Pursuit-Evasion Differential Game with Many Inertial Players

Abstract

We consider pursuit-evasion differential game of countable number inertial players in Hilbert space with integral constraints on the control functions of players. Duration of the game is fixed. The payoff functional is the greatest lower bound of distances between the pursuers and evader when the game is terminated. The pursuers try to minimize the functional, and the evader tries to maximize it. In this paper, we find the value of the game and construct optimal strategies of the players.

1. Introduction and Preliminaries

Many books have been devoted to differential games, such as books by Isaacs [1], Pontryagin [2], Friedman [3], Krasovskii and Subbotin [4].

Constructing the player's optimal strategies and finding the value of the game are of specific interest in studying of differential games.

The pursuit-evasion differential games involving several objects with simple motions take the attention of many authors. Ivanov and Ledyaev [5] studied simple motion differential game of several players with geometric constraints. They obtained sufficient conditions to find optimal pursuit time in ℝ^𝑛, by using the method of the Lyapunov function for an auxiliary problem.

Levchenkov and Pashkov [6] investigated differential game of optimal approach of two identical inertial pursuers to a noninertial evader on a fixed time interval. Control parameters were subject to geometric constraints. They constructed the value function of the game and used necessary and sufficient conditions which a function must satisfy to be the value function [7].

Chodun [8] examined evasion differential game with many pursuers and geometric constraints. He found a sufficient condition for avoidance.

Ibragimov [9] obtained the formula for optimal pursuit time in differential game described by an infinite system of differential equations. In [10] simple motion differential game of many pursuers with geometric constraints was investigated in the Hilbert space 𝑙₂.

In the present paper, we consider a pursuit-evasion differential game of infinitely many inertial players with integral constraints on control functions. The duration of the game 𝜃 is fixed. The payoff functional of the game is the greatest lower bound of the distances between the evader and the pursuers at 𝜃. The pursuer's goal is to minimize the payoff, and the evader's goal is to maximize it. This paper is close in spirit to [10]. We obtain a sufficient condition to find the value of the game and constructed the optimal strategies of players.

2. Formulation of the Problem

In the space 𝑙₂ consisting of elements 𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑘,…), with ∑^∞_𝑘=1𝛼²_𝑘<∞, and inner product (𝛼,𝛽)=∑^∞_𝑘=1𝛼_𝑘𝛽_𝑘, the motions of the countably many pursuers 𝑃_𝑖 and the evader 𝐸 are defined by the equations 𝑃_𝑖∶𝑥̈_𝑖=𝑢_𝑖,𝑥_𝑖(0)=𝑥⁰_𝑖,𝑥̇_𝑖(0)=𝑥¹_𝑖,𝐸∶𝑦̈=𝑣,𝑦(0)=𝑦⁰,𝑦(̇0)=𝑦¹, (2.1) where 𝑥_𝑖,𝑥⁰_𝑖,𝑥¹_𝑖,𝑢_𝑖,𝑦,𝑦⁰,𝑦¹,𝑣∈𝑙₂,𝑢_𝑖=(𝑢_𝑖1,𝑢_𝑖2,…,𝑢_𝑖𝑘,…) is the control parameter of the pursuer 𝑃_𝑖, and 𝑣=(𝑣₁,𝑣₂,…,𝑣_𝑘,…) is that of the evader 𝐸; here and throughout the following, 𝑖=1,2,…,𝑚,…. Let 𝜃 be a given positive number, and let 𝐼={1,2,…,𝑚,…}.

As a real life example, one may consider the case of a missile catching an aircraft. If the initial positions and speeds (first derivative) of both missile and aircraft are given and the constraints of both missile and aircraft are their available fuel, which could be mathematically interpreted as the mean average of their acceleration function (second derivative), then the corresponding pursuit-evasion problem is described by (2.1).

A ball (resp., sphere) of radius 𝑟 and center at the point 𝑥₀ is denoted by 𝐻(𝑥₀,𝑟)={𝑥∈𝑙₂∶‖𝑥−𝑥₀‖≤𝑟} (resp., by 𝑆(𝑥₀,𝑟)={𝑥∈𝑙₂∶‖𝑥−𝑥₀‖=𝑟}).

Definition 2.1. A function 𝑢_𝑖(⋅),𝑢_𝑖∶[0,𝜃]→𝑙₂, such that 𝑢_𝑖𝑘∶[0,𝜃]→𝑅¹,𝑘=1,2,…, are Borel measurable functions and ‖𝑢_𝑖(⋅)‖₂=(^𝜃₀‖𝑢_𝑖(𝑠)‖²𝑑𝑠)^1/2≤𝜌_𝑖,‖𝑢_𝑖‖=(^∞_𝑘=1𝑢²_𝑖𝑘)^1/2, (2.2) where 𝜌_𝑖 is given positive number, is called an admissible control of the ith pursuer.

Definition 2.2. A function 𝑣(⋅),𝑣∶[0,𝜃]→𝑙₂, such that 𝑣_𝑘∶[0,𝜃]→𝑅¹,𝑘=1,2,…, are Borel measurable functions and ‖𝑣(⋅)‖₂=(^𝜃₀‖𝑣(𝑠)‖²𝑑𝑠)^1/2≤𝜎, (2.3) where 𝜎 is a given positive number, is called an admissible control of the evader.

