971, 350 p.
114. Ho Y.C., Bryson A., Baron S. Differential games and optimal pursuit-evasion strategies, IEEE Trans. Autom. Contr., 1965, vol. 10, issue 4, pp. 385-389. DOI: 10.1109/TAC.1965.1098197
115. Ibragimov G.I., Azamov A.A., Khakestari M. Solution of a linear pursuit-evasion game with integral constraints, ANZIAM Journal, 2010, vol. 52, pp. E59-E75. DOI: 10.21914/anziamj.v52i0.3605
116. Krasovskii A.N., Choi Y.S. Stochastic control with the leaders-stabilizers, Yekaterinburg: Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 2001, 51 p.
117. Krasovskii A.N., Krasovskii N.N. Control under lack of information, Berlin etc.: Birkhäuser, 1995. 322 p.
118. Kumkov S.S., Patsko V.S. Construction of singular surfaces in linear differential games, Advances in dynamic games and applications, Annals of the International Society of Dynamic Games, vol. 6, Basel: Birkhäuser, 2001. pp. 185-202. DOI: 10.1007/978-1-4612-0155-7
119. Kurzhanski A.B., Valyi I. Ellipsoidal calculus for estimation and control, Boston: Birkhäuser, 1996, XV+321 p.
120. Kurzhanski A.B., Varaiya P. Dynamics and control of trajectory tubes, Springer, 2014, 445 p.
121. Lewin J. Differential games: Theory and methods for solving game problems with singular surfaces, New York: Springer-Verlag, 1994, 242 p.
122. Lions P.-L., Souganidis P.E. Differential games, optimal control and directional derivatives of viscosity solutions of Bellman's and Isaacs' equations, SIAM Journal on Control and Optimization, 1985, vol. 23, issue 4, pp. 566-583. DOI: 10.1137/0323036
123. Melikyan A.A. Generalaized characteristics of first order PDEs: Applications in optimal control and differential games, Boston: Birkhäuser, 1998, 310 p.
124. Miller B., Rubinovich E.Y. Impulsive control in continuous and discrete-continuous systems, New York: Kluwer Academic / Plenum Publishers, 2003, 447 p.
125. Motta M., Rampazzo F. Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential and Integral Equations, 1995, vol. 8, no. 2, pp. 269-288.
126. Roxin E. Axiomatic approach in differential games, Journal of Optimization Theory and Applications, 1969, vol. 3, issue 3, pp. 153-163. DOI: 10.1007/BF00929440
127. Sethian J.A. Level set methods and fast marching methods: Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, Cambridge University Press, 1999, 404 p.
128. Subbotin A.I. Generalized solutions of first-order PDEs: The dynamical optimization perspective, Boston etc.: Birkhäuser, 1995, 312 p.
129. Varaiya P., Lin J. Existence of saddle points in differential games, SIAM Journal on Control, 1969, vol. 7, issue 1, pp. 141-157. DOI: 10.1137/0307011
130. Williams J.D. The compleat strategyst, New York: McGraw-Hill, 1966, 268 p.
131. Gomoyunov M.I., Kornev D.V. On calculating the value of a differential game in the class of counterstrategies, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2013, vol. 19, no. 1, pp 59-68 (in Russian).
132. Gomoyunov M.I., Kornev D.V., Lukoyanov N.Yu. To the problem of positional guarantee optimization under conditions of delayed control, Trudy XII Vserossiiskogo soveshchaniya po problemam upravleniya (VSPU-2014) (Proceedings of XII All-Russian Conference on Control Problems (RCCP–2014)), Moscow: Institute of Control Sciences of RAS, 2014, pp. 1268-1279 (in Russian).
133. Gomoyunov M.I., Kornev D.V., Lukoyanov N.Yu. On the numerical solution of a minimax control problem with a positional functional, Proceedings of the Steklov Institute of Mathematics, 2015, vol. 291, suppl. 1, pp. 77-95. DOI: 10.1134/S0081543815090060
134. Kornev D.V. On the question of program realization of a solution of a differential game with the nonterminal payoff, Sovremennye problemy matematiki: Trudy 42-i Regional'noi molodezhnoi konferentsii (Contemporary Problems of Mathematics: Proceedings of the 42th Regional Youth Conference), Yekaterinburg: Ural Branch of the Russian Academy of Sciences, 2011, pp. 34-37 (in Russian).
135. Kornev D.V. On numerical solving differential games on minimax of a positional functional in classes of mixed strategies, Systems Dynamics and Control Processes (SDCP'2014), Proceedings of International Conference dedicated to the 90th anniversary of Academician N.N. Krasovskii, Russia, Yekaterinburg, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ural Federal University, September 15-20, 2014, pp. 111-113 (in Russian).
136. Kornev D.V. On numerical solving differential games with nonterminal payoff, Sovremennye problemy matematiki: Tezisy Mezhdunarodnoi (43-i Vserossiiskoi) molodezhnoi shkoly-konferentsii (Contemporary Problems of Mathematics: Abstracts of the 43th All-Russian Youth Conference), Yekaterinburg: Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 2012, pp. 136-138 (in Russian).
