Differential games on minmax of the positional quality index. (English) Zbl 1431.91037

Summary: The paper goes back to the research of N. N. Krasovskii devoted to two-person zero-sum positional differential games on minmax of non-terminal quality indices, which evaluate a set of system’s states realized at given times. The first part of the paper gives a survey of the results concerning existence of the value and saddle point in such differential games. A special attention is paid to the case when the quality index has a certain positional structure. The second part of the paper overviews a method for constructing the value and optimal strategies in the case when the dynamical system is linear in the state vector, and the quality index has the appropriate convexity properties. The method is based on the recurrent procedure of constructing the upper convex hulls of certain auxiliary functions. To illustrate that this method can be numerically realized on modern computers, a model example is considered.


91A23 Differential games (aspects of game theory)
91A24 Positional games (pursuit and evasion, etc.)
49J35 Existence of solutions for minimax problems
91A05 2-person games
Full Text: DOI


[1] Fleming WH (1961) The convergence problem for differential games. J Math Anal Appl 3:102-116 · Zbl 0113.14705
[2] Friedman A (1971) Differential games. Wiley, New York, p 368
[3] Gomoyunov MI (2013) The optimization of a guaranteed result with a delay in the control. J Appl Math Mech 77(5):459-469 · Zbl 1432.49052
[4] Gomoyunov MI, Kornev DV (2016) On calculating the value of a differential game in the class of counter strategies. Ural Math J 2(1):38-47 · Zbl 1398.91087
[5] Gomoyunov MI, Kornev DV, Lukoyanov NYu (2014) Game theory applications to guarantee optimization in dynamical systems with control delays. Int Game Theory Rev 16(2):1440010 (19 pages) · Zbl 1287.49039
[6] Gomoyunov MI, Kornev DV, Lukoyanov NYu (2015) On the numerical solution of a minimax control problem with a positional functional. Proc Steklov Inst Math 291(Suppl. 1):S77-S95 · Zbl 1336.49037
[7] Gomoyunov MI, Lukoyanov NYu (2012) Guarantee optimization in functional-differential systems with a control aftereffect. J Appl Math Mech 76(4):369-377 · Zbl 1423.49031
[8] Gomoyunov MI, Lukoyanov NYu (2015) On the stability of a procedure for solving a minimax control problem for a positional functional. Proc Steklov Inst Math 288(Suppl. 1):S54-S69 · Zbl 1317.49007
[9] Gomoyunov MI, Lukoyanov NY (2018) On the numerical solution of differential games for neutral-type linear systems. Proc Steklov Inst Math 301(Suppl. 1):44-56. https://doi.org/10.1134/S0081543818050048 · Zbl 1401.49056
[10] Isaacs R (1965) Differential games. Wiley, New York, p 384 · Zbl 1233.91001
[11] Kornev DV (2012) On numerical solution of positional differential games with nonterminal payoff. Autom Remote Control 73(11):1808-1821 · Zbl 1270.91014
[12] Kornev DV, Lukoyanov NYu (2014) On numerical solution of differential games in classes of mixed strategies. IFAC Proc Vol (IFAC-Papers Online) 19:1550-1555
[13] Kornev DV, Lukoyanov NYu (2016) On a minimax control problem for a positional functional under geometric and integral constraints on control actions. Proc Steklov Inst Math 293(Suppl. 1):S85-S100 · Zbl 1351.49008
[14] Krasovskii AN (1980) On minimax position control. J Appl Math Mech 44(4):425-430 · Zbl 0489.90096
[15] Krasovskii AN (1982) A non-linear differential game with value given by an integral. Differ Equ 18(8):911-916
[16] Krasovskii AN (1987) Construction of mixed strategies on the basis of stochastic programs. J Appl Math Mech 51(2):144-149
[17] Krasovskii AN, Krasovskii NN (1995) Control under lack of information. Birkhäuser, Berlin, p 322 · Zbl 0827.93001
[18] Krasovskii AN, Krasovskii NN, Tretyakov VE (1981) Stochastic programmed design for a deterministic positional differential game. J Appl Math Mech 45(4):425-430
[19] Krasovskii NN (1979) Differential games. Approximation and formal models. Math USSR-Sb 35(6):795-822
[20] Krasovskii NN (1982) On stochastic programmed design of strategies in a differential game. J Appl Math Mech 46(6):709-714 · Zbl 0528.90102
[21] Krasovskii NN (1985) Control of a dynamical system: problem on the minimum of guaranteed result. Nauka Publ., Moscow, p 516 [in Russian]
[22] Krasovskii NN, Lukoyanov NYu (1996) Problem of conflict control with hereditary information. J Appl Math Mech 60(6):869-882 · Zbl 1040.93522
[23] Krasovskii NN, Reshetova TN (1988) On the program synthesis of a guaranteed control. Probl Control Inf Theory 17(6):333-343
[24] Krasovskii NN, Subbotin AI (1988) Game-theoretical control problems. Springer, New York, p 517 · Zbl 0649.90101
[25] Lokshin MD (1990) Differential games with integral constraints on disturbances. J Appl Math Mech 54(3):331-337 · Zbl 0739.90082
[26] Lokshin MD (1992) Differential games with integral restrictions on the controlling actions. Differ Equ 28(11):1602-1610 · Zbl 0806.90138
[27] Lukoyanov NYu (1995) A conflict control problem under mixed restrictions on control actions. Differ Equ 31(9):1427-1436 · Zbl 0864.90144
[28] Lukoyanov NYu (1998) The problem of computing the value of a differential game for a positional functional. J Appl Math Mech 62(2):177-186 · Zbl 0970.49028
[29] Lukoyanov NYu (2001) How to compute the value of a positional differential game. Differ Equ 37(1):17-26 · Zbl 0996.91021
[30] Lukoyanov NYu (2003) Functional Hamilton-Jacobi type equations with ci-derivatives in control problems with hereditary information. Nonlinear Funct Anal Appl 8(4):535-556 · Zbl 1085.49037
[31] Lukoyanov NYu, Reshetova TN (1998) Problems of conflict control of high dimensionality functional systems. J Appl Math Mech 62(4):545-554
[32] Pontryagin LS (1981) Linear Differential Games of Pursuit. Math. USSR-Sbornik 40(3):285-303 · Zbl 0468.90099
[33] Pschenichnyi BN, Sagaidak MI (1970) Differential games of prescribed duration. Cybernetics 6(2):72-83
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.