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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 460171, 12 pages
http://dx.doi.org/10.1155/2012/460171
Research Article

A Multiplayer Pursuit Differential Game on a Closed Convex Set with Integral Constraints

1Institute for Mathematical Research and Department of Mathematics, Faculty of Science (FS), Universiti Putra Malaysia, Selangor, 43400 Serdang, Malaysia
2Department of Mathematics, National University of Uzbekistan, Vuzgorodok, 100174 Tashkent, Uzbekistan

Received 5 July 2012; Revised 30 September 2012; Accepted 30 September 2012

Academic Editor: Jianquan Lu

Copyright © 2012 Gafurjan Ibragimov and Nu'man Satimov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study a simple motion pursuit differential game of many pursuers and many evaders on a nonempty convex subset of . In process of the game, all players must not leave the given set. Control functions of players are subjected to integral constraints. Pursuit is said to be completed if the position of each evader , , coincides with the position of a pursuer , , at some time , that is, . We show that if the total resource of the pursuers is greater than that of the evaders, then pursuit can be completed. Moreover, we construct strategies for the pursuers. According to these strategies, we define a finite number of time intervals and on each interval only one of the pursuers pursues an evader, and other pursuers do not move. We derive inequalities for the resources of these pursuer and evader and, moreover, show that the total resource of the pursuers remains greater than that of the evaders.

1. Related Work

Linear differential games with integral constraints on controls were examined in many works, for example, [116].

Satimov et al. [10] studied a linear pursuit differential game of many pursuers and one evader with integral constraints on controls of players in . Game is described by the following equations: where is the control parameter of the th pursuer and is that of the evader. The eigenvalues of the matrices are assumed to be real numbers. They proved that if the total resource of controls of the pursuers is greater than that of the evader, then under certain conditions pursuit can be completed.

Ibragimov [15] examined a pursuit differential game of pursuers and evaders with integral constraints described by the following systems of differential equations where is the control parameter of the th pursuer and is that of the th evader. Different from the previous work, here eigenvalues of matrices are not necessarily real, and, moreover, the number of evaders can be any. If the total resource of controls of the pursuers is greater than that of the evaders, that is, and real parts of all eigenvalues of the matrices are nonpositive, then it was proved that pursuit can be completed from any initial position. Here, game was considered in without any state constraint.

Ivanov [17] considered generalized Lion and Man problem in the case of geometric constraints. All players have equal dynamic possibilities. Motions of the players are described by the following equations: where , are control parameters of the pursuers and is control parameter of the evader. During the game, all players may not leave a given compact subset of . It was shown that if the number of pursuers does not exceed the dimension of the space , then evasion is possible; otherwise pursuit can be completed. In other words, Ivanov [17] derived necessary and sufficient condition of evasion for the multiple Lion and Man game in .

Ibragimov [16] studied a differential game problem of one pursuer and one evader with integral constraints. Game occurs on a closed convex subset of , and movements of the players are described by the following equations: Evasion and pursuit problems were investigated. In the latter case, a formula for optimal pursuit time was found.

Leong and Ibragimov [14] studied simple motion pursuit differential game of pursuers and one evader on a closed convex subset of the Hilbert space . Control functions of the players are subjected to integral constraints. The total resource of the pursuers is assumed to be greater than that of the evader. Strategies of pursuers were constructed to complete the pursuit from any initial position.

In the present paper, we study a pursuit differential game of many pursuers and many evaders on a nonempty convex subset of . In process of the game, all players must not leave the set . Control functions of the players are subjected to integral constraints. We will show that if the total resource of the pursuers is greater than that of the evaders, then pursuit can be completed. In Table 1, we can compare the cases studied in the works of the previous researches and the present paper.

tab1
Table 1: Brief summary of obtained results.

2. Statement of the Problem

We consider a differential game described by the following equations: where is control parameter of the pursuer , is that of the evader , , and is a scalar measurable function that satisfies the following conditions:

Definition 2.1. A measurable function , , is called admissible control of the pursuer if where , are given positive numbers. We denote the set of all admissible controls of the pursuer by .

Definition 2.2. A measurable function , is called an admissible control of the evader if where , are given positive numbers.

Definition 2.3. A Borel measurable function , , is called a strategy of the pursuer if for any control of the evader , the initial value problem has a unique solution and the inequality holds.

Definition 2.4. We say that pursuit can be completed from the initial position for the time in the game (2.1)–(2.4), if there exist strategies , of the pursuers such that for any controls of the evaders and numbers , the equality holds for some at some time .
Given nonempty convex subset of , according to the rule of the game all players must not leave the set , that is, , , , . This information describes a differential game of many players with integral constraints on control functions of players.

Problem 1. Find a sufficient condition of completing pursuit in the game (2.1)–(2.4).

3. The Main Result

Since the control parameters of the players can take the values of opposite signs, without loss of generality, we may assume that . Moreover, without loss of generality, we may assume that is not identically zero on every open interval. Otherwise, if for instance , then instead of we consider

We now formulate the main result of the paper.

Theorem 3.1. If then pursuit can be completed for a finite time in the game (2.1)–(2.4) from any initial position.

