Fixed Duration Pursuit-Evasion Differential Game with Integral Constraints

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. 1. Statement of the problem. The study of two person zero-sum differential games was initiated by Isaacs [16]. Berkovitz [3], Fleming [5], Friedman [6], Hajek [8], Krasovskii, [19], Petrosyan [22], Pontryagin [23], Subbotin [26] and others developed mathematical foundations for the theory of differential games. 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]. Constructing the optimal strategies, and finding the value of the game are of interest in differential games, e.g., see [10], [11], [17], [24], [26], [27]. Such problems in the case of many pursuers were studied, for example, in [10] and [11]. In [10], a differential game of optimal approach of countably many pursuers to one evader was studied in Hilbert space with geometric constraints on controls of players. In [11], such differential game was studied for inertial players with integral constraints under the assumption that the control resource of the evader less than that of each pursuer. In the present paper, we also discuss an optimal pursuit problem with countably many pursuers and one evader in Hilbert space l2, and control resource of the evader σ can be greater than that of any pursuer. In the space l2 with elements α = (α1, α2, ..., αk, ...), ı ∑ k=1 α k <∞, and inner product and norm (α, β) = ı ∑ k=1 αkβk, ||α|| = ( ı ∑ k=1 α k )1/2 , 2012 iCAST: Contemporary Mathematics, Mathematical Physics and their Applications IOP Publishing Journal of Physics: Conference Series 435 (2013) 012017 doi:10.1088/1742-6596/435/1/012017 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 motions of the pursuers Pi and the evader E are described by the equations Pi : ẋi = ui, xi(0) = xi0, E : ẏ = v, y(0) = y0, (1) where xi, x0, ui, y, y0, v ∈ l2, ui = (ui1, ui2, ..., uik, ...) is control parameter of the pursuer Pi, and v = (v1, v2, ..., vk, ...) is that of the evader E; throughout, i = 1, 2, ...,m, ... Let θ be a fixed time, I = {1, 2, ...,m, ...}, and H(x0, r) = {x ∈ l2 : ||x− x0|| ≤ r} S(x0, r) = {x ∈ l2 : ||x− x0|| = r}. Definition 1. A function ui = ui(t), 0 ≤ t ≤ θ, with the Borel measurable coordinates uik : [0, θ]→ R1, k = 1, 2, ..., subjected to the condition  θ ∫ 0 ||ui(t)||dt 1/2 ≤ ρi, is called an admissible control of the pursuer Pi, where ρi are given positive numbers. Definition 2. A function v = v(t), 0 ≤ t ≤ θ, with the Borel measurable coordinates vk : [0, θ]→ R1, k = 1, 2, ..., subjected to the condition  θ ∫ 0 ||v(t)||dt 1/2 ≤ σ, is called an admissible control of the evader E, where σ is given positive number. If it has been chosen admissible controls ui(·), v(·) of players, then corresponding to them motions xi(·), y(·) are defined by formulas xi(t) = (xi1(t), xi2(t), ..., xik(t), ...), y = (y1(t), y2(t), ..., yk(t), ...),