1. Introduction
The differential game has been an area of great interest to many applied mathematicians due to its application in solving real life problems in knowledge areas such as economics, engineering, missile guidance, behavioral biology. The first to study differential game was Rufus Isaacs [
1], and one of the games analyzed was “the homicidal chauffeur game”. For the fundamental concepts of differential games, see [
1,
2,
3,
4,
5,
6,
7,
8].
There are many different types of differential game problems, and one type is called pursuit-evasion differential game. Pursuit-evasion differential game is a game involving two players, called pursuer and evader, with conflicting goals. The aim of the pursuer is to complete the game in a finite time, whereas that of the evader is contrary. Strategies of pursuit and evasion play a role in many areas of life, such as missile launched at enemy aircraft, coastguard saving shipwrecked sailors, etc. The problem of constructing optimal strategies and finding the value of the game in a pursuit-evasion differential game motivated a lot of researchers to study this class of differential game problems, and fundamental results have been obtained, see [
9,
10,
11,
12,
13].
Differential games of many pursuers with the integral and geometry constrained were studied by [
9,
14,
15,
16,
17,
18,
19]. The case where the state variables are constraints was studied in [
20], they considered a nonempty closed convex set in a plane, with the pursuers and evader movement restricted within the set during the game. Conditions under which pursuit could be completed were obtained and the strategies for the pursuers were constructed.
The evasion differential game of two dimensions, which involves one evader and several pursuers, was studied in [
11]. The control functions of the players were subject with integral constraints. The game is solved by presenting explicit strategy for the evader, which guarantees evasion under the condition that there is no relation between the energy resource of the players.
In [
21] Levchenko and Pashkov considered differential games described by simple differential equations, where the controls obeyed integral constraints. However, they showed that irrespective of the resources for controls of an individual, the completion of the game remains doable.
Ibragimov and Satimov [
22] obtained sufficient condition for the completion of pursuit in a differential game problem of several pursuers and evaders in the space
, with integral constraints on the control functions of the players. The results were obtained under the condition that the energy resource of the pursuers is greater than that of the evaders.
Ibragimov [
9] studied a differential game of a countable number of pursuers pursuing one evader in Hilbert space
, with geometric constraints on the control functions of the pursuers and evader. Optimal strategies of the players were constructed and optimal pursuit time was found, under the assumption that the energy resource of the pursuers is greater than that of the evader.
Ibragimov and Kuchkarov [
10] considered the same problem in [
9], with integral constraints imposed on the control functions of the players. In this case, optimal strategies were constructed and value of the game was found under the assumption that energy resource of the evader is greater than that of any pursuers.
Salimi and Ferrara [
12] studied a simple motion differential game with finite number of pursuers and one evader with integral constraints imposed on control of the players in Hilbert space
. The equations of motion are described by
where
is the control function of the
pursuers and
v is that of the evader. The authors solved the problem and found the value of the game under the assumption that the energy resource of each pursuer is not necessarily greater than that of the evader, and optimal strategies of the pursuers were also constructed.
Inspired by the results in [
9,
10,
11,
12,
22] and some known results on optimal pursuit problem in a Hilbert space
, the objective of this paper is to construct the optimal strategies and finding value of the game such that there is no relation between the energy resource of the players.
This paper is sectioned as follows. The second section present statement of the problems and some useful definitions which will be required for the later sections. In the third section, attainability domain of the players, optimal strategies and value of the game and some examples are given to show the application of the obtained results. In the last section, we give the concluding part of the paper.
2. Statement of the Problem
In this section, we present the statement of the problem and some useful definitions that will be used to prove our main theorem.
Here, we will consider an optimal pursuit problem with finite or countably many pursuers and one evader in a Hilbert space
, in such away that there is no relation between the their energy resources. In the space
, with elements
the inner product and norm are defined as
Let
and E denote the motions of the pursuers and the evader, whose equations are described by
where
,
and
are the control parameters of pursuer
and evader E, respectively. Throughout this paper,
.
Let
be a fixed time, and the function
be nonzero on any open interval—scalar measurable and square integrable over the interval
It is also assumed to satisfy the following conditions:
denote the ball and sphere, respectively, in the space
with center
and radius
r. We now give some useful definitions.
Definition 1. The admissible control of the pursuer is a function , defined asprovided that , , are Borel measurable functions and is a fixed positive number . Let denote the set of all admissible controls of the pursuer Definition 2. The admissible control of the evader is a function , defined aswhere , , are Borel measurable functions and σ is a fixed positive number. Let denote the set of all admissible controls of the evader y. Once the players admissible controls
and
are chosen, the corresponding motion
and
of the players are defined as
It is not difficult to see that
, where
is the space of functions
such that
- (i)
, are absolutely continuous functions;
- (ii)
, is continuous with respect to the norm on space.