Proof. (10) An auxiliary differential game: To prove this theorem, we first study an auxiliary differential game of one pursuer and one evader , which is described by the following equations: where is control parameter of the pursuer , and is that of the evader . Assume that . Pursuit is completed if at some time . Here the players move in without any state constraint.
Set where is an arbitrary fixed number satisfying Let
We will now prove the following statement.

Lemma 3.2. Let the pursuer use the strategy (3.4). (i) If then pursuit can be completed in the game (3.3) for the time , and, moreover,
(ii) If , then either or

Proof of the Lemma. Let . In this case, we show that the control (3.4) is admissible and ensures the equality . Indeed, clearly
To show the admissibility of the strategy (3.4), we use the Cauchy-Schwartz inequality as follows:
Then by this inequality we have
According to (3.5) and the condition , the right-hand side of (3.11) is less than or equal to
At this point, we have proved admissibility of the control (3.4).
In particular, using (3.11) we obtain
Thus, (3.7) holds. We now turn to the part (ii) of the Lemma 3.2. Let and the pursuer use the strategy (3.4) on the interval . If for a control of the evader , the following inequality is satisfied: then, clearly, the control (3.4) is admissible, and similar to (3.9) we obtain that and the proof of the lemma follows. Hence, if , then for the control (3.4) we must have
From this inequality by using calculations similar to (3.11), we then obtain and hence
This completes the proof of the lemma.

The last inequality can be interpreted as follows. Though , the pursuer can force to expend the evader’s energy more than . At the same time, using the strategy (3.4), the pursuer will spend all his energy by a time

Then, of course, the pursuer cannot move anymore and automatically .

(20) Fictitious pursuers (FP): We now prove the theorem. To this purpose, introduce fictitious pursuers by equations where is control parameter of the pursuer . FPs may go out of the set . They move without any state constraint in . The aim of the FPs is to complete the pursuit as earlier as possible. Set where is any number satisfying inequalities

Since by the assumption as , it follows from (3.2) that such exists. Equations (3.20), (3.21) mean that all FPs do not move on the time interval , and only one FP moves according to (3.20).

We will now show that if (3.2) holds and the FPs use the strategies (3.20), (3.21), then the pursuit problem with the pursuers and evaders is reduced to a pursuit problem with the pursuers and evaders , for which and . Hence, by the time , the number of pursuers is reduced to .

Indeed, we consider two possible cases: (i) ; (ii) . In the former case, that is, , if the equality holds, then we eliminate the pursuer and the evader and then consider the pursuit problem with pursuers and evaders under the condition In case of , according to (3.8), we obtain that

Then in view of (3.23) we obtain and at the time we consider the pursuit problem with the pursuers and evaders .

We now turn to the case (ii), that is, . Then the pursuer certainly ensures the equality and according to Lemma 3.2 (see, (3.7))

Then with the aid of (3.23) we obtain and therefore at the time we arrive at the pursuit problem with the pursuers and evaders .

Let now be an arbitrary fixed number satisfying inequalities where and are the numbers of pursuers and evaders, respectively, which take part in the pursuit problem at time . Set

Observe that according to (3.30) all pursuers except for will not move on the time interval .

Let the pursuers use the strategies (3.30). Applying the same arguments above, we arrive at the following conclusion.(1)If and , then starting from the time we consider a pursuit problem with the pursuers and evaders under the condition (2)If and , then starting from the time we consider the pursuit problem with the pursuers and evaders under the condition (3)If , then by Lemma 3.2 the equality holds. In this case, starting from the time the pursuit problem with the pursuers and evaders is considered under the condition

Repeated application of this procedure enables us to complete the pursuit for some finite time since the number of players is decreasing. Thus, we have proved that FPs can complete the pursuit.

(20) Completion of the proof of the Theorem: We will now show that the actual pursuers also can complete the pursuit. Define the controls of the pursuers by the controls of the FPs . We denote by the projection of the point on the set , that is,

Note that if . It is familiar that for any point there exists a unique point . Moreover, for any , and hence the operator relates any absolute continuous function , to an absolute continuous function where is the time in which pursuit can be completed by FPs. Define the control to satisfy

We first show admissibility of such defined control . Indeed, from (3.35),

Hence the inequality holds almost everywhere and therefore

Since for any evader , the equality holds for some and , and the evader is in for any ; in particular, , then FP is also in . Consequently,

This means differential game (2.1)–(2.4) can be completed for the time . The proof of the theorem is complete.

We give an illustrative example.

Example 3.3. We consider a differential game described by the following equations:
It is assumed that for all . The control functions satisfy the integral constraints (2.3) and (2.4). Pursuit is said to be completed if for any numbers the equality holds for some at some time . Players may not leave the half space , where is a nonzero vector. Since where then the equality is equivalent to the one and inclusions and are equivalent to ones and , respectively. Therefore, differential game described by (3.41) is equivalent to the game described by the following equations:
Denoting , we obtain a differential game of the form (2.1).

4. Conclusion

We have obtained a sufficient condition (3.2) to complete the pursuit in the differential game of pursuers and evaders with integral and state constraints. We have constructed strategies of the pursuers and showed that pursuit can be completed from any initial position in . In the case of and (one evader), the condition (3.2) of the theorem takes the form . This condition is sharp since if , then evasion is possible [13].

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