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Collective pursuit with integral constraints on the controls of players.
Abstract
Under consideration is a linear differential game of several persons with integral constraints on the controls of the players. Pursuit terminates if a solution to at least one of the equations describing the differential game hits the origin at some instant of time. We establish a necessary and sufficient condition of terminating pursuit from all points of the space in the case of a single pursuer and a sufficient condition in the case of many pursuers. This is an English translation of the author’s article [Mat. Tr. 6, No. 2, 66-79 (2003; Zbl 1060.91032)].
Supplementary resource (1)
... Linear differential games with integral constraints on controls has been a subject of interest for many years (see eg. [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. More specifically, there were a few works on the related problem in the case of many pursuers and one evader. ...
... His work was generalized by Ibragimov [8], in which he examined a pursuit differential game of m pursuers and k evaders with integral constraints described by the following systems of differential equations: ...
... This is because we can pass a two-dimensional plane through the initial positions 0 10 20 ,, x y y of the players in n , and then consider evasion from the projection x of the pursuer on this plane. According to the theorem, the evaders moving on the plane can avoid to coincide with the projection of the pursuer 12 , then pursuit can be completed (see, for example, [8], [13]). The results of the paper can be developed towards the evasion differential game with many evaders. ...
A simple motion evasion differential game of two evaders and one pursuer
was studied. Control functions of all players are subjected to integral
constraints. We say that evasion is possible if the state of at least
one of the evaders does not coincide with that of the pursuer. We proved
that if the total energy of the evaders is greater than or equal to the
energy of the pursuer, then evasion is possible. Though the game is
considered in a plane, the results of the paper can be easily extended
to Rn.
... A large number of works are devoted to differential games where the position of the players changes continuously in time (see, e.g.,123456789101112131415161718). Zero-sum differential games were first considered in the book of Isaacs [5] who derived the main equation of the theory of differential games. ...
We consider a linear pursuit game of one pursuer and one evader whose motions are described by different-type linear discrete systems. Controls of the players satisfy total constraints. Terminal set M is a subset of R íµí± and it is assumed to have nonempty interior. Game is said to be completed if íµí±¦ (íµí±) − íµí±¥ (íµí±) ∈ íµí± at some step k. To construct the control of the pursuer, at each step i, we use positions of the players from step 1 to step i and the value of the control parameter of the evader at the step i. We give sufficient conditions of completion of pursuit and construct the control for the pursuer in explicit form. This control forces the evader to expend some amount of his resources on a period consisting of finite steps. As a result, after several such periods the evader exhausted his energy and then pursuit will be completed.
... Many investigations were devoted to study the differential games with integral constraints; e.g., [1], [2], [4], [7], [9] [15], [18], [20], [21], [25], [27]. ...
We investigate a pursuit-evasion differential game of countably many
pursuers and one evader. Integral constraints are imposed on control
functions of the players. Duration of the game is fixed and the payoff
of the game is infimum of the distances between the evader and pursuers
when the game is completed. Purpose of the pursuers is to minimize the
payoff and that of the evader is to maximize it. Optimal strategies of
the players are constructed, and the value of the game is found. It
should be noted that energy resource of any pursuer may be less than
that of the evader.
The objective of this paper is to study a pursuit differential game with finite or countably number of pursuers and one evader. The game is described by differential equations in l2-space, and integral constraints are imposed on the control function of the players. The duration of the game is fixed and the payoff functional is the greatest lower bound of distances between the pursuers and evader when the game is terminated. However, we discuss the condition for finding the value of the game and construct the optimal strategies of the players which ensure the completion of the game. An important fact to note is that we relaxed the usual conditions on the energy resources of the players. Finally, some examples are provided to illustrate our result.
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