Once the players' admissible controls 𝑢_𝑖(⋅) and 𝑣(⋅) are chosen, the corresponding motions 𝑥_𝑖(⋅) and 𝑦(⋅) of the players are defined as

𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑘(𝑡),…),𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑘𝑥(𝑡),…),_𝑖𝑘(𝑡)=𝑥⁰_𝑖𝑘+𝑡𝑥¹_𝑖𝑘+^𝑡₀^𝑠₀𝑢_𝑖𝑘(𝑟)𝑑𝑟𝑑𝑠,𝑦_𝑘(𝑡)=𝑦⁰_𝑘+𝑡𝑦¹_𝑘+^𝑡₀^𝑠₀𝑣_𝑘(𝑟)𝑑𝑟𝑑𝑠. (2.4)

One can readily see that 𝑥_𝑖(⋅),𝑦(⋅)∈𝐶(0,𝜃;𝑙₂), where 𝐶(0,𝜃;𝑙₂) is the space of functions 𝑓(𝑡)=(𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑘(𝑡),…)∈𝑙₂,𝑡≥0, (2.5) such that the following conditions hold:

(1)𝑓_𝑘(𝑡),0≤𝑡≤𝜃,𝑘=1,2,…, are absolutely continuous functions;(2)𝑓(𝑡),0≤𝑡≤𝜃, is a continuous function in the norm of 𝑙₂.

Definition 2.3. A function 𝑈_𝑖(𝑡,𝑥_𝑖,𝑦,𝑣),𝑈_𝑖∶[0,∞)×𝑙₂×𝑙₂×𝑙₂→𝑙₂, such that the system 𝑥̈_𝑖=𝑈_𝑖(𝑡,𝑥_𝑖,𝑦,𝑣),𝑥_𝑖(0)=𝑥⁰_𝑖,𝑥̇_𝑖(0)=𝑥¹_𝑖,𝑦̈=𝑣,𝑦(0)=𝑦⁰,𝑦(̇0)=𝑦¹, (2.6) has a unique solution (𝑥_𝑖(⋅),𝑦(⋅)), with 𝑥_𝑖(⋅),𝑦(⋅)∈𝐶(0,𝜃;𝑙₂), for an arbitrary admissible control 𝑣=𝑣(𝑡),0≤𝑡≤𝜃, of the evader 𝐸, is called a strategy of the pursuer 𝑃_𝑖. A strategy 𝑈_𝑖 is said to be admissible if each control formed by this strategy is admissible.

Definition 2.4. Strategies 𝑈_𝑖0 of the pursuers 𝑃_𝑖 are said to be optimal if inf_{𝑈1,…,𝑈𝑚,…}Γ₁(𝑈₁,…,𝑈_𝑚,…)=Γ₁(𝑈₁₀,…,𝑈_𝑚0,…), (2.7) where Γ₁(𝑈₁,…,𝑈_𝑚,…)=sup_𝑣(⋅)inf_𝑖∈𝐼‖𝑥_𝑖(𝜃)−𝑦(𝜃)‖,𝑈_𝑖 are admissible strategies of the pursuers 𝑃_𝑖, and 𝑣(⋅) is an admissible control of the evader 𝐸.

Definition 2.5. A function 𝑉(𝑡,𝑥₁,…,𝑥_𝑚,…,𝑦),𝑉∶[0,∞)×𝑙₂×⋯×𝑙₂×⋯×𝑙₂→𝑙₂, such that the countable system of equations 𝑥̈_𝑘=𝑢_𝑘,𝑥_𝑘(0)=𝑥⁰_𝑘,𝑥̇_𝑘(0)=𝑥¹_𝑘̈,𝑘=1,2,…,𝑚,…,𝑦=𝑉(𝑡,𝑥₁,…,𝑥_𝑚,…,𝑦),𝑦(0)=𝑦⁰,𝑦(̇0)=𝑦¹, (2.8) has a unique solution (𝑥₁(⋅),…,𝑥_𝑚(⋅),…,𝑦(⋅)), with 𝑥_𝑖(⋅),𝑦(⋅)∈𝐶(0,𝜃,𝑙₂), for arbitrary admissible controls 𝑢_𝑖=𝑢_𝑖(𝑡),0≤𝑡≤𝜃, of the pursuers 𝑃_𝑖, is called a strategy of the evader 𝐸. If each control formed by a strategy 𝑉 is admissible, then the strategy 𝑉 itself is said to be admissible.

Definition 2.6. A strategy 𝑉₀ of the evader 𝐸 is said to be optimal if sup_𝑉Γ₂(𝑉)=Γ₂(𝑉₀), where Γ₂(𝑉)=inf_{𝑢1(⋅),…,𝑢𝑚(⋅),…}inf_𝑖∈𝐼‖𝑥_𝑖(𝜃)−𝑦(𝜃)‖, where 𝑢_𝑖(⋅) are admissible controls of the pursuers 𝑃_𝑖, and 𝑉 is an admissible strategy of the evader 𝐸.

If Γ₁(𝑈₁₀,…,𝑈_𝑚0,…)=Γ₂(𝑉₀)=𝛾, then we say that the game has the value 𝛾 [7].

