Numerical methods for solving differential games with nonterminal payoff

D. V. Kornev

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 48 (2016) no. 2, pp. 82-151.

Résumé

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.

Détail

Keywords: control theory, differential games, feedback strategies, resource constraints.

Bibliographie

[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, 31–35 (in Russian)

[2] Isaacs R., Differential games, John Wiley and Sons, New York, 1965 | Zbl

[3] Al'brekht E. G., “Constructing the approximate solutions of some quasilinear differential games”, Proc. Steklov Inst. Math., 2000, suppl. 1, S24–S34 | MR | Zbl | Zbl

[4] Balashov M. V., “On the $P$-property of compact convex sets”, Mathematical Notes, 71:3 (2002), 295–304 | DOI | DOI | MR | Zbl

[5] Batukhtin V. D., “Extremal aiming in a nonlinear convergence game”, Soviet Mathematics. Doklady, 13 (1972), 1427–1431 | Zbl

[6] Berdyshev Yu. I., “Problem of successive optimization without time decomposition”, Cybernetics, 23:4 (1987), 474–479 | DOI | Zbl

[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., 66:5 (2002), 709–718 | DOI | MR | Zbl

[8] Berdyshev Yu. I., Chentsov A. G., “Optimization of a weighted criterion function in one control problem”, Cybernetics, 22:1 (1986), 67–74 | DOI | MR | Zbl

[9] Botkin N. D., “Differential pursuit game with mixed control constraints”, Automation and Remote Control, 53:6-1 (1992), 793–800 | MR | Zbl

[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, 288, suppl. 1 (2015), 54–69 | DOI | MR

[11] Gornov A. Yu., Computational technologies of solving optimal control problems, Nauka, Novosibirsk, 2009, 279 pp.

[12] Gornov A. Yu., Tyatyushkin A. I., Finkelstein E. A., “Numerical methods for solving applied optimal control problems”, Computational Mathematics and Mathematical Physics, 53:12 (2013), 1825–1838 | DOI | DOI | MR | Zbl

[13] Grigorenko N. L., “On structure of one class of differential games with common integral constraints”, Upravlyaemye Sistemy, 1974, no. 12, 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, 275, suppl. 1 (2011), 49–56 | DOI | MR | Zbl

[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, 7:1 (1996), 101–116 | DOI | MR | Zbl

[16] Dar'in A. N., Kurzhanskii A. B., “Method of dynamical programming in problems of control synthesis under polytypic and double contstraints”, Control problems and applications (engineering, manufacturing, economics), Proceedings of international conference (Minsk, 2005), v. 2, 51–65 (in Russian)

[17] Dar'in A. N., Kurzhanskii A. B., “Control under indeterminacy and double constraints”, Differential Equations, 39:11 (2003), 1554–1567 | DOI | MR | Zbl

[18] Dvurechenskii P. E., Ivanov G. E., “Algorithms for computing Minkowski operators and their application in differential games”, Computational Mathematics and Mathematical Physics, 54:2 (2014), 235–264 | DOI | DOI | MR | Zbl

[19] Dykhta V. A., Samsonyuk O. N., Optimal impulse control and its applications, Fizmatlit, M., 2003, 256 pp.

[20] Zhukovskii V. I., Chikrii A. A., Linear quadratic differential games, Naukova dumka, Kiev, 1994, 241 pp.

[21] Zavalishchin S. T., Sesekin A. N., Impulse procesess. Models and applications, Nauka, M., 1991, 256 pp.

[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, 51:4 (2011), 550–574 | DOI | MR | Zbl

[23] Ivanov G. E., Polovinkin E. S., “On strongly convex linear differential games”, Differentsial'nye uravneniya, 31:10 (1995), 1641–1648 (in Russian) | Zbl

[24] Ignatenko A. P., “On one approach problem with integral-geometry constraints”, Teoriya optimal'nykh reshenii, 2007, no. 6, 74–79 (in Russian)

[25] Ioffe A. D., Tikhomirov V. M., Theory of extremal problems, Nauka, M., 1974, 479 pp.

[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, Ural Scientific Center, Sverdlovsk, 1984, 127–158 (in Russian) | Zbl

[27] Kleimenov A. F., Nonantagonistic positional differential games, Nauka, Yekaterinburg, 1993, 185 pp.