137. Kornev D.V. On numerical solution of positional differential games with nonterminal payoff, Automation and Remote Control, 2012, vol. 73, issue 11, pp. 1808-1821. DOI: 10.1134/S0005117912110045
138. Kornev D.V. On a numerical method of solving conflict control problems, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2012, no. 1 (39), pp. 67-68 (in Russian).
139. Kornev D.V. About on numerical method for solving positional differential games in mixed strategies, Vestn. Tambov. Univ. Ser. Estestv. Tekh. Nauki, 2013, vol. 18, no. 5-2, pp. 2556-2558 (in Russian).
140. Kornev D.V. About guarantee optimization under integral constraints on control actions and nonterminal quality index, Trudy XII Vserossiiskogo soveshchaniya po problemam upravleniya (VSPU-2014) (Proceedings of XII All-Russian Conference on Control Problems (RCCP–2014)), Moscow: Institute of Control Sciences of RAS, 2014, pp. 2059-2070 (in Russian).
141. Kornev D.V. Programmnyi kompleks dlya resheniya pozitsionnykh differentsial'nykh igr s neterminal'noi platoi (The program complex for solving positional differential games with nonterminal payoff). Patent RF, no. 2015614531, 2015.
142. Kornev D.V., Lukoyanov N.Yu. To the problem of dynamical guarantee optimization under geometrical and integral constraints on control capabilities, Absracts of II International Seminar “Control Theory and Theory of Generalized Solutions of Hamilton-Jacobi Equations” (CGS'2015) dedicated to the 70th anniversary of Academician A.I. Subbotin, Russia, Yekaterinburg, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ural Federal University, April 1-3, 2015, pp. 94-95 (in Russian).
143. Kornev D.V., Lukoyanov N.Yu. On a minimax control problem for a positional functional under geometric and integral constraints on control actions, Proceedings of the Steklov Institute of Mathematics, 2015, vol. 293, issue 2, pp. 85-100. DOI: 10.1134/S0081543816050096
144. Kornev D.V., Lukoyanov N.Yu. On numerical solution of differential games with nonterminal payoff in classes of mixed strategies, Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki, 2013, issue 3, pp. 34-48 (in Russian). DOI: 10.20537/vm130304
145. Kornev D.V., Lukoyanov N.Yu. Numerical methods of solving linear positional differential games with nonterminal payoff, Algoritmicheskii analiz neustoichivykh zadach: Tezisy dokladov Mezhdunarodnoi konferentsii, posvyashchennoi pamyati V.K. Ivanova (Algorithmic analysis of problems: abstacts of international conference dedicated to the memory of V.K. Ivanov), Yekaterinburg: Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 2011, p. 248.
146. Gomoyunov M.I., Kornev D.V., Lukoyanov N.Yu. Game theory applications to guarantee optimization in dynamical systems with control delays, International Game Theory Review, 2014, vol. 16, issue 02, 1440010, 19 p. DOI: 10.1142/S0219198914400106
147. Kornev D., Lukoyanov N. On numerical solution of differential games in classes of mixed strategies, IFAC Proceedings Volumes, 2014, vol. 47, issue 3, pp. 1550-1555. DOI: 10.3182/20140824-6-ZA-1003.02208
148. Kornev D., Lukoyanov N. On numerical solving of differential games with nonterminal payoff, IFAC Proceedings Volumes, 2012, vol. 45, issue 25, pp. 71-76. DOI: 10.3182/20120913-4-IT-4027.00026
2. Isaacs R. Differential games, New York: John Wiley and Sons, 1965. Translated under the title Differentsial'nye igry, Moscow: Mir, 1967, 479 p.
3. Al'brekht E.G. Constructing the approximate solutions of some quasilinear differential games, Proc. Steklov Inst. Math., 2000, suppl. 1, pp. S24-S34.
4. Balashov M.V. On the P-property of compact convex sets, Mathematical Notes, 2002, vol. 71, issue 3, pp. 295-304. DOI: 10.1023/A:1014864822124
5. Batukhtin V.D. Extremal aiming in a nonlinear convergence game, Soviet Mathematics. Doklady, 1972, vol. 13, pp. 1427-1431.
6. Berdyshev Yu.I. Problem of successive optimization without time decomposition, Cybernetics, 1987, vol. 23, issue 4, pp. 474-479. DOI: 10.1007/BF01078903
7. Berdyshev Yu.I. A problem of the sequential approach to a group of moving points by a third-order non-linear control system, J. Appl. Math. Mech., 2002, vol. 66, issue 5, pp. 709-718. DOI: 10.1016/S0021-8928(02)90001-8
8. Berdyshev Yu.I., Chentsov A.G. Optimization of a weighted criterion function in one control problem, Cybernetics, 1986, vol. 22, issue 1, pp. 67-74. DOI: 10.1007/BF01078629
9. Botkin N.D. Differential pursuit game with mixed control constraints, Automation and Remote Control, 1992, vol. 53, no. 6, part 1, pp. 793-800.
10. Gomoyunov M.I., Lukoyanov N.Yu. On the stability of a procedure for solving a minimax control problem for a positional functional, Proceedings of the Steklov Institute of Mathematics, 2015, vol. 288, suppl. 1, pp. 54-69. DOI: 10.1134/S0081543815020078
11. Gornov A.Yu. Vychislitel'nye tekhnologii resheniya zadach optimal'nogo upravleniya (Computational technologies of solving optimal control problems), Novosibirsk: Nauka, 2009, 279 p.
