On procedures for constructing solutions in differential games on a finite interval of time |
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Authors: | A M Taras’ev T B Tokmantsev A S Uspenskii V N Ushakov |
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Institution: | A. M. Taras’ev, T. B. Tokmantsev, A. S. Uspenskii and V. N. Ushakov |
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Abstract: | Many problems of conflict-control theory can be reduced to approach-avoidance games with a certain terminal set. One of the
main approaches to solution of such problems is the approach suggested by N. N. Krasovskii, which is based on positional constructions.
The basis of these constructions consists of the extremal aiming principle at stable bridges. In this connection, the problem
of constructing the maximal stable bridge, the set of all positions from which the problem of approaching the terminal set
(which is the main task of one of the problem) is solvable, is important. The paper considers the approach-avoidance game
on a finite interval of time in which the first player must ensure the attainment by the state vector of the control system
of the terminal set on this interval, and the second player must ensure the avoidance of the terminal set. The main subject
of the study is the maximal stable bridge, which is the set of positional consumption in this game. The problem of exactly
constructing the set of positional consumption is solvable only in simple cases, and it is more realistic to consider and
solve the problem of approximate construction of this set. The paper proposes approaches to approximate construction of the
set of positional consumption based on sampling the time interval of the game and the technique of backward constructions,
which has been developed at the scientific school of N. N. Krasovskii since the 1980s.
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 23, Optimal
Control, 2005. |
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