On calculating the value of a differential game in the class of counter strategies. (English) Zbl 1398.91087

Summary: For a linear dynamical system with control and disturbance, a feedback control problem is considered, in which the Euclidean norm of a set of deviations of the system’s motion from given targets at given instants of time is optimized. The problem is formalized into a differential game in “strategy-counter strategy” classes. A game value computing procedure which reduces the problem to a recursive construction of upper convex hulls of auxiliary functions is justified. Results of numerical simulations are presented.


91A23 Differential games (aspects of game theory)
49N70 Differential games and control
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[1] [1] Krasovskii N.N., Control of a Dynamic System. Problem about the Minimum of the Guaranteed Result, Nauka, Moscow, 1985 (in Russian)
[2] [2] Isaacs R., Differential Games, John Wiley, New York, 1965
[3] [3] Krasovskii N.N., “On the problem of unification of differential games”, Doklady AN SSSR, 226:6 (1976), 1260-1263
[4] [4] Krasovskii A.N., “Construction of mixed strategies on the basis of stochastic programs”, J. Appl. Math. Mech, 51:2 (1987), 144-149
[5] [5] Krasovskii A.N., Control under Lack of Information, Birkhauser, Berlin etc., 1995 · Zbl 0827.93001
[6] [6] Subbotin A.I., Minimax Inequalities and Hamilton-Jacobi Equations, Nauka, Moscow, 1991 (in Russian) · Zbl 0733.70014
[7] [7] Subbotin A.I., Generalized Solutions of First Order PDEs. The Dynamical Optimization Perspective, Birkhauser, Boston etc., 1995
[8] [8] Subbotin A.I., “Existence and Uniqueness Results for Hamilton-Jacobi Equations”, Nonlinear Anal., 16:7/8 (1991), 683-699 · Zbl 0739.35011
[9] [9] Lukoyanov N.Yu., “One differential game with nonterminal payoff”, Izvestiya akademii nauk. Teoriya i sistemy upravleniya, 1997, no. 1, 85-90
[10] [10] Lukoyanov N.Yu., “The problem of computing the value of a differential game for a positional functional”, J. Appl. Math. Mech, 62:2 (1998), 177-186 · Zbl 0970.49028
[11] [11] Blagodatskikh V.I., Filippov A.F., “Differential inclusions and optimal control”, Proc. Steklov Inst. Math., 169 (1986), 199-259 · Zbl 0608.49027
[12] [12] Krasovskii A.N., Reshetova T.N., Control under Information Deficiency: Study Guide, UrGU, Sverdlovsk, 1990 (in Russian)
[13] [13] Kornev D.V., “On numerical solution of positional differential games with nonterminal payoff”, Autom. Remote Control, 73:11 (2012), 1808-1821 · Zbl 1270.91014
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