12. Gornov A.Yu., Tyatyushkin A.I., Finkelstein E.A. Numerical methods for solving applied optimal control problems, Computational Mathematics and Mathematical Physics, 2013, vol. 53, issue 12, pp. 1825-1838. DOI: 10.1134/S0965542513120063
13. Grigorenko N.L. On structure of one class of differential games with common integral constraints, Upravlyaemye Sistemy, 1974, no. 12, pp. 23-31 (in Russian).
14. Grigorenko N.L., Kamzolkin D.V., Luk'yanova L.N. A numerical algorithm for solving a nonstationary problem of optimal control, Proceedings of the Steklov Institute of Mathematics, 2011, vol. 275, suppl. 1, pp. 49-56. DOI: 10.1134/S0081543811090045
15. Grigorenko N.L., Kiselev Yu.N., Lagunova N.V., Silin D.B., Trin'ko N.G. Solution methods for differential games, Computational Mathematics and Modeling, 1996, vol. 7, issue 1, pp. 101-116. DOI: 10.1007/BF01128750
16. Dar'in A.N., Kurzhanskii A.B. Method of dynamical programming in problems of control synthesis under polytypic and double contstraints, Problemy upravleniya i prilozheniya (tekhnika, proizvodstvo, ekonomika): trudy mezhdunarodnoi konferentsii (Control problems and applications (engineering, manufacturing, economics): proceedings of international conference), vol. 2, Minsk, 2005, pp. 51-65 (in Russian).
17. Dar'in A.N., Kurzhanskii A.B. Control under indeterminacy and double constraints, Differential Equations, 2003, vol. 39, issue 11, pp. 1554-1567. DOI: 10.1023/B:DIEQ.0000019347.24930.a3
18. Dvurechenskii P.E., Ivanov G.E. Algorithms for computing Minkowski operators and their application in differential games, Computational Mathematics and Mathematical Physics, 2014, vol. 54, issue 2, pp. 235-264. DOI: 10.1134/S0965542514020055
19. Dykhta V.A., Samsonyuk O.N. Optimal'noe impul'snoe upravlenie s prilozheniyami (Optimal impulse control and its applications), Moscow: Fizmatlit, 2003, 256 p.
20. Zhukovskii V.I., Chikrii A.A. Lineino-kvadratichnye differentsial'nye igry (Linear quadratic differential games), Kiev: Naukova dumka, 1994, 241 p.
21. Zavalishchin S.T., Sesekin A.N. Impul'snye protsessy. Modeli i prilozheniya (Impulse procesess. Models and applications), Moscow: Nauka, 1991, 256 p.
22. Ivanov G.E., Kazeev V.A. Minimax algorithm for constructing an optimal control strategy in differential games with a Lipschitz payoff, Computational Mathematics and Mathematical Physics, 2011, vol. 51, issue 4, pp. 550-574. DOI: 10.1134/S0965542511040075
23. Ivanov G.E., Polovinkin E.S. On strongly convex linear differential games, Differentsial'nye uravneniya, 1995, vol. 31, no. 10, pp. 1641-1648 (in Russian).
24. Ignatenko A.P. On one approach problem with integral-geometry constraints, Teoriya optimal'nykh reshenii, 2007, no. 6, pp. 74-79 (in Russian).
25. Ioffe A.D., Tikhomirov V.M. Teoriya ekstremal'nykh zadach (Theory of extremal problems), Moscow: Nauka, 1974, 479 p.
26. Isakova E.A., Logunova G.V., Patsko V.S. Computation of stable bridges for linear differential games with fixed time of termination, Algoritmy i programmy resheniya lineinykh differentsial'nykh igr, Sverdlovsk: Ural Scientific Center, 1984, pp. 127-158 (in Russian).
27. Kleimenov A.F. Neantagonisticheskie pozitsionnye differentsial'nye igry (Nonantagonistic positional differential games), Yekaterinburg: Nauka, 1993, 185 p.
28. Kovrizhnykh A.Yu. On the problem of conflict control with a quasipositional functional, Proceedings of the Steklov Institute of Mathematics, 2000, suppl. 2, pp. S79-S93.
29. Cormen T.H., Leiserson C.E., Rivest R.L., Stein C. Introduction to Algorithms, MIT press, 2001, 1202 p. Translated under the title Algoritmy: postroenie i analiz, Moscow: Vil'yams, 2005, 1296 p.
30. Krasovskii A.A., Krasovskii A.N. Nonlinear positional differential game in the class of mixed strategies, Proceedings of the Steklov Institute of Mathematics, 2012, vol. 277, issue 1, pp. 137-143. DOI: 10.1134/S0081543812040098
31. Krasovskii A.N. On minimax position control, Journal of Applied Mathematics and Mechanics, 1980, vol. 44, issue 4, pp. 425-430. DOI: 10.1016/0021-8928(80)90028-3
32. Krasovskii A.N. Construction of mixed strategies on the basis of stochastic programs, Journal of Applied Mathematics and Mechanics, 1987, vol. 51, issue 2, pp. 144-149. DOI: 10.1016/0021-8928(87)90055-4
33. Krasovskii A.N. Sintez smeshannykh strategii upravleniya (Synthesis of mixed control strategies), Sverdlovsk: Ural State University, 1988, 151 p.