It is to find optimal strategies 𝑈_𝑖0 and 𝑉₀ of the players 𝑃_𝑖 and 𝐸, respectively, and the value of the game. Instead of differential game described by (2.1) we can consider an equivalent differential game with the same payoff function and described by the following system: 𝑃_𝑖∶𝑥̇_𝑖(𝑡)=(𝜃−𝑡)𝑢_𝑖(𝑡),𝑥_𝑖(0)=𝑥_𝑖0=𝑥¹_𝑖𝜃+𝑥⁰_𝑖,𝐸∶𝑦(̇𝑡)=(𝜃−𝑡)𝑣(𝑡),𝑦(0)=𝑦₀=𝑦¹𝜃+𝑦⁰. (2.9) Indeed, if the pursuer 𝑃_𝑖 uses an admissible control 𝑢_𝑖(𝑡)=(𝑢_𝑖1(𝑡),𝑢_𝑖2(𝑡),…),0≤𝑡≤𝜃, then according to (2.1) we have 𝑥_𝑖(𝜃)=𝑥⁰_𝑖+𝑥¹_𝑖𝜃+^𝜃₀^𝑡₀𝑢_𝑖(𝑠)𝑑𝑠𝑑𝑡=𝑥⁰_𝑖+𝑥¹_𝑖𝜃+^𝜃₀(𝜃−𝑡)𝑢_𝑖(𝑡)𝑑𝑡, (2.10) and the same result can be obtained by (2.9) 𝑥_𝑖(𝜃)=𝑥_𝑖0+^𝜃₀(𝜃−𝑡)𝑢_𝑖(𝑡)𝑑𝑡=𝑥⁰_𝑖+𝑥¹_𝑖𝜃+^𝜃₀(𝜃−𝑡)𝑢_𝑖(𝑡)𝑑𝑡. (2.11) Also, for the evader the same argument can be made, therefore in the distance‖𝑥_𝑖(𝜃)−𝑦(𝜃)‖ we can take either the solution of (2.1) or the solution of (2.9).

The attainability domain of the pursuer 𝑃_𝑖 at time 𝜃 from the initial state 𝑥_𝑖0 at time 𝑡₀=0 is the closed ball 𝐻(𝑥_𝑖0,𝜌_𝑖(𝜃³/3)^1/2). Indeed, by Cauchy-Schwartz inequality ‖𝑥_𝑖(𝜃)−𝑥_𝑖0‖=‖^𝜃₀(𝜃−𝑠)𝑢_𝑖≤(𝑠)𝑑𝑠‖_𝜃0(𝜃−𝑠)‖𝑢_𝑖(𝑠)‖𝑑𝑠≤(^𝜃₀(𝜃−𝑠)²𝑑𝑠)^1/2⋅(^𝜃₀‖𝑢_𝑖(𝑠)‖²𝑑𝑠)^1/2≤𝜌_𝑖(𝜃³3)^1/2. (2.12)

On the other hand, if 𝑥∈𝐻(𝑥_𝑖0,𝜌_𝑖(𝜃³/3)^1/2), that is, ‖𝑥−𝑥_𝑖0‖≤𝜌_𝑖(𝜃³/3)^1/2, then for the pursuer's control 𝑢_𝑖(𝑡)=3(𝜃−𝑡)𝜃³(𝑥−𝑥_𝑖0),0≤𝑡≤𝜃, (2.13) we obtain 𝑥_𝑖(𝜃)=𝑥.

The pursuer's control is admissible because ^𝜃₀‖𝑢_𝑖(𝑡)‖²3𝑑𝑡=(𝜃³)²𝜃³3‖𝑥−𝑥_𝑖0‖²≤(𝜌_𝑖(𝜃³3)^1/2)²⋅3𝜃³=𝜌²_𝑖. (2.14) Likewise, the attainability domain of the evader 𝐸 at time 𝜃 from the initial state 𝑦₀ at time 𝑡₀=0 is the closed ball 𝐻(𝑦₀,𝜎(𝜃³/3)^1/2).

3. An Auxiliary Game

In this section we fix the index 𝑖 and study an auxiliary differential game of two players 𝑃_𝑖 and 𝐸, also for simplicity we drop the index 𝑖 and use the notion 𝜌_𝑖=𝜌,𝑥_𝑖0=𝑥₀ and 𝑥_𝑖=𝑥. Let 𝑋={𝑧∈𝑙₂∶2(𝑦₀−𝑥₀𝜃,𝑧)≤³3(𝜌²−𝜎²)+‖𝑦₀‖²−‖𝑥₀‖²},𝜌≥𝜎, (3.1) if 𝑥₀≠𝑦₀; if 𝑥₀=𝑦₀, then 𝑋={𝑧∈𝑙₂∶(𝑝,𝑧−𝑦₀𝜃)≤𝜌(³3)^1/2}, (3.2) where 𝑝 is an arbitrary fixed unit vector.

Consider the one-pursuer game described by the equations 𝑃∶𝑥̇=(𝜃−𝑡)𝑢(𝑡),𝑥(0)=𝑥₀,𝐸∶𝑦̇=(𝜃−𝑡)𝑣(𝑡),𝑦(0)=𝑦₀, (3.3) with the state of the evader 𝐸 being subject to 𝑦(𝜃)∈𝑋. The goal of the pursuer 𝑃 is to realize the equality 𝑥(𝜏)=𝑦(𝜏) at some 𝜏,0≤𝜏≤𝜃, and that of the evader 𝐸 is opposite.

