An Optimal Pursuit Differential Game Problem with One Evader and Many Pursuers

Abstract

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 l 2 -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.

A Differential Game Problem of Many Pursuers and One Evader in the Hilbert Space $$\ell_2$$

“…Motivated by the results in [21], we study the same problem for a differential game described by (1) in the Hilbert space 2 (coinciding with 2 r and r = 0) ; which amounts to solving the pursuit version of the problem in [3] and evasion version of the problem considered in [25]. More precisely, we investigate a differential game problem of avoidance of contacts (evasion) and completion of pursuit described by (1), but reduced to (2) in the Hilbert space 2 . With all players control parameters subject to integral constraint, we will show that if the total energy resources of the pursuers is less than that of the evader, then avoidance of contact is guaranteed.…”

“…al. [1] wherein the authors, using the pursuers strategies (22) estimated the value of the game described by (8). The game value is value of the payoff function at the instant of the termination of the game and when players are using their optimal strategies.…”

“…The game value is value of the payoff function at the instant of the termination of the game and when players are using their optimal strategies. To estimate this value, the authors [1] not only constructed the optimal strategies (22) but also presented the tools (that is, players attainability domain and also the strategies of some fictitious pursuers) required to construct (22). In this paper, we present a more detailed application of the strategy in showing completion of pursuit and moreover, we address evasion problem under the same dynamic equations of players.…”

“…In [22] and [23], pursuit-evasion differential game described by (1) was studied in the Hilbert space 2 with geometric and integral constraints on control functions of the players respectively, where a(t) = − t, and is the duration of the game. In both papers, sufficient conditions for completion of pursuit as well as value of the game were obtained.…”

Simple motion pursuit differential game problem of many players with integral and geometric constraints on controls function.

“…In many studies of differential game, motion of each player are explicitly stated and considered to be a system of differential equations of the same order. In the papers [2,5,[7][8][9]13,16], motion of each of the player is considered to obey first order differential equation. In other studies such as Refs.…”