34. Krasovskii A.N., Reshetova T.N. Upravlenie pri defitsite informatsii: Uchebnoe posobie (Control under lack of information: textbook), Sverdlovsk: Ural State University, 1990, 104 p.
35. Krasovskii N.N. Igrovye zadachi o vstreche dvizhenii (Game-theoretical problems of meeting of motions), Moscow: Nauka, 1970, 420 p.
36. Krasovskii N.N. On the problem of damping of a linear system under minimum control intensity, Journal of Applied Mathematics and Mechanics, 1965, vol. 29, issue 2, pp. 245-253. DOI: 10.1016/0021-8928(65)90028-6
37. Krasovskii N.N. Upravlenie dinamicheskoi sistemoi (Control of a dynamic system), Moscow: Nauka, 1985, 516 p.
38. Krasovskii N.N., Lukoyanov N.Yu. Problem of conflict control with hereditary information, Journal of Applied Mathematics and Mechanics, 1996, vol. 60, no. 6, pp. 869-882. DOI: 10.1016/S0021-8928(96)00109-8
39. Krasovskii N.N., Reshetova T.N. On the program synthesis of a guaranteed control, Problems of Control and Information Theory, 1988, vol. 17, no. 6, pp. 1-11.
40. Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial'nye igry (Positional differential games), Moscow: Nauka, 1974, 456 p.
41. Krasovskii N.N., Tret'yakov V.E. To a pursuit problem under constraints on impulses of control forces, Differentsial'nye Uravneniya, 1966, vol. 2, no. 5, pp. 587-599 (in Russian).
42. Krasovskii A.N., Krasovskii N.N., Tret'yakov V.E. Stochastic programmed design for a deterministic positional differential game, Journal of Applied Mathematics and Mechanics, 1981, vol. 45, issue 4, pp. 425-430. DOI: 10.1016/0021-8928(81)90081-2
43. Kryazhimskii A.V. On the theory of positional differential games of approach-evasion, Doklady Akad. Nauk SSSR, 1978, vol. 239, no. 4, pp. 779-782 (in Russian).
44. Kurzhanskii A.B. Upravlenie i nablyudenie v usloviyakh neopredelennosti (Control and observation under conditions of uncertainty), Moscow: Nauka, 1977, 392 p.
45. Ledyaev Yu.S. Regular differential games with mixed constraints on the controls, Proceedings of the Steklov Institute of Mathematics, 1986, vol. 167, pp. 233-242.
46. Lokshin M.D. Differential games with integral restrictions on the controlling actions, Differential Equations, 1992, vol. 28, no. 11, pp. 1602-1610.
47. Lukoyanov N.Yu. The problem of computing the value of a differential game for a positional functional, Journal of Applied Mathematics and Mechanics, 1998, vol. 62, issue 2, pp. 177-186. DOI: 10.1016/S0021-8928(98)00023-9
48. Lukoyanov N.Yu. The problem of conflicting control with mixed constraints, Journal of Applied Mathematics and Mechanics, 1995, vol. 59, no. 6, pp. 911-919. DOI: 10.1016/0021-8928(95)00124-7
49. Lukoyanov N.Yu. A conflict control problem under mixed restrictions on control actions, Differential Equations, 1995, vol. 31, no. 9, pp. 1427-1436.
50. Lukoyanov N.Yu. How to compute the value of a positional differential game, Differential Equations, 2001, vol. 37, issue 1, pp. 17-26. DOI: 10.1023/A:1019260029465
51. Lukoyanov N.Yu. A differential game with nonterminal cost, Journal of Computer and Systems Sciences International, 1997, vol. 36, no. 1, pp. 77-82.
52. Maksimov V.I. On the existence of a saddle point in a difference-differential encounter-evasion game, Journal of Applied Mathematics and Mechanics, 1978, vol. 42, no. 1, pp. 13-20. DOI: 10.1016/0021-8928(78)90083-7
53. Mamadaliev N. On a pursuit problem with integral constraints on the players’ controls, Siberian Mathematical Journal, 2015, vol. 56, issue 1, pp. 107-124. DOI: 10.1134/S0037446615010115
54. Mezentsev A.V. Differentsial'nye igry s integral'nymi ogranicheniyami na upravleniya (Differential games with integral control constraints), Moscow: Moscow State University, 1988, 135 p.
55. Mikhalev D.K., Ushakov V.N. Two algorithms for approximate construction of the set of positional absorption in the game problem of pursuit, Automation and Remote Control, 2007, vol. 68, issue 11, pp. 2056-2070. DOI: 10.1134/S0005117907110136
56. Mishchenko E.F. Pursuit and evasion problems in differential games theory, Izvestiya Akademii Nauk SSSR. Tekhnicheskaya Kibernetika, 1971, no. 5, pp. 3-9 (in Russian).
57. Nikol'skii M.S. Linear differential pursuit games with integral constraints, Differential Equations, 1992, vol. 28, no. 2, pp. 190-193.