We define the pursuer's strategy as follows: if 𝑥₀=𝑦₀, then we set 𝑢(𝑡)=𝑣(𝑡),0≤𝑡≤𝜃, (3.4) and if 𝑥₀≠𝑦₀, then we set 3𝑢(𝑡)=𝑣(𝑡)−(𝑣(𝑡),𝑒)𝑒+𝑒(𝜃³(𝜃−𝑡)²(𝜌²−𝜎²)+(𝑣(𝑡),𝑒)²)^1/2,0≤𝑡≤𝜏, (3.5) where 𝑒=(𝑦₀−𝑥₀)/‖𝑦₀−𝑥₀‖, and 𝑢(𝑡)=𝑣(𝑡),𝜏<𝑡≤𝜃, (3.6) where 𝜏,0≤𝜏≤𝜃, is the time instant at which 𝑥(𝜏)=𝑦(𝜏) for the first time.

Lemma 3.1. If 𝜎≤𝜌 and 𝑦(𝜃)∈𝑋, then the pursuer's strategy (3.4), (3.5), and (3.6) in the game (3.3) ensures that 𝑥(𝜃)=𝑦(𝜃).

Proof. If 𝑥₀=𝑦₀, then from (3.4) we have 𝑥(𝑡)=𝑦(𝑡),0≤𝑡≤𝜃, because 𝑥(𝑡)=𝑥₀+^𝑡₀(𝜃−𝑠)𝑢(𝑠)𝑑𝑠=𝑦₀+^𝑡₀(𝜃−𝑠)𝑣(𝑠)𝑑𝑠=𝑦(𝑡). (3.7) In particular, 𝑥(𝜃)=𝑦(𝜃).
Let 𝑥₀≠𝑦₀. By (3.5) and (3.6), we have 𝑦(𝑡)−𝑥(𝑡)=𝑒𝑓(𝑡), where 𝑓(𝑡)=‖𝑦₀−𝑥₀‖+^𝑡₀−(𝜃−𝑠)(𝑣(𝑠),𝑒)𝑑𝑠_𝑡03(𝜃−𝑠)(𝜃³(𝜃−𝑠)²(𝜌²−𝜎²)+(𝑣(𝑠),𝑒)²)^1/2𝑑𝑠. (3.8) Obviously 𝑓(0)=‖𝑦₀−𝑥₀‖>0. Now we show that 𝑓(𝜃)≤0. This will imply that 𝑓(𝜏)=0 for some 𝜏∈[0,𝜃].
To this end we consider the following two-dimensional vector function: 3𝑔(𝑡)=((𝜃³)^1/2(𝜃−𝑡)²(𝜌²−𝜎²)^1/2,(𝜃−𝑡)(𝑣(𝑡),𝑒)),0≤𝑡≤𝜃. (3.9) For the last integral of (3.8) we have ^𝜃₀3(𝜃−𝑠)(𝜃³(𝜃−𝑠)²(𝜌²−𝜎²)+(𝑣(𝑠),𝑒)²)^1/2=𝑑𝑠_𝜃0(3𝜃³(𝜃−𝑠)⁴(𝜌²−𝜎²)+(𝜃−𝑠)²(𝑣(𝑠),𝑒)²)^1/2=𝑑𝑠_𝜃0|𝑔(𝑠)|𝑑𝑠≥|^𝜃₀𝑔(𝑠)𝑑𝑠|=|(^𝜃₀(3𝜃³)^1/2(𝜃−𝑠)²(𝜌²−𝜎²)^1/2𝑑𝑠,^𝜃₀𝜃(𝜃−𝑠)(𝑣(𝑠),𝑒)𝑑𝑠)|=|((³3)^1/2(𝜌²−𝜎²)^1/2,^𝜃₀𝜃(𝜃−𝑠)(𝑣(𝑠),𝑒)𝑑𝑠)|=(³3(𝜌²−𝜎²)+(^𝜃₀(𝜃−𝑠)(𝑣(𝑠),𝑒)𝑑𝑠)²)^1/2. (3.10) Then 𝑓(𝜃)≤‖𝑦₀−𝑥₀‖+^𝜃₀𝜃(𝜃−𝑠)(𝑣(𝑠),𝑒)𝑑𝑠−(³3(𝜌²−𝜎²)+(^𝜃₀(𝜃−𝑠)(𝑣(𝑠),𝑒)𝑑𝑠)²)^1/2. (3.11)
By assumption, 𝑦(𝜃)∈𝑋, therefore 2(𝑦₀−𝑥₀𝜃,𝑦(𝜃))≤³3(𝜌²−𝜎²)+‖𝑦₀‖²−‖𝑥₀‖², (3.12) so (𝑒,𝑦(𝜃))≤𝑑, where 𝑑=(𝜃³/3)(𝜌²−𝜎²)+‖𝑦₀‖²−‖𝑥₀‖²2‖𝑦₀−𝑥₀‖. (3.13) As (𝑒,𝑦(𝜃))=(𝑒,𝑦₀+∫^𝜃₀(𝜃−𝑠)𝑣(𝑠)𝑑𝑠)≤𝑑, then we obtain ^𝜃₀(𝜃−𝑠)(𝑣(𝑠),𝑒)𝑑𝑠≤𝑑−(𝑦₀,𝑒). (3.14) On the other hand, 𝜓(𝑡)=‖𝑦₀−𝑥₀‖+𝑡−((𝜃³/3)(𝜌²−𝜎²)+𝑡²)^1/2 is an increasing function on (−∞,∞). Then it follows from (3.11) and (3.14) that 𝑓(𝜃)≤‖𝑦₀−𝑥₀‖+𝑑−(𝑦₀𝜃,𝑒)−(³3(𝜌²−𝜎²)+(𝑑−(𝑦₀,𝑒))²)^1/2. (3.15) Now we show that the right-hand side of the last inequality is equal to zero. We show ‖𝑦₀−𝑥₀‖+𝑑−(𝑦₀𝜃,𝑒)=(³3(𝜌²−𝜎²)+(𝑑−(𝑦₀,𝑒))²)^1/2. (3.16) The left part of this equality is positive, since ‖𝑦₀−𝑥₀‖+𝑑−(𝑦₀=,𝑒)(𝜃³/3)(𝜌²−𝜎²)+‖𝑦₀‖²−‖𝑥₀‖²+2(‖𝑥₀‖²−2(𝑥₀,𝑦₀)+‖𝑦₀‖²)−2‖𝑦₀‖²+2(𝑥₀,𝑦₀)2‖𝑦₀−𝑥₀‖=(𝜃³/3)(𝜌²−𝜎²)+‖𝑥₀‖²−2(𝑥₀,𝑦₀)+‖𝑦₀‖²2‖𝑦₀−𝑥₀‖=(𝜃³/3)(𝜌²−𝜎²)+‖𝑥₀−𝑦₀‖²2‖𝑦₀−𝑥₀‖>0. (3.17) Therefore taking square we have ‖𝑦₀−𝑥₀‖²+(𝑑−(𝑦₀,𝑒))²+2‖𝑦₀−𝑥₀‖(𝑑−(𝑦₀𝜃,𝑒))=³3(𝜌²−𝜎²)+(𝑑−(𝑦₀,𝑒))², (3.18) then ‖𝑦₀−𝑥₀‖²+2‖𝑦₀−𝑥₀‖((𝜃³/3)(𝜌²−𝜎²)+‖𝑦₀‖²−‖𝑥₀‖²2‖𝑦₀−𝑥₀‖−(𝑦₀𝜃,𝑒))=³3(𝜌²−𝜎²). (3.19) The above equality is true since ‖𝑦₀−𝑥₀‖²+‖𝑦₀‖²−‖𝑥₀‖²−2‖𝑦₀−𝑥₀‖‖𝑦₀‖=0. (3.20) So 𝑓(𝜃)=0, consequently 𝑓(𝜏)=0 for some 𝜏,0≤𝜏≤𝜃. Therefore, 𝑥(𝜏)=𝑦(𝜏). Further, by (3.6), 𝑢(𝑡)=𝑣(𝑡) at 𝜏<𝑡≤𝜃. Then 𝑥(𝜃)=𝑥(𝜏)+^𝜃_𝜏(𝜃−𝑠)𝑢(𝑠)𝑑𝑠=𝑦(𝜏)+^𝜃_𝜏(𝜃−𝑠)𝑣(𝑠)𝑑𝑠=𝑦(𝜃), (3.21) and the proof of the lemma is complete.