[28] Kovrizhnykh A. Yu., “On the problem of conflict control with a quasipositional functional”, Proceedings of the Steklov Institute of Mathematics, 2000, suppl. 2, S79–S93 | MR | Zbl

[29] Cormen T. H., Leiserson C. E., Rivest R. L., Stein C., Introduction to Algorithms, MIT press, 2001, 1202 pp. | MR | Zbl

[30] Krasovskii A. A., Krasovskii A. N., “Nonlinear positional differential game in the class of mixed strategies”, Proceedings of the Steklov Institute of Mathematics, 277:1 (2012), 137–143 | DOI | MR | Zbl

[31] Krasovskii A. N., “On minimax position control”, Journal of Applied Mathematics and Mechanics, 44:4 (1980), 425–430 | DOI | MR | Zbl

[32] Krasovskii A. N., “Construction of mixed strategies on the basis of stochastic programs”, Journal of Applied Mathematics and Mechanics, 51:2 (1987), 144–149 | DOI | MR | Zbl

[33] Krasovskii A. N., Synthesis of mixed control strategies, Ural State University, Sverdlovsk, 1988, 151 pp.

[34] Krasovskii A. N., Reshetova T. N., Control under lack of information, Textbook, Ural State University, Sverdlovsk, 1990, 104 pp.

[35] Krasovskii N. N., Game-theoretical problems of meeting of motions, Nauka, M., 1970, 420 pp.

[36] Krasovskii N. N., “On the problem of damping of a linear system under minimum control intensity”, Journal of Applied Mathematics and Mechanics, 29:2 (1965), 245–253 | DOI | MR | Zbl

[37] Krasovskii N. N., Control of a dynamic system, Nauka, M., 1985, 516 pp.

[38] Krasovskii N. N., Lukoyanov N. Yu., “Problem of conflict control with hereditary information”, Journal of Applied Mathematics and Mechanics, 60:6 (1996), 869–882 | DOI | MR | Zbl

[39] Krasovskii N. N., Reshetova T. N., “On the program synthesis of a guaranteed control”, Problems of Control and Information Theory, 17:6 (1988), 1–11 | DOI | MR

[40] Krasovskii N. N., Subbotin A. I., Positional differential games, Nauka, M., 1974, 456 pp.

[41] Krasovskii N. N., Tret'yakov V. E., “To a pursuit problem under constraints on impulses of control forces”, Differentsial'nye Uravneniya, 2:5 (1966), 587–599 (in Russian) | Zbl

[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, 45:4 (1981), 425–430 | DOI | MR | Zbl

[43] Kryazhimskii A. V., “On the theory of positional differential games of approach-evasion”, Doklady Akad. Nauk SSSR, 239:4 (1978), 779–782 (in Russian) | Zbl

[44] Kurzhanskii A. B., Control and observation under conditions of uncertainty, Nauka, M., 1977, 392 pp.

[45] Ledyaev Yu. S., “Regular differential games with mixed constraints on the controls”, Proceedings of the Steklov Institute of Mathematics, 167 (1986), 233–242 | MR | Zbl | Zbl

[46] Lokshin M. D., “Differential games with integral restrictions on the controlling actions”, Differential Equations, 28:11 (1992), 1602–1610 | MR | Zbl

[47] Lukoyanov N. Yu., “The problem of computing the value of a differential game for a positional functional”, Journal of Applied Mathematics and Mechanics, 62:2 (1998), 177–186 | DOI | MR | Zbl

[48] Lukoyanov N. Yu., “The problem of conflicting control with mixed constraints”, Journal of Applied Mathematics and Mechanics, 59:6 (1995), 911–919 | DOI | MR | Zbl

[49] Lukoyanov N. Yu., “A conflict control problem under mixed restrictions on control actions”, Differential Equations, 31:9 (1995), 1427–1436 | MR | Zbl

[50] Lukoyanov N. Yu., “How to compute the value of a positional differential game”, Differential Equations, 37:1 (2001), 17–26 | DOI | MR | Zbl

[51] Lukoyanov N. Yu., “A differential game with nonterminal cost”, Journal of Computer and Systems Sciences International, 36:1 (1997), 77–82 | Zbl

[52] Maksimov V. I., “On the existence of a saddle point in a difference-differential encounter-evasion game”, Journal of Applied Mathematics and Mechanics, 42:1 (1978), 13–20 | DOI | MR

[53] Mamadaliev N., “On a pursuit problem with integral constraints on the players' controls”, Siberian Mathematical Journal, 56:1 (2015), 107–124 | DOI | MR | Zbl

[54] Mezentsev A. V., Differential games with integral control constraints, Moscow State University, M., 1988, 135 pp.