58. Nikol'skii M.S. Pervyi pryamoi metod L.S. Pontryagina v differentsial'nykh igrakh (The first direct method of L.S. Pontryagin in differential games), Moscow: Moscow State University, 1984, 65 p.
59. Nikol'skii M.S. A direct method in linear differential games with integral constraints, Upravlyaemye Sistemy, 1969, no. 2, pp. 49-58 (in Russian).
60. Nikol'skii M.S. A direct method in linear differential games with general integral constraints, Differentsial'nye Uravneniya, 1972, vol. 8, no. 6, pp. 964-971 (in Russian).
61. Osipov Yu.S. Differential games of systems with aftereffect, Doklady Akad. Nauk SSSR, 1971, vol. 196, no. 4, pp. 779-782 (in Russian).
62. Osipov Yu.S., Pimenov V.G. On positional control under aftereffect in the controlling forces, Journal of Applied Mathematics and Mechanics, 1981, vol. 45, issue 2, pp. 159-163. DOI: 10.1016/0021-8928(81)90028-9
63. Patsko V.S., Turova V.L. Chislennoe reshenie differentsial'nykh igr na ploskosti (Numerical solving of differential games on a plain), Yekaterinburg: Ural Branch of the Russian Academy of Sciences, 1995, 77 p.
64. Petrov N.N. About existence of the value of a pursuit game, Doklady Akad. Nauk SSSR, 1970, vol. 190, no. 6, pp. 621-624 (in Russian).
65. Petrov N.N. On a group pursuit problem, Automation and Remote Control, 1996, vol. 57, no. 6, pp. 808-813.
66. Petrosyan L.A. Differentsial'nye igry presledovaniya (Pursuit differential games), Leningrad: Leningrad State University, 1977, 222 p.
67. Polovinkin E.S., Ivanov G.E., Balashov M.V., Konstantinov R.V., Khorev A.V. An algorithm for the numerical solution of linear differential games, Sbornik: Mathematics, 2001, vol. 192, no. 10, pp. 1515-1542. DOI: 10.1070/SM2001v192n10ABEH000604
68. Pontryagin L.S. About linear differential games. I, Doklady Akad. Nauk SSSR, 1967, vol. 174, no. 6, pp. 1278-1280 (in Russian).
69. Pontryagin L.S. About linear differential games. II, Doklady Akad. Nauk SSSR, 1967, vol. 175, no. 4, pp. 764-766 (in Russian).
70. Pshenichnyi B.N. Structure of differential games, Doklady Akad. Nauk SSSR, 1969, vol. 184, no. 2, pp. 285-187 (in Russian).
71. Pshenichnyi B.N., Onopchuk Yu.N. Linear differential games with integral constraints, Izvestiya Akademii Nauk SSSR. Tekhnicheskaya Kibernetika, 1968, no. 1, pp. 13-22 (in Russian).
72. Pshenichnyi B.N., Sagaidak M.I. Differential games of prescribed duration, Cybernetics, 1970, vol. 6, issue 2, pp. 72-83. DOI: 10.1007/BF01070503
73. Rokafellar R. Convex analysis, Princeton: Princeton University Press, 1970. Translated under the title Vypuklyi analiz, Moscow: Mir, 1973, 469 p.
74. Samatov B.T. The pursuit-evasion problem under integral-geometric constraints on pursuer controls, Automation and Remote Control, 2013, vol. 74, issue 7, pp. 1072-1081. DOI: 10.1134/S0005117913070023
75. Solomatin A.M., Ushakov V.N. Construction of a positional absorption set in a linear game with integral constraints, Upravlenie i Otsenivanie v Dinamicheskikh Sistemakh, 1982, pp. 74-89 (in Russian).
76. Straustrup B. The C++ programming language: special edition, Addison Wesley, 2000, 1029 p. Translated under the title Yazyk programmirovaniya C++. Spetsial'noe izdanie, Moscow: Binom, 2011, 1136 p.
77. Subbotin A.I., Ushakov V.N. Alternative for an encounter-evasion differential game with integral constraints on the players'controls, Journal of Applied Mathematics and Mechanics, 1975, vol. 39, issue 3, pp. 367-375. DOI: 10.1016/0021-8928(75)90001-5
78. Subbotin A.I., Chentsov A.G. Optimizatsiya garantii v zadachakh upravleniya (Optimization of guarantee in control problems), Moscow: Nauka, 1981, 288 p.
79. Subbotin A.I. Minimaksnye neravenstva i uravneniya Gamil'tona-Yakobi (Minimax inequalities and Hamilton-Jacobi equations), Moscow: Nauka, 1991, 216 p.
80. Subbotina N.N., Subbotin A.I. Alternative for the encounter-evasion differential game with constraints on the momenta of the players' controls, Journal of Applied Mathematics and Mechanics, 1975, vol. 39, issue 3, pp. 376-385. DOI: 10.1016/0021-8928(75)90002-7
81. Subbotina N.N. The method of characteristics for Hamilton-Jacobi equations and applications to dynamical optimization, Journal of Mathematical Sciences, 2004, vol. 135, issue 3, pp. 2955-3091. DOI: 10.1007/s10958-006-0146-2
82. Taras'ev A.M. Approximation schemes for constructing minimax solutions of Hamilton-Jacobi equations, Journal of Applied Mathematics and Mechanics, 1994, vol. 58, issue 2, pp. 207-221. DOI: 10.1016/0021-8928(94)90049-3
83. Taras'ev A.M., Uspenskii A.A., Ushakov V.N. Approximation schemes and finite-difference operators for constructing generalized solutions of Hamilton-Jacobi equations, Izv. Ross. Akad. Nauk, Tekh. Kibern., 1994, no. 3, pp. 173-185 (in Russian).
