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.

Many books have been devoted to differential games, such as books by Isaacs [

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 [

Levchenkov and Pashkov [

Chodun [

Ibragimov [

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

In the space

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 (

A ball (resp., sphere) of radius

A function

A function

Once the players' admissible controls

One can readily see that

A function

Strategies

A function

A strategy

If

It is to find optimal strategies

The attainability domain of the pursuer

On the other hand, if

The pursuer's control is admissible because

In this section we fix the index

Consider the one-pursuer game described by the equations

We define the pursuer's strategy as follows: if

If

If

Let

To this end we consider the following two-dimensional vector function:

By assumption,

Now consider the game (

There exists a nonzero vector

Let

If Assumption

Proof of the above theorem relies on the following lemmas.

Consider the sphere

Let

Let

We prove this theorem in three parts.

The strategies of the counterfeit pursuers

Now let us show that the strategies (

The strategies of the pursuers

By the definition of

By the assumption of the theorem,

Indeed, if

For the second integral in (

Thus if the pursuers use the strategies (

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., [

It should be noted that the condition given by Assumption

The present work can be extended by considering higher-order differential equations instead of (

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.