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

Abstract

In this paper, we investigate a differential game problem of multiple number of pursuers and a single evader with motions governed by a certain system of first-order differential equations. The problem is formulated in the Hilbert space 2 , with control functions of players subject to integral constraints. Avoidance of contact is guaranteed if the geometric position of the evader and that of any of the pursuers fails to coincide for all time t. On the other hand, pursuit is said to be completed if the geometric position of at least one of the pursuers coincides with that of the evader. We obtain sufficient conditions that guarantees avoidance of contact and construct evader's strategy. Moreover, we prove completion of pursuit subject to some sufficient conditions. Finally, we demonstrate our results with some illustrative examples.

Pursuit and Evasion Linear Differential Game Problems with Generalized Integral Constraints

“…Remark 2. It is worth noting that if the matrices A(t) = 0 and B(t) = η(t) (some scalar function), the result obtained here for an evasion problem reduces to that of Rilwan et al [6] irrespective of the space considered. Additionally, if A(t) = 1, B(t) = −λ i where λ i > 0, i = 1, 2, • • • , m and p = 2, the evasion problem considered here reduces to that of Ibragimov and Hasim [16] and also Ibragimov et al [4].…”

“…This transformation and also the fact that the integral constraints (3) generalize the existing integral constraints in the literature motivated the following research question: is it possible to solve a pursuit and evasion problem with players' dynamics described in [9] with the integral constraints (3) imposed on players' control function? Answering this question will indeed generalize some results on pursuit and evasion problems in the literature (see, for example, [3][4][5][6][8][9][10][11][12]14,16,17]).…”

“…Motivated by the results in [2], some authors [3][4][5][6][7][8][9][10][11][12][13][14] studied this class of problem and proposed various methods of solving the pursuit and evasion problems with integral or other form of the aforementioned constraints on the players' control functions. For instance, Rakhmanov et al [3] studied a linear pursuit differential game of one pursuer and one evader.…”

“…In [6], the authors obtained sufficient conditions that guarantee a pursuit and also an evasion in a differential game with integral constraints. To this end, pursuers' and evader's optimal strategies are constructed.…”

“…The authors [10] obtained a sufficient condition for the completion of a pursuit. Rilwan et al [6] also studied an evasion version of the problem [10] with p = 2 and obtained sufficient conditions that guaranteed the avoidance of contact of the evader from the pursuers. To this end, they constructed the evader's optimal strategy.…”