4. Main Result

Now consider the game (2.9). We will solve the optimal pursuit problem under the following assumption.

Assumption 4.1. There exists a nonzero vector 𝑝₀ such that (𝑦₀−𝑥_𝑖0,𝑝₀)≥0 for all 𝑖∈𝐼.
Let
𝛾=inf{𝑙≥0∶𝐻(𝑦₀𝜃,𝜎(³3)^1/2)⊂^∞_𝑖=1𝐻(𝑥_𝑖0,𝜌_𝑖(𝜃³3)^1/2+𝑙)}. (4.1)

Theorem 4.2. If Assumption 4.1 is true and 𝜎≤𝜌_𝑖+𝛾(3/𝜃³)^1/2 for all 𝑖∈𝐼, then the number 𝛾 given by (4.1) is the value of the game (2.9).

Proof of the above theorem relies on the following lemmas.

Consider the sphere 𝑆(𝑦₀,𝑟) and finitely or countably many balls 𝐻(𝑥_𝑖0,𝑅_𝑖) and 𝐻(𝑦₀,𝑟), where 𝑥_𝑖0≠𝑦₀ and 𝑟 and 𝑅_𝑖,𝑖∈𝐼 are positive numbers.

Lemma 4.3 (see [10]). Let 𝑋_𝑖={𝑧∈𝑙₂∶2(𝑦₀−𝑥_𝑖0,𝑧)≤𝑅²_𝑖−𝑟²+‖𝑦₀‖²−‖𝑥_𝑖0‖²}, (4.2) if 𝑥_𝑖0≠𝑦₀, and 𝑋_𝑖={𝑧∈𝑙₂∶(𝑧−𝑦₀,𝑝₀)≤𝑅_𝑖}, (4.3) if 𝑥_𝑖0=𝑦₀. If Assumption 4.1 is valid and 𝐻(𝑦₀,𝑟)⊂_𝑖∈𝐼𝐻(𝑥_𝑖0,𝑅_𝑖), (4.4) then 𝐻(𝑦₀,𝑟)⊂⋃_𝑖∈𝐼𝑋_𝑖.

Lemma 4.4 (see [10]). Let inf_𝑖∈𝐼𝑅_𝑖=𝑅₀>0. If Assumption 4.1 is true and for any 0<𝜀<𝑅₀ the set ⋃_𝑖∈𝐼𝐻(𝑥_𝑖0,𝑅_𝑖−𝜀) does not contain the ball 𝐻(𝑦₀,𝑟), then there exists a point 𝑦∈𝑆(𝑦₀,𝑟) such that ‖𝑦−𝑥_𝑖0‖≥𝑅_𝑖 for all 𝑖∈𝐼.