Definition 3. The strategy of the pursuer is a function , such that the system of equationshas a unique solution for any and admissible control , , of the evader E. We said that the strategy is admissible if each control formed by strategy is admissible. Definition 4. The optimal strategies of the pursuers are defined assuch thatwhere and are admissible strategies of the pursuers and evader E, respectively. Definition 5. The strategy of evader E is a function , such that the system of equationshas a unique solution for any , and admissible controls , of the pursuers . We said that strategy V is admissible if each control formed by strategy V is admissible. Definition 6. The optimal strategy of the evader E is defined asprovided that are admissible control of the pursuers and V is that of the evader E. If
, then, problem (
2) has a value
.
Our aim is to find the optimal strategies , of the players and value of the game, respectively.
3. Auxiliary Game
It is easily to see that the attainability domain of the pursuer
from the initial position
at time
is the closed ball
. Indeed
Moreover, if
, that is,
, then, for the control
of the pursuer, we get
Therefore, the admissibility of this control follows from the relation
Moreover, applying the same procedure one can see that the attainability domain of the evader E at time from the initial state is the ball
In this section, we study a differential game of one pursuer
x and one evader
y. For simplicity, we use the notation
,
, and
. Then, dynamics of
x and
y are described by
The target of the pursuer P is to perceive the equality at some , ; and that of the evader E is contrary.
Lemma 1. If , then, there exists an admissible strategy of the pursuer P which ensures in the game (4). Proof. Suppose the assumption of Lemma 1 holds, construct the strategy of the pursuer as follows:
Then, admissibility of this strategy can be proved as follows. Since
then,
Hence from the strategy (
5) and inequality (
6), we have
This shows that the strategy (
5) is admissible. Therefore,
This proves the lemma. ☐
Remark 1. It should be noted that in the construction of the pursuer’s strategy we do not require the inequality
4. Main Result
We recall the following lemmas in order to prove our main theorem.
Lemma 2. (see Ibragimov et al. 2005. Lemma 9). Suppose r and , , are fixed positive real numbers and , , and are collections of finitely or a countable number of closed balls. Let If , for a nonzero vector , andthen, Lemma 3. (See Ibragimov 2005. Assertion 5). Let and , for a nonzero vector , if for any the set does not contain the ball , then, there exists a point such that , for all
Theorem 1. Let for all , for a nonzero vector , then, the numberis the value of the game (2). Proof. The proof of the theorem in divided into three parts:
It can be shown easily that the attainability domain of the fictitious pursuers
at time
from the initial position
is the ball
Next, we define the strategy of the fictitious pursuers
, as follows:
where
is the time for which
Note that such time may not exist.
We now define the strategies of the real pursuers
, by
where
- 2.
The value is guaranteed for the pursuers.
We now show that the strategies (
12) satisfy the inequality
Thus, it follows from definition of
that
Then, it follows from Lemma 2 where
and
that
where
Accordingly, the point
belongs to some half-space
, with
and hence,
By Lemma 1, for the strategies of the pursuers
, we obtain
and
By strategy (
12), we have
Using Cauchy–Schwatz inequality, we obtain
Therefore, from inequalities (
17) and (
18) we obtain
Hence, the value is guaranteed by the actual pursuers.
- 3.
The value is guaranteed for the evader.
Define the evader’s strategy that satisfies
where
are the admissible control of the pursuers. If
, then, the result follows from (
20). Suppose that
, then, by definition of
, for any
, the set
does not include the ball
. According to Lemma 3, there exists a point
such that
. Therefore, by the inequality
Hence, if the evader comes to the point
at the time
, the inequality (
20) is guaranteed. This is achievable using the control function
that is,
Hence, the value of the game is not less than and the inequality is satisfied. This complete the proof of Theorem 1. ☐
We now present some examples to demonstrate our result.
Example 1. Consider the differential game problem (2) with , and denoting Consider the following initial positions of the players: We can easily see that by simple computation we obtain that and
Next, we show that is value of the game. Firstly, it is enough to show that
- i)
Given any , the insertionholds, where O is the origin. - ii)
Since the ball is not contained in the set that is, let be an arbitrary point of the ball . This implies that The following two cases are possible, either z has a non negative coordinate or all coordinates of z are negative. Suppose that there exists a non-negative coordinate of the vector z, then,hence, Now, suppose that all the coordinates of z are negative. This implies that the inequalityis satisfied for sufficiently large k, since the series , as is convergent. Additionally, any point with negative coordinates does not belong to the set , since for any number i, Therefore, the numberis the value of the game by the theorem.