84. Taras'ev A.M., Uspenskii A.A., Ushakov V.N. Finite-difference method of constructing functions of optimal guaranteed result, Gagarinskie nauchnye chteniya po kosmonavtike i aviatsii. 1991 (Gagarin Readings on space and aviation), Moscow: Nauka, 1992, pp. 166-172.
85. Ukhobotov V.I. On a class of differential games with an integral constraint, Journal of Applied Mathematics and Mechanics, 1977, vol. 41, issue 5, pp. 838-844. DOI: 10.1016/0021-8928(77)90166-6
86. Ukhobotov V.I. A type of linear game with mixed constraints on the controls, Journal of Applied Mathematics and Mechanics, 1987, vol. 51, issue 2, pp. 139-144. DOI: 10.1016/0021-8928(87)90054-2
87. Ukhobotov V.I., Gushchin D.V. About one class of similar differential game with mixed limitations, Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki, 2010, issue 3, pp. 81-86 (in Russian). DOI: 10.20537/vm100310
88. Ushakov V.N. Extremal strategies in differential games with integral constraints, Journal of Applied Mathematics and Mechanics, 1972, vol. 36, issue 1, pp. 12-19. DOI: 10.1016/0021-8928(72)90076-7
89. Ushakov V.N., Khripunov A.P. Approximate construction of solutions in game-theoretic control problems, Journal of Applied Mathematics and Mechanics, 1997, vol. 61, issue 3, pp. 401-408. DOI: 10.1016/S0021-8928(97)00051-8
90. Fan Ky. Minimax theorems, Proceedings of the National Academy of Sciences of the United States of America, 1953, vol. 39, no. 1, pp. 42-47.
91. Chentsov A.G. On a game problem of converging at a given instant of time, Mathematics of the USSR-Sbornik, 1976, vol. 28, no. 3, pp. 353-376. DOI: 10.1070/SM1976v028n03ABEH001657
92. Chernous'ko F.L. Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem. Metod ellipsoidov (Phase state estimation in dynamical systems. Method of ellipsoids), Moscow: Nauka, 1988, 319 p.
93. Chernous'ko F.L., Melikyan A.A. Igrovye zadachi upravleniya i poiska (Game problems of control and search), Moscow: Nauka, 1978, 270 p.
94. Chikrii A.A. Conflict controlled processes, Springer Netherlands, 1997, XX+404 p. DOI: 10.1007/978-94-017-1135-7
95. Chikrii A.A, Belousov A.A. On linear differential games with convex integral constraints, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2013, vol. 19, no. 4, pp. 308-319 (in Russian).
96. Chistyakov S.V. On solving pursuit game problems, Journal of Applied Mathematics and Mechanics, 1977, vol. 41, issue 5, pp. 845-852. DOI: 10.1016/0021-8928(77)90167-8
97. Alton K., Mitchell I.M. Fast marching methods for stationary Hamilton-Jacobi equations with axis-aligned anisotropy, SIAM J. Numer. Anal., 2008, vol. 47, no. 1, pp. 363-385.
98. Bardi M., Falcone M., Soravia P. Numerical methods for pursuit-evasion games via viscosity solutions, Stochastic and differential games, Annals of the International Society of Dynamic Games, vol. 4, Boston: Birkhäuser, 1999, pp. 105-175. DOI: 10.1007/978-1-4612-1592-9_3
99. Barron E.N. Differential games with maximum cost, Nonlinear Analysis: Theory, Methods and Applications, 1990, vol. 14, issue 11, pp. 971-989. DOI: 10.1016/0362-546X(90)90113-U
100. Basar T., Bernhard P. H-infinity optimal control and related minimax design problems: a dynamic game approach, Boston: Birkhäuser, 1995, 428 p.
101. Berkovitz L.D. Characterization of the values of differential games, Applied Mathematics and Optimization, 1988, vol. 17, issue 1, pp. 177-183. DOI: 10.1007/BF01448365
102. Blaquiere A., Gerard F., Leitmann G. Quantitative and qualitative games, New York etc.: Academic Press, 1969. 172 p.
103. Botkin N.D., Hoffmann K.-H., Turova V.L. Stable numerical schemes for solving Hamilton-Jacobi-Bellman-Isaacs equations, SIAM J. Sci. Comput., 2011, vol. 33, issue 2, pp. 992-1007. DOI: 10.1137/100801068
104. Cardaliaguet P., Quincampoix M., Saint-Pierre P. Numerical methods for optimal control and differential games, Ceremade CNRS URA 749, University of Paris Dauphine, 1995.