Proof of Theorem 4.2. We prove this theorem in three parts.
(1) Construction of the Pursuers' Strategies. We introduce counterfeit pursuers 𝑧_𝑖, whose motions are described by the equations 𝑧̇_𝑖=(𝜃−𝑡)𝑤^𝜀_𝑖,𝑧_𝑖(0)=𝑥_𝑖0,(^𝜃₀‖𝑤^𝜀_𝑖(𝑠)‖²𝑑𝑠)^1/2≤𝜌_𝑖(𝜀)=𝜌_𝑖3+𝛾(𝜃³)^1/2+𝜀𝑘_𝑖(3𝜃³)^1/2, (4.5) where 𝑘_𝑖=max{1,𝜌_𝑖} and 𝜀,0<𝜀<1, is an arbitrary positive number. It is obvious that the attainability domain of the counterfeit pursuer 𝑧_𝑖 at time 𝜃 from an initial state 𝑥_𝑖0 is the ball 𝐻(𝑥_𝑖0,𝜌_𝑖(𝜀)(𝜃³/3)^1/2)=𝐻(𝑥_𝑖0,𝜌_𝑖(𝜃³/3)^1/2+𝛾+𝜀/𝑘_𝑖).
The strategies of the counterfeit pursuers 𝑧_𝑖 are defined as follows: if 𝑥_𝑖0=𝑦₀, then we set 𝑤^𝜀_𝑖(𝑡)=𝑣(𝑡),0≤𝑡≤𝜃, (4.6) and if 𝑥_𝑖0≠𝑦₀, then we set 𝑤^𝜀_𝑖(𝑡)=𝑣(𝑡)−(𝑣(𝑡),𝑒_𝑖)𝑒_𝑖+𝑒_𝑖(3𝜃³(𝜃−𝑡)²(𝜌²_𝑖(𝜀)−𝜎²)+(𝑣(𝑡),𝑒_𝑖)²)^1/2,0≤𝑡≤𝜏_𝑖, (4.7) where 𝑒_𝑖=(𝑦₀−𝑥_𝑖0)/‖𝑦₀−𝑥_𝑖0‖, and 𝑤^𝜀_𝑖(𝑡)=𝑣(𝑡),𝜏_𝑖<𝑡≤𝜃, (4.8) where 𝜏_𝑖,0≤𝜏_𝑖≤𝜃, is the time instant at which 𝑧_𝑖(𝜏_𝑖)=𝑦(𝜏_𝑖) for the first time if it exists. Note that 𝜏_𝑖 need not to exist in [0,𝜃].
Now let us show that the strategies (4.6), (4.7), and (4.8) are admissible. If 𝑥_𝑖0=𝑦₀ and 0≤𝑡≤𝜃, then ^𝜃₀‖𝑤^𝜀_𝑖(𝑠)‖²𝑑𝑠=^𝜃₀‖𝑣(𝑠)‖²𝑑𝑠≤𝜎²≤(𝜌_𝑖3+𝛾(𝜃³)^1/2)²≤(𝜌_𝑖3+𝛾(𝜃³)^1/2+𝜀𝑘_𝑖(3𝜃³)^1/2)²=𝜌²_𝑖(𝜀). (4.9) If 𝑥_𝑖0≠𝑦₀ we have ^𝜃₀‖𝑤^𝜀_𝑖(𝑠)‖²𝑑𝑠=^𝜏𝑖₀‖𝑤^𝜀_𝑖(𝑠)‖²𝑑𝑠+^𝜃_𝜏𝑖‖𝑤^𝜀_𝑖(𝑠)‖²=𝑑𝑠_𝜏𝑖0‖𝑣(𝑠)‖²3𝑑𝑠+𝜃³(𝜌²_𝑖(𝜀)−𝜎²)^𝜏𝑖₀(𝜃−𝑠)²𝑑𝑠+^𝜃_𝜏𝑖‖𝑣(𝑠)‖²≤𝑑𝑠_𝜃0‖𝑣(𝑠)‖²3𝑑𝑠+𝜃³(𝜌²_𝑖(𝜀)−𝜎²)^𝜃₀(𝜃−𝑠)²𝑑𝑠≤𝜎²+𝜌²_𝑖(𝜀)−𝜎²=𝜌²_𝑖(𝜀). (4.10)
The strategies of the pursuers 𝑥_𝑖 are defined as follows: 𝑢_𝑖𝜌(𝑡)=^𝑖𝜌_𝑖𝑤_𝑖(𝑡),0≤𝑡≤𝜃, (4.11) where 𝜌_𝑖=𝜌_𝑖(0)=𝜌_𝑖+𝛾(3/𝜃³)^1/2 and 𝑤_𝑖(𝑡)=𝑤⁰_𝑖(𝑡); that is, 𝑤_𝑖(𝑡) is given by (4.6), (4.7), and (4.8) with 𝜀=0 and the same 𝜏_𝑖.?
(2) The value 𝛾is guaranteed for the pursuers. Let us show that the above-constructed strategies of the pursuers satisfy the inequalities sup_𝑣(⋅)inf_𝑖∈𝐼‖𝑦(𝜃)−𝑥_𝑖(𝜃)‖≤𝛾. (4.12)
By the definition of 𝛾, we have 𝐻(𝑦₀𝜃,𝜎(³3)^1/2)⊂^∞_𝑖=1𝐻(𝑥_𝑖0,𝜌_𝑖(𝜃³3)^1/2𝜀+𝛾+𝑘_𝑖). (4.13) By Assumption 4.1 the inequality (𝑦₀−𝑥_𝑖0,𝑝₀)≥0 holds for all 𝑖∈𝐼. Then it follows from Lemma 4.3 that 𝐻(𝑦₀𝜃,𝜎(³3)^1/2)⊂^∞_𝑖=1𝑋^𝜀_𝑖, (4.14) where 𝑋^𝜀_𝑖={𝑧∶2(𝑦₀−𝑥_𝑖0,𝑧)≤(𝜌_𝑖(𝜃³3)^1/2𝜀+𝛾+𝑘_𝑖)²−𝜎²𝜃³3+‖𝑦₀‖²−‖𝑥_𝑖0‖²}, (4.15) if 𝑥_𝑖0≠𝑦₀, and 𝑋^𝜀_𝑖={𝑧∶(𝑧−𝑦₀,𝑝₀)≤𝜌_𝑖(𝜃³3)^1/2𝜀+𝛾+𝑘_𝑖}, (4.16) if 𝑥_𝑖0=𝑦₀. Consequently, the point 𝑦(𝜃)∈𝐻(𝑦₀,𝜎(𝜃³/3)^1/2) belongs to some half-space 𝑋^𝜀_𝑠,𝑠=𝑠(𝜀)∈𝐼.