[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, 68:11 (2007), 2056–2070 | DOI | MR | Zbl

[56] Mishchenko E. F., “Pursuit and evasion problems in differential games theory”, Izvestiya Akademii Nauk SSSR. Tekhnicheskaya Kibernetika, 1971, no. 5, 3–9 (in Russian)

[57] Nikol'skii M. S., “Linear differential pursuit games with integral constraints”, Differential Equations, 28:2 (1992), 190–193 | MR

[58] Nikol'skii M. S., The first direct method of L.S. Pontryagin in differential games, Moscow State University, M., 1984, 65 pp.

[59] Nikol'skii M. S., “A direct method in linear differential games with integral constraints”, Upravlyaemye Sistemy, 1969, no. 2, 49–58 (in Russian)

[60] Nikol'skii M. S., “A direct method in linear differential games with general integral constraints”, Differentsial'nye Uravneniya, 8:6 (1972), 964–971 (in Russian) | Zbl

[61] Osipov Yu. S., “Differential games of systems with aftereffect”, Doklady Akad. Nauk SSSR, 196:4 (1971), 779–782 (in Russian) | Zbl

[62] Osipov Yu. S., Pimenov V. G., “On positional control under aftereffect in the controlling forces”, Journal of Applied Mathematics and Mechanics, 45:2 (1981), 159–163 | DOI | MR | Zbl

[63] Patsko V. S., Turova V. L., Numerical solving of differential games on a plain, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 1995, 77 pp.

[64] Petrov N. N., “About existence of the value of a pursuit game”, Doklady Akad. Nauk SSSR, 190:6 (1970), 621–624 (in Russian)

[65] Petrov N. N., “On a group pursuit problem”, Automation and Remote Control, 57:6 (1996), 808–813 | MR | Zbl

[66] Petrosyan L. A., Pursuit differential games, Leningrad State University, Leningrad, 1977, 222 pp.

[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, 192:10 (2001), 1515–1542 | DOI | DOI | MR | Zbl

[68] Pontryagin L. S., “About linear differential games. I”, Doklady Akad. Nauk SSSR, 174:6 (1967), 1278–1280 (in Russian) | Zbl

[69] Pontryagin L. S., “About linear differential games. II”, Doklady Akad. Nauk SSSR, 175:4 (1967), 764–766 (in Russian) | Zbl

[70] Pshenichnyi B. N., “Structure of differential games”, Doklady Akad. Nauk SSSR, 184:2 (1969), 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, 13–22 (in Russian)

[72] Pshenichnyi B. N., Sagaidak M. I., “Differential games of prescribed duration”, Cybernetics, 6:2 (1970), 72–83 | DOI

[73] Rokafellar R., Convex analysis, Princeton University Press, Princeton, 1970 | MR

[74] Samatov B. T., “The pursuit-evasion problem under integral-geometric constraints on pursuer controls”, Automation and Remote Control, 74:7 (2013), 1072–1081 | DOI | MR | Zbl

[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, 74–89 (in Russian)

[76] Straustrup B., The C++ programming language, Special edition, Addison Wesley, 2000, 1029 pp.

[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, 39:3 (1975), 367–375 | DOI | MR | Zbl

[78] Subbotin A. I., Chentsov A. G., Optimization of guarantee in control problems, Nauka, M., 1981, 288 pp.

[79] Subbotin A. I., Minimax inequalities and Hamilton–Jacobi equations, Nauka, M., 1991, 216 pp.

[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, 39:3 (1975), 376–385 | DOI | MR | Zbl

[81] Subbotina N. N., “The method of characteristics for Hamilton–Jacobi equations and applications to dynamical optimization”, Journal of Mathematical Sciences, 135:3 (2004), 2955–3091 | DOI | MR | Zbl

[82] Taras'ev A. M., “Approximation schemes for constructing minimax solutions of Hamilton–Jacobi equations”, Journal of Applied Mathematics and Mechanics, 58:2 (1994), 207–221 | DOI | MR | Zbl

[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, 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”, Gagarin Readings on space and aviation (1991), Nauka, M., 1992, 166–172

[85] Ukhobotov V. I., “On a class of differential games with an integral constraint”, Journal of Applied Mathematics and Mechanics, 41:5 (1977), 838–844 | DOI | MR

[86] Ukhobotov V. I., “A type of linear game with mixed constraints on the controls”, Journal of Applied Mathematics and Mechanics, 51:2 (1987), 139–144 | DOI | MR | Zbl

[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, no. 3, 81–86 (in Russian)

[88] Ushakov V. N., “Extremal strategies in differential games with integral constraints”, Journal of Applied Mathematics and Mechanics, 36:1 (1972), 12–19 | DOI | MR | Zbl

