#### 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 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 .

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 is fixed. The payoff functional of the game is the greatest lower bound of the distances between the evader and the pursuers at . 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 consisting of elements with , and inner product , the motions of the countably many pursuers and the evader are defined by the equations where is the control parameter of the pursuer , and is that of the evader ; here and throughout the following, . Let be a given positive number, and let

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 and center at the point is denoted by (resp., by ).

*Definition 2.1. *A function such that are Borel measurable functions and
where is given positive number, is called an *admissible control of the ith pursuer*.

*Definition 2.2. *A function such that are Borel measurable functions and
where is a given positive number, is called an *admissible control of the evader*.

Once the players' admissible controls and are chosen, the corresponding motions and of the players are defined as

One can readily see that where is the space of functions such that the following conditions hold:

(1) are absolutely continuous functions;(2) is a continuous function in the norm of*Definition 2.3. *A function such that the system
has a unique solution with for an arbitrary admissible control of the evader is called a *strategy of the pursuer *. A strategy is said to be *admissible* if each control formed by this strategy is admissible.

*Definition 2.4. *Strategies of the pursuers are said to be *optimal* if
where are admissible strategies of the pursuers and is an admissible control of the evader .

*Definition 2.5. *A function such that the countable system of equations
has a unique solution with for arbitrary admissible controls of the pursuers is called a *strategy* of the evader If each control formed by a strategy is admissible, then the strategy itself is said to be *admissible*.

*Definition 2.6. *A strategy of the evader is said to be *optimal* if where where are admissible controls of the pursuers and is an admissible strategy of the evader

If then we say that the game has the value [7].

It is to find optimal strategies and of the players and , 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: Indeed, if the pursuer uses an admissible control , then according to (2.1) we have and the same result can be obtained by (2.9) Also, for the evader the same argument can be made, therefore in the distance we can take either the solution of (2.1) or the solution of (2.9).

The attainability domain of the pursuer at time from the initial state at time is the closed ball Indeed, by Cauchy-Schwartz inequality

On the other hand, if that is, then for the pursuer's control we obtain

The pursuer's control is admissible because Likewise, the attainability domain of the evader at time from the initial state at time is the closed ball

#### 3. An Auxiliary Game

In this section we fix the index and study an auxiliary differential game of two players and , also for simplicity we drop the index and use the notion and Let if if then where is an arbitrary fixed unit vector.

Consider the one-pursuer game described by the equations with the state of the evader being subject to . The goal of the pursuer is to realize the equality at some and that of the evader is opposite.

We define the pursuer's strategy as follows: if then we set and if then we set where , and where is the time instant at which for the first time.

Lemma 3.1. *If and then the pursuer's strategy (3.4), (3.5), and (3.6) in the game (3.3) ensures that *

*Proof. *If then from (3.4) we have because
In particular,

Let By (3.5) and (3.6), we have where
Obviously Now we show that This will imply that for some

To this end we consider the following two-dimensional vector function:
For the last integral of (3.8) we have
Then

By assumption, therefore
so where
As , then we obtain
On the other hand, is an increasing function on Then it follows from (3.11) and (3.14) that
Now we show that the right-hand side of the last inequality is equal to zero. We show
The left part of this equality is positive, since
Therefore taking square we have
then
The above equality is true since
So , consequently for some Therefore, Further, by (3.6), at Then
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 such that for all .

Let

Theorem 4.2. *If Assumption 4.1 is true and for all then the number 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 and finitely or countably many balls and where and and are positive numbers.

Lemma 4.3 (see [10]). * Let
**
if and
**
if If Assumption 4.1 is valid and
**
then *

Lemma 4.4 (see [10]). *Let If Assumption 4.1 is true and for any the set does not contain the ball then there exists a point such that for all *

*Proof of Theorem 4.2. * We prove this theorem in three parts. * Construction of the Pursuers' Strategies*. We introduce counterfeit pursuers whose motions are described by the equations
where and , is an arbitrary positive number. It is obvious that the attainability domain of the counterfeit pursuer at time from an initial state is the ball

The strategies of the counterfeit pursuers are defined as follows: if then we set
and if then we set
where and
where is the time instant at which for the first time if it exists. Note that need not to exist in

Now let us show that the strategies (4.6), (4.7), and (4.8) are admissible. If and then
If we have

The strategies of the pursuers are defined as follows:
where and that is, is given by (4.6), (4.7), and (4.8) with and the same .?* The value **is guaranteed for the pursuers*. Let us show that the above-constructed strategies of the pursuers satisfy the inequalities

By the definition of we have
By Assumption 4.1 the inequality holds for all Then it follows from Lemma 4.3 that
where
if and
if Consequently, the point belongs to some half-space

By the assumption of the theorem, then it follows from Lemma 3.1 that if uses the strategy (4.6), (4.7), and (4.8), then By taking account of (4.11) we obtain
Now we put aside the right-hand side of the last inequality. Let us show that

Indeed, if then by (4.6), and the validity of (4.18) is obvious. Now let If there exists mentioned in (4.7) and (4.8), then
In the last inequality, we have used the facts that , and the inequality
So
where is some positive number.

For the second integral in (4.17) we have
Then it follows from (4.17) that

Thus if the pursuers use the strategies (4.11), the inequality (4.12) is true.* The value?? ** ??is guaranteed for the evader*. Let us construct the evader's strategy ensuring that
where are arbitrary admissible controls of the pursuers. If then inequality (4.23) is obviously valid for any admissible control of the evader. Let . By the definition of for any the set
does not contain the ball Then, by Lemma 4.4 there exists a point that is, such that On the other hand
Consequently
Now by using the control
we obtain
Then the value of the game is not less than 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 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.