By the assumption of the theorem, 𝜌_𝑖(𝜀)>𝜎; then it follows from Lemma 3.1 that if 𝑧_𝑖 uses the strategy (4.6), (4.7), and (4.8), then 𝑧_𝑠(𝜃)=𝑦(𝜃). By taking account of (4.11) we obtain ‖𝑦(𝜃)−𝑥_𝑠(𝜃)‖=‖𝑧_𝑠(𝜃)−𝑥_𝑠(𝜃)‖=‖^𝜃₀(𝜃−𝑡)(𝑤^𝜀_𝑠𝜌(𝑡)−^𝑠𝜌_𝑠𝑤_𝑠≤(𝑡))𝑑𝑡‖_𝜃0‖(𝜃−𝑡)(𝑤^𝜀_𝑠(𝑡)−𝑤_𝑠(𝑡))‖𝑑𝑡+^𝜃₀‖(𝜃−𝑡)(𝑤_𝑠𝜌(𝑡)−^𝑠𝜌_𝑠𝑤_𝑠(𝑡))‖𝑑𝑡. (4.17) Now we put aside the right-hand side of the last inequality. Let us show that lim_𝜀→0sup_𝑖∈𝐼^𝜃₀‖(𝜃−𝑡)(𝑤^𝜀_𝑖(𝑡)−𝑤_𝑖(𝑡))‖𝑑𝑡=0. (4.18)
Indeed, if 𝑥_𝑖0=𝑦₀, then by (4.6), 𝑤^𝜀_𝑖(𝑡)=𝑤_𝑖(𝑡)=𝑣(𝑡), and the validity of (4.18) is obvious. Now let 𝑥_𝑖0≠𝑦₀. If there exists 𝜏_𝑖∈[0,𝜃], mentioned in (4.7) and (4.8), then ^𝜏𝑖₀‖𝑤^𝜀_𝑖(𝑡)−𝑤_𝑖(𝑡)‖²𝑑𝑡=^𝜏𝑖₀3((𝜃³(𝜃−𝑡)²(𝜌²_𝑖(𝜀)−𝜎²)+(𝑣(𝑡),𝑒_𝑖)²)^1/23−(𝜃³(𝜃−𝑡)²(𝜌²_𝑖−𝜎²)+(𝑣(𝑡),𝑒_𝑖)²)^1/2)²≤𝑑𝑡_𝜏𝑖03((𝜃³(𝜃−𝑡)²(𝜌²_𝑖(𝜀)−𝜎²))^1/23−(𝜃³(𝜃−𝑡)²(𝜌²_𝑖−𝜎²))^1/2)²≤𝑑𝑡_𝜃03((𝜃³(𝜃−𝑡)²)^1/2((𝜌²_𝑖(𝜀)−𝜎²)^1/2−(𝜌²_𝑖−𝜎²)^1/2))²𝑑𝑡=((𝜌²_𝑖(𝜀)−𝜎²)^1/2−(𝜌²_𝑖−𝜎²)^1/2)²=((2𝜌_𝑖𝜀𝑘_𝑖(3𝜃³)^1/2+3𝜃³(𝜀𝑘_𝑖)²+𝜌²_𝑖−𝜎²)^1/2−(𝜌²_𝑖−𝜎²)^1/2)²≤2𝜌_𝑖𝜀𝑘_𝑖(3𝜃³)^1/2+3𝜃³(𝜀𝑘_𝑖)²3≤(2(𝜃³)^1/2+6𝛾𝜃³+3𝜃³)𝜀. (4.19) In the last inequality, we have used the facts that 0<𝜀<1, 𝑘_𝑖≥1 and the inequality 𝜌_𝑖𝑘_𝑖=𝜌_𝑖𝑘_𝑖+𝛾(3/𝜃³)^1/2𝑘_𝑖3≤1+𝛾(𝜃³)^1/2. (4.20) So ^𝜃₀‖(𝜃−𝑡)(𝑤^𝜀_𝑖(𝑡)−𝑤_𝑖(𝑡))‖𝑑𝑡=^𝜏𝑖₀‖(𝜃−𝑡)(𝑤^𝜀_𝑖(𝑡)−𝑤_𝑖(𝑡))‖𝑑𝑡≤(^𝜏𝑖₀(𝜃−𝑡)²𝑑𝑡)^1/2(^𝜏𝑖₀‖𝑤^𝜀_𝑖(𝑡)−𝑤_𝑖(𝑡)‖²𝑑𝑡)^1/2≤𝐾𝜀, (4.21) where 𝐾 is some positive number.
For the second integral in (4.17) we have ^𝜃₀𝜌‖(𝜃−𝑡)(1−^𝑠𝜌_𝑠)𝑤_𝑠𝜌(𝑡)‖𝑑𝑡=(1−^𝑠𝜌_𝑠)^𝜃₀‖(𝜃−𝑡)𝑤_𝑠𝜌(𝑡)‖𝑑𝑡≤(1−^𝑠𝜌_𝑠)(^𝜃₀(𝜃−𝑡)²𝑑𝑡)^1/2(^𝜃₀‖𝑤_𝑠(𝑡)‖²𝑑𝑡)^1/2𝜌=(1−^𝑠𝜌_𝑠𝜃)(³3)^1/2𝜌_𝑠=𝛾. (4.22) Then it follows from (4.17) that ‖𝑦(𝜃)−𝑥_𝑠(𝜃)‖≤𝛾+𝐾𝜀.
Thus if the pursuers use the strategies (4.11), the inequality (4.12) is true.