105. Cardaliaguet P., Quincampoix M., Saint-Pierre P. Set-valued numerical analysis for optimal control and differential games, Stochastic and differential games, Annals of the International Society of Dynamic Games, vol. 4, Boston: Birkhäuser, 1999, pp. 177-247. DOI: 10.1007/978-1-4612-1592-9_4
106. Chernousko F.L., Ananievski I.M., Reshmin S.A. Control of nonlinear dynamical systems. Methods and applications, Berlin-Heidelberg: Springer, 2008, 396 p. DOI: 10.1007/978-3-540-70784-4
107. Elliot R.J., Kalton N.J. The existence of value for differential games, Memoirs of the American Mathematical Society, 1972, no. 126, 67 p. DOI: 10.1090/memo/0126
108. Evans L.C., Ishii H. Differential games and nonlinear first order PDE on bounded domains, Manuscripta Mathematica, 1984, vol. 49, issue 2, pp. 109-139. DOI: 10.1007/BF01168747
109. Falcone M. Recent results in the approximation of nonlinear optimal control problems, Large-scale scientific computing, Lecture Notes in Computer Science, vol. 8353, Berlin-Heidelberg: Springer, 2014, pp. 15-32. DOI: 10.1007/978-3-662-43880-0_2
110. Fleming W.H. A note on differential games of prescribed duration, Contributions to the Theory of Games, Annals of Mathematical Studies, 1957, vol. 3, pp. 407-416.
111. Fleming W.H. The convergence problem for differential games, Journal of Mathematical Analysis and Applications, 1961, vol. 3, issue 1, pp. 102-116. DOI: 10.1016/0022-247X(61)90009-9
112. Fleming W.H. The convergence problem for differential games, Advances in Game Theory. (AM-52) (Annals of Mathematics Studies), Princeton: Princeton University Press, 1964, pp. 195-210.
113. Friedman A. Differential games, New York: Wiley Interscience, 1971, 350 p.
114. Ho Y.C., Bryson A., Baron S. Differential games and optimal pursuit-evasion strategies, IEEE Trans. Autom. Contr., 1965, vol. 10, issue 4, pp. 385-389. DOI: 10.1109/TAC.1965.1098197
115. Ibragimov G.I., Azamov A.A., Khakestari M. Solution of a linear pursuit-evasion game with integral constraints, ANZIAM Journal, 2010, vol. 52, pp. E59-E75. DOI: 10.21914/anziamj.v52i0.3605
116. Krasovskii A.N., Choi Y.S. Stochastic control with the leaders-stabilizers, Yekaterinburg: Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 2001, 51 p.
117. Krasovskii A.N., Krasovskii N.N. Control under lack of information, Berlin etc.: Birkhäuser, 1995. 322 p.
118. Kumkov S.S., Patsko V.S. Construction of singular surfaces in linear differential games, Advances in dynamic games and applications, Annals of the International Society of Dynamic Games, vol. 6, Basel: Birkhäuser, 2001. pp. 185-202. DOI: 10.1007/978-1-4612-0155-7
119. Kurzhanski A.B., Valyi I. Ellipsoidal calculus for estimation and control, Boston: Birkhäuser, 1996, XV+321 p.
120. Kurzhanski A.B., Varaiya P. Dynamics and control of trajectory tubes, Springer, 2014, 445 p.
121. Lewin J. Differential games: Theory and methods for solving game problems with singular surfaces, New York: Springer-Verlag, 1994, 242 p.
122. Lions P.-L., Souganidis P.E. Differential games, optimal control and directional derivatives of viscosity solutions of Bellman's and Isaacs' equations, SIAM Journal on Control and Optimization, 1985, vol. 23, issue 4, pp. 566-583. DOI: 10.1137/0323036
123. Melikyan A.A. Generalaized characteristics of first order PDEs: Applications in optimal control and differential games, Boston: Birkhäuser, 1998, 310 p.
124. Miller B., Rubinovich E.Y. Impulsive control in continuous and discrete-continuous systems, New York: Kluwer Academic / Plenum Publishers, 2003, 447 p.
125. Motta M., Rampazzo F. Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential and Integral Equations, 1995, vol. 8, no. 2, pp. 269-288.
126. Roxin E. Axiomatic approach in differential games, Journal of Optimization Theory and Applications, 1969, vol. 3, issue 3, pp. 153-163. DOI: 10.1007/BF00929440
127. Sethian J.A. Level set methods and fast marching methods: Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, Cambridge University Press, 1999, 404 p.
128. Subbotin A.I. Generalized solutions of first-order PDEs: The dynamical optimization perspective, Boston etc.: Birkhäuser, 1995, 312 p.
129. Varaiya P., Lin J. Existence of saddle points in differential games, SIAM Journal on Control, 1969, vol. 7, issue 1, pp. 141-157. DOI: 10.1137/0307011
130. Williams J.D. The compleat strategyst, New York: McGraw-Hill, 1966, 268 p.
131. Gomoyunov M.I., Kornev D.V. On calculating the value of a differential game in the class of counterstrategies, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2013, vol. 19, no. 1, pp 59-68 (in Russian).
132. Gomoyunov M.I., Kornev D.V., Lukoyanov N.Yu. To the problem of positional guarantee optimization under conditions of delayed control, Trudy XII Vserossiiskogo soveshchaniya po problemam upravleniya (VSPU-2014) (Proceedings of XII All-Russian Conference on Control Problems (RCCP–2014)), Moscow: Institute of Control Sciences of RAS, 2014, pp. 1268-1279 (in Russian).