[89] Ushakov V. N., Khripunov A. P., “Approximate construction of solutions in game-theoretic control problems”, Journal of Applied Mathematics and Mechanics, 61:3 (1997), 401–408 | DOI | MR | Zbl

[90] Fan Ky, “Minimax theorems”, Proceedings of the National Academy of Sciences of the United States of America, 39:1 (1953), 42–47 | DOI | MR

[91] Chentsov A. G., “On a game problem of converging at a given instant of time”, Mathematics of the USSR-Sbornik, 28:3 (1976), 353–376 | DOI | Zbl

[92] Chernous'ko F. L., Phase state estimation in dynamical systems. Method of ellipsoids, Nauka, M., 1988, 319 pp.

[93] Chernous'ko F. L., Melikyan A. A., Game problems of control and search, Nauka, M., 1978, 270 pp.

[94] Chikrii A. A., Conflict controlled processes, Springer Netherlands, 1997, XX+404 pp. | DOI | MR

[95] Chikrii A. A., Belousov A. A., “On linear differential games with convex integral constraints”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 19, no. 4, 2013, 308–319 (in Russian)

[96] Chistyakov S. V., “On solving pursuit game problems”, Journal of Applied Mathematics and Mechanics, 41:5 (1977), 845–852 | DOI | MR

[97] Alton K., Mitchell I. M., “Fast marching methods for stationary Hamilton–Jacobi equations with axis-aligned anisotropy”, SIAM J. Numer. Anal., 47:1 (2008), 363–385 | DOI | MR

[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, 4, Birkhäuser, Boston, 1999, 105–175 | DOI | MR | Zbl

[99] Barron E. N., “Differential games with maximum cost”, Nonlinear Analysis: Theory, Methods and Applications, 14:11 (1990), 971–989 | DOI | MR | Zbl

[100] Basar T., Bernhard P., H-infinity optimal control and related minimax design problems: a dynamic game approach, Birkhäuser, Boston, 1995, 428 pp. | MR

[101] Berkovitz L. D., “Characterization of the values of differential games”, Applied Mathematics and Optimization, 17:1 (1988), 177–183 | DOI | MR | Zbl

[102] Blaquiere A., Gerard F., Leitmann G., Quantitative and qualitative games, Academic Press, New York etc., 1969, 172 pp. | MR

[103] Botkin N. D., Hoffmann K.-H., Turova V. L., “Stable numerical schemes for solving Hamilton–Jacobi–Bellman–Isaacs equations”, SIAM J. Sci. Comput., 33:2 (2011), 992–1007 | DOI | MR | Zbl

[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, 4, Birkhäuser, Boston, 1999, 177–247 | DOI | MR | Zbl

[106] Chernousko F. L., Ananievski I. M., Reshmin S. A., Control of nonlinear dynamical systems. Methods and applications, Springer, Berlin–Heidelberg, 2008, 396 pp. | DOI | Zbl

[107] Elliot R. J., Kalton N. J., The existence of value for differential games, Memoirs of the American Mathematical Society, 126, 1972, 67 pp. | DOI | MR

[108] Evans L. C., Ishii H., “Differential games and nonlinear first order PDE on bounded domains”, Manuscripta Mathematica, 49:2 (1984), 109–139 | DOI | MR | Zbl

[109] Falcone M., “Recent results in the approximation of nonlinear optimal control problems”, Large-scale scientific computing, Lecture Notes in Computer Science, 8353, Springer, Berlin–Heidelberg, 2014, 15–32 | DOI | MR

[110] Fleming W. H., “A note on differential games of prescribed duration”, Contributions to the Theory of Games, Annals of Mathematical Studies, 3, 1957, 407–416 | MR

[111] Fleming W. H., “The convergence problem for differential games”, Journal of Mathematical Analysis and Applications, 3:1 (1961), 102–116 | DOI | MR | Zbl

[112] Fleming W. H., “The convergence problem for differential games”, Advances in Game Theory, Annals of Mathematics Studies, 52, Princeton University Press, Princeton, 1964, 195–210 | MR

[113] Friedman A., Differential games, Wiley Interscience, New York, 1971, 350 pp. | MR | Zbl

[114] Ho Y. C., Bryson A., Baron S., “Differential games and optimal pursuit-evasion strategies”, IEEE Trans. Autom. Contr., 10:4 (1965), 385–389 | DOI | MR

[115] Ibragimov G. I., Azamov A. A., Khakestari M., “Solution of a linear pursuit-evasion game with integral constraints”, ANZIAM Journal, 52 (2010), E59–E75 | DOI | MR

[116] Krasovskii A. N., Choi Y. S., Stochastic control with the leaders-stabilizers, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 2001, 51 pp.