(3) The value?? 𝛾 ??is guaranteed for the evader. Let us construct the evader's strategy ensuring that inf_{𝑢1(⋅),…,𝑢𝑚(⋅),…}inf_𝑖∈𝐼‖𝑦(𝜃)−𝑥_𝑖(𝜃)‖≥𝛾, (4.23) where 𝑢₁(⋅),…,𝑢_𝑚(⋅),… are arbitrary admissible controls of the pursuers. If 𝛾=0, then inequality (4.23) is obviously valid for any admissible control of the evader. Let 𝛾>0. By the definition of 𝛾, for any 𝜀>0, the set ^∞_𝑖=1𝐻(𝑥_𝑖0,𝜌_𝑖(𝜃³3)^1/2+𝛾−𝜀), (4.24) does not contain the ball 𝐻(𝑦₀,𝜎(𝜃³/3)^1/2). Then, by Lemma 4.4 there exists a point 𝑦∈𝑆(𝑦₀,𝜎(𝜃³/3)^1/2), that is, ‖𝑦−𝑦₀‖=𝜎(𝜃³/3)^1/2 such that ‖𝑦−𝑥_𝑖0‖≥𝜌_𝑖(𝜃³/3)^1/2+𝛾. On the other hand ‖𝑥_𝑖(𝜃)−𝑥_𝑖0𝜃‖≤(³3)^1/2(^𝜃₀‖𝑢_𝑖(𝑡)‖²𝑑𝑡)^1/2=𝜌_𝑖(𝜃³3)^1/2. (4.25) Consequently ‖𝑦−𝑥_𝑖(𝜃)‖≥‖𝑦−𝑥_𝑖0‖−‖𝑥_𝑖(𝜃)−𝑥_𝑖0‖≥𝜌_𝑖(𝜃³3)^1/2+𝛾−𝜌_𝑖(𝜃³3)^1/2=𝛾. (4.26) Now by using the control 3𝑣(𝑡)=𝜎(𝜃³)^1/2(𝜃−𝑡)𝑒,0≤𝑡≤𝜃,𝑒=𝑦−𝑦₀‖𝑦−𝑦₀‖, (4.27) we obtain 𝑦(𝜃)=𝑦₀+^𝜃₀(𝜃−𝑠)𝑣(𝑠)𝑑𝑠=𝑦₀+^𝜃₀(𝜃−𝑠)²3𝜎(𝜃³)^1/2𝑒𝑑𝑠=𝑦₀𝜃+𝜎(³3)^1/2𝑒=𝑦. (4.28) Then the value of the game is not less than 𝛾, and inequality (4.23) holds. The proof of the theorem is complete.

5. Conclusion

A pursuit-evasion differential game of fixed duration with countably many pursuers has been studied. Control functions satisfy integral constraints. Under certain conditions, the value of the game has been found, and the optimal strategies of players have been constructed.

The proof of the main result relies on the solution of an auxiliary differential game problem in the half-space. Such method was used by many authors (see, e.g., [5, 6]), but the method used here for this auxiliary problem is different from those of others and requires only basic knowledge of calculus.

It should be noted that the condition given by Assumption 4.1 is relevant. If this condition does not hold, then, in general, we do not have a solution of the pursuit-evasion problem even in a finite dimensional space with a finite number of pursuers.

The present work can be extended by considering higher-order differential equations instead of (2.1). Then differential game can be reduced to an equivalent game, described by (2.1), with 𝜃−𝑡 replaced by another function.

Acknowledgments

The authors would like to thank the referee for giving useful comments and suggestions for the improvement of this paper. The present research was supported by the National Fundamental Research Grant Scheme (FRGS) of Malaysia, no. 05-10-07-376FR.