On a game problem for point control near the surface of the Moon. (Russian. English summary) Zbl 1459.91031
Summary: A game control problem in which the first player controls the material point of variable composition is considered. The second player controls the point that can move with a limited speed. It is assumed that the material point of variable composition, along with the controlled reactive power, is exposed to a constant force, the value of which is proportional to the mass of the point. This situation occurs, for example, when we consider the motion of a material point near the surface of the Moon, where there is no atmospheric resistance. It is considered that the point of variable composition has constant relative velocity of separating fuel particles, and the value of thrust is limited from above with a given positive number. The first player tries to minimize the distance between the points in a set moment, consuming as little resources as possible. The formulated two-criterion problem, with the use of weight coefficients, gets reduced to a differential game, the payoff of which is the sum of both terminal and integral components. By changing variables, the problem is reduced to a single-type game in which vectograms of players are balls with time-dependent radii. The function of the game price is calculated, and optimal control of the players is determined.
MSC:
| 91A80 | Applications of game theory |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 93C73 | Perturbations in control/observation systems |
References:
| [1] | Krasovskiy N. N., Motion control theory, Nauka Publ., M., 1970, 420 pp. (in Russ.) |
| [2] | Isaacs R., Differential games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, John Wiley and Sons, Inc., New York, 1965, 384 pp. · Zbl 0125.38001 |
| [3] | Ukhobotov V. I., “Modification of the game “Isotropic missiles”, Multicriterial systems with uncertainty and their applications, Interuniversity collection of scientific papers, Chelyabinskiy gosudarstvennyy universitet Publ., Chelyabinsk; Izd-vo Bashkirskogo universiteta Publ., 1988, 123-130 (in Russ.) · Zbl 0693.00013 |
| [4] | Ukhobotov V. I., Zaytseva O. V., “About one problem of impulse pursuit at the limited velocity of the escaping”, Bulletin of the South Ural State University. Series of “Computer Technologies, Automatic Control & Radioelectronics”, 2 (178):11 (2010), 29-32 (in Russ.) |
| [5] | Pozharitskiy G. K., “Game problem of impulse encounter with an opponent limited in energy”, Journal of Applied Mathematics and Mechanics, 39:4 (1975), 579-589 (in Russ.) · Zbl 0375.90097 |
| [6] | Ukhobotov V. I., Troitsky A. A., “Problem about pursuer with pulse control near surface of the Moon”, Matematicheskaya Teoriya Igr i Ee Prilozheniya, 5:4 (2013), 105-118 (in Russ.) · Zbl 1287.91020 |
| [7] | Ukhobotov V. I., Gushchin D. V., “Single-type differential games with convex integral payoff”, Proceedings of the Steklov Institute of Mathematics, 275, suppl. 1 (2011), S178-S185 · Zbl 1305.49058 · doi:10.1134/S0081543811090136 |
| [8] | Ioffe A. D., Tikhomirov V. M., Theory of extremum problems, Nauka Publ., M., 1974, 479 pp. (in Russ.) |
| [9] | Ukhobotov V. I., “A linear control problem under interference with a payoff depending on the modulus of a linear function”, Trudy Instituta Matematiki I Mekhaniki UrO RAN, 23, no. 1 (2017), 251-261 (in Russ.) · doi:10.21538/0134-4889-2017-23-1-251-261 |
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