[117] Krasovskii A. N., Krasovskii N. N., Control under lack of information, Birkhäuser, Berlin etc., 1995, 322 pp. | MR

[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, 6, Birkhäuser, Basel, 2001, 185–202 | DOI | MR | Zbl

[119] Kurzhanski A. B., Valyi I., Ellipsoidal calculus for estimation and control, Birkhäuser, Boston, 1996, XV+321 pp. | MR

[120] Kurzhanski A. B., Varaiya P., Dynamics and control of trajectory tubes, Springer, 2014, 445 pp. | MR | Zbl

[121] Lewin J., Differential games: Theory and methods for solving game problems with singular surfaces, Springer-Verlag, New York, 1994, 242 pp. | MR

[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, 23:4 (1985), 566–583 | DOI | MR | Zbl

[123] Melikyan A. A., Generalaized characteristics of first order PDEs: Applications in optimal control and differential games, Birkhäuser, Boston, 1998, 310 pp. | MR

[124] Miller B., Rubinovich E. Y., Impulsive control in continuous and discrete-continuous systems, Kluwer Academic Plenum Publishers, New York, 2003, 447 pp. | MR | Zbl

[125] Motta M., Rampazzo F., “Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls”, Differential and Integral Equations, 8:2 (1995), 269–288 | MR | Zbl

[126] Roxin E., “Axiomatic approach in differential games”, Journal of Optimization Theory and Applications, 3:3 (1969), 153–163 | DOI | MR | Zbl

[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 pp. | MR | Zbl

[128] Subbotin A. I., Generalized solutions of first-order PDEs: The dynamical optimization perspective, Birkhäuser, Boston etc., 1995, 312 pp. | MR

[129] Varaiya P., Lin J., “Existence of saddle points in differential games”, SIAM Journal on Control, 7:1 (1969), 141–157 | DOI | Zbl

[130] Williams J. D., The compleat strategyst, McGraw-Hill, New York, 1966, 268 pp.

[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, 19, no. 1, 2013, 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”, Proceedings of XII All-Russian Conference on Control Problems, RCCP-2014, Institute of Control Sciences of RAS, M., 2014, 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, 291, suppl. 1 (2015), 77–95 | DOI | MR

[134] Kornev D. V., “On the question of program realization of a solution of a differential game with the nonterminal payoff”, Contemporary Problems of Mathematics, Proceedings of the 42th Regional Youth Conference, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 2011, 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), 111–113 (in Russian)

[136] Kornev D. V., “On numerical solving differential games with nonterminal payoff”, Contemporary Problems of Mathematics, Abstracts of the 43th All-Russian Youth Conference, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 2012, 136–138 (in Russian)

[137] Kornev D. V., “On numerical solution of positional differential games with nonterminal payoff”, Automation and Remote Control, 73:11 (2012), 1808–1821 | DOI | MR | Zbl

[138] Kornev D. V., “On a numerical method of solving conflict control problems”, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2012, no. 1(39), 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, 18:5-2 (2013), 2556–2558 (in Russian)

[140] Kornev D. V., “About guarantee optimization under integral constraints on control actions and nonterminal quality index”, Proceedings of XII All-Russian Conference on Control Problems, RCCP-2014, Institute of Control Sciences of RAS, M., 2014, 2059–2070 (in Russian)

[141] Kornev D. V., 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), 2015, 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, 293:2 (2015), 85–100 | DOI | MR

[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, no. 3, 34–48 (in Russian) | DOI

[145] Kornev D. V., Lukoyanov N. Yu., “Numerical methods of solving linear positional differential games with nonterminal payoff”, Algorithmic analysis of problems, Abstracts of international conference dedicated to the memory of V. K. Ivanov, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 2011, 248

[146] Gomoyunov M. I., Kornev D. V., Lukoyanov N. Y., “Game theory applications to guarantee optimization in dynamical systems with control delays”, International Game Theory Review, 16:02 (2014), 1440010, 19 pp. | DOI | MR | Zbl

[147] Kornev D., Lukoyanov N., “On numerical solution of differential games in classes of mixed strategies”, IFAC Proceedings Volumes, 47:3 (2014), 1550–1555 | DOI

[148] Kornev D., Lukoyanov N., “On numerical solving of differential games with nonterminal payoff”, IFAC Proceedings Volumes, 45:25 (2012), 71–76 | DOI