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On game problems for second-order evolution equations. (English. Russian original) Zbl 1207.49048
Russ. Math. 51, No. 1, 49-57 (2007); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2007, No. 1, 54-62 (2007).
From the introduction: In this paper, we consider certain problems of the theory of differential games in systems with distributed parameters. The players influence on the system with the use of control parameters contained in the right-hand side of the equation. Controls of players are chosen in the form of functions on which various constraints are imposed, so-called geometric, integral, and mixed constraints.
In the first three games, the goal of the first player is to bring the system into an unperturbed state. In the fourth game, the goal of the first player is to bring the system and its velocity into an arbitrary $$\ell$$-neighborhood of zero. The second player in all the games has the opposite goal.We present conditions which are sufficient in order that the first player can reach the goal in a finite time. For the third game, we also consider the encounter-evasion problem.

##### MSC:
 49N75 Pursuit and evasion games 91A23 Differential games (aspects of game theory) 35J25 Boundary value problems for second-order elliptic equations 35L05 Wave equation 35L15 Initial value problems for second-order hyperbolic equations
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##### References:
 [1] V. A. Il’in and V. V. Tikhomirov, ”The Wave Equation with a Boundary Control at Both Endpoints and the Complete Vibration Dumping Problem,” Differents. uravneniya 35(4), 692–704 (1999). [2] V. A. Il’in, ”Two-Endpoint Boundary Control of Vibration Described by a Finite-Energy Generalized Solution of the Wave Equation,” Differents. uravneniya 36(11), 1513–1528 (2000). [3] V. A. Il’in, ”Boundary Control of a String Oscillating at One End, with the Other End Fixed and Under the Condition of the Existence of Finite Energy,” Dokl. RAN 378(6), 743–747 (2001). · Zbl 1090.93531 [4] Yu. S. Osipov, ”On the Theory of Differential Games in Systems with Distributed Parameters,” Dokl. Akad. Nauk SSSR 223(6), 1314–1317 (1975). [5] Yu. S. Osipov, ”Positional Control in Parabolic Systems,” Prikl. Mat. Mekh. 41(2), 195–201 (1977). [6] Yu. S. Osipov and S. P. Okheizin, ”On the Theory of Positional Control in Hyperbolic Systems,” Dokl. Akad. Nauk SSSR 233(4), 551–554 (1977). [7] J. L. Lions, Contrôle Optimal de Systèmes Gouvernès par des Équation aux Dériveès Partielles (Paris, Dunod, 1968; Mir, Moscow, 1972). · Zbl 0179.41801 [8] F. L. Chernous’ko, ”Bounded Controls in Systems with Distributed Parameters,” Prikl. Mat. Mekh. 56(5), 810–826 (1992). [9] S. A. Avdonin and S. A. Ivanov, Controllability of Systems with Distributed Parameters and Families of Exponents (UMKVO, Kiev, 1989) [in Russian]. [10] A. G. Butkovskii, Methods of Control of Systems with Distributed Parameters (Nauka, Moscow, 1975) [in Russian]. [11] M. S. Nikol’skii, ”A Pursuit Problem for Various Constraints on Pursuer and Evader Controls,” in Theory of Optimal Decisions (Kiev, 1975), pp. 59–66. [12] N. Yu. Satimov, B. B. Rikhsiev, and G. I. Ibragimov, ”A Many-Person Differential Game with Integral Constraints,” in Current Problems of the Theory of Optimal Control and Differential Games (Fan, Tashkent, 1999), pp. 89–94. [13] M. Tukhtasinov, ”Some Problems in the Theory of Differential Pursuit Games in Systems with Distributed Parameters,” Prikl. Mat. Mekh. 59(6), 979–984 (1995). · Zbl 0885.90142 [14] G. I. Ibragimov, ”A Problem of Optimal Pursuit in Systems with Distributed Parameters,” Prikl. Mat. Mekh. 66(5), 753–759 (2002). · Zbl 1037.49039 [15] S. G. Mikhlin, Linear Partial Differential Equations (Vysshaya Shkola, Moscow, 1977) [in Russian].
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