133. Gomoyunov M.I., Kornev D.V., Lukoyanov N.Yu. On the numerical solution of a minimax control problem with a positional functional, Proceedings of the Steklov Institute of Mathematics, 2015, vol. 291, suppl. 1, pp. 77-95. DOI: 10.1134/S0081543815090060
134. Kornev D.V. On the question of program realization of a solution of a differential game with the nonterminal payoff, Sovremennye problemy matematiki: Trudy 42-i Regional'noi molodezhnoi konferentsii (Contemporary Problems of Mathematics: Proceedings of the 42th Regional Youth Conference), Yekaterinburg: Ural Branch of the Russian Academy of Sciences, 2011, pp. 34-37 (in Russian).
135. Kornev D.V. On numerical solving differential games on minimax of a positional functional in classes of mixed strategies, Systems Dynamics and Control Processes (SDCP'2014), Proceedings of International Conference dedicated to the 90th anniversary of Academician N.N. Krasovskii, Russia, Yekaterinburg, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ural Federal University, September 15-20, 2014, pp. 111-113 (in Russian).
136. Kornev D.V. On numerical solving differential games with nonterminal payoff, Sovremennye problemy matematiki: Tezisy Mezhdunarodnoi (43-i Vserossiiskoi) molodezhnoi shkoly-konferentsii (Contemporary Problems of Mathematics: Abstracts of the 43th All-Russian Youth Conference), Yekaterinburg: Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 2012, pp. 136-138 (in Russian).
137. Kornev D.V. On numerical solution of positional differential games with nonterminal payoff, Automation and Remote Control, 2012, vol. 73, issue 11, pp. 1808-1821. DOI: 10.1134/S0005117912110045
138. Kornev D.V. On a numerical method of solving conflict control problems, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2012, no. 1 (39), pp. 67-68 (in Russian).
139. Kornev D.V. About on numerical method for solving positional differential games in mixed strategies, Vestn. Tambov. Univ. Ser. Estestv. Tekh. Nauki, 2013, vol. 18, no. 5-2, pp. 2556-2558 (in Russian).
140. Kornev D.V. About guarantee optimization under integral constraints on control actions and nonterminal quality index, Trudy XII Vserossiiskogo soveshchaniya po problemam upravleniya (VSPU-2014) (Proceedings of XII All-Russian Conference on Control Problems (RCCP–2014)), Moscow: Institute of Control Sciences of RAS, 2014, pp. 2059-2070 (in Russian).
141. Kornev D.V. Programmnyi kompleks dlya resheniya pozitsionnykh differentsial'nykh igr s neterminal'noi platoi (The program complex for solving positional differential games with nonterminal payoff). Patent RF, no. 2015614531, 2015.
142. Kornev D.V., Lukoyanov N.Yu. To the problem of dynamical guarantee optimization under geometrical and integral constraints on control capabilities, Absracts of II International Seminar “Control Theory and Theory of Generalized Solutions of Hamilton-Jacobi Equations” (CGS'2015) dedicated to the 70th anniversary of Academician A.I. Subbotin, Russia, Yekaterinburg, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ural Federal University, April 1-3, 2015, pp. 94-95 (in Russian).
143. Kornev D.V., Lukoyanov N.Yu. On a minimax control problem for a positional functional under geometric and integral constraints on control actions, Proceedings of the Steklov Institute of Mathematics, 2015, vol. 293, issue 2, pp. 85-100. DOI: 10.1134/S0081543816050096
144. Kornev D.V., Lukoyanov N.Yu. On numerical solution of differential games with nonterminal payoff in classes of mixed strategies, Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki, 2013, issue 3, pp. 34-48 (in Russian). DOI: 10.20537/vm130304
145. Kornev D.V., Lukoyanov N.Yu. Numerical methods of solving linear positional differential games with nonterminal payoff, Algoritmicheskii analiz neustoichivykh zadach: Tezisy dokladov Mezhdunarodnoi konferentsii, posvyashchennoi pamyati V.K. Ivanova (Algorithmic analysis of problems: abstacts of international conference dedicated to the memory of V.K. Ivanov), Yekaterinburg: Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 2011, p. 248.
146. Gomoyunov M.I., Kornev D.V., Lukoyanov N.Yu. Game theory applications to guarantee optimization in dynamical systems with control delays, International Game Theory Review, 2014, vol. 16, issue 02, 1440010, 19 p. DOI: 10.1142/S0219198914400106
147. Kornev D., Lukoyanov N. On numerical solution of differential games in classes of mixed strategies, IFAC Proceedings Volumes, 2014, vol. 47, issue 3, pp. 1550-1555. DOI: 10.3182/20140824-6-ZA-1003.02208
148. Kornev D., Lukoyanov N. On numerical solving of differential games with nonterminal payoff, IFAC Proceedings Volumes, 2012, vol. 45, issue 25, pp. 71-76. DOI: 10.3182/20120913-4-IT-4027.00026
Received
2016-10-01
Published 2016-11-20
Published 2016-11-20