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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 717124, 13 pages
http://dx.doi.org/10.1155/2012/717124
Research Article

On Some Pursuit and Evasion Differential Game Problems for an Infinite Number of First-Order Differential Equations

1Department of Mathematics, Universiti Putra Malaysia, Serdang, 43400 Selangor, Malaysia
2Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, Serdang, 43400 Selangor, Malaysia

Received 25 January 2012; Accepted 6 May 2012

Academic Editor: Debasish Roy

Copyright © 2012 Abbas Badakaya Ja'afaru and Gafurjan Ibragimov. 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 pursuit and evasion differential game problems described by infinite number of first-order differential equations with function coefficients in Hilbert space l2. Problems involving integral, geometric, and mix constraints to the control functions of the players are considered. In each case, we give sufficient conditions for completion of pursuit and for which evasion is possible. Consequently, strategy of the pursuer and control function of the evader are constructed in an explicit form for every problem considered.

1. Introduction

The books by Friedman [1], Isaacs [2], Krasovskiĭ and Subbotin [3], Lewin [4], Petrosyan [5], and Pontryagin [6] among others are fundamental to the study of differential games.

Many works are devoted to differential game problems described by both ordinary differential equations in 𝑅𝑛 and partial differential equations. In particular, pursuit and evasion differential game problems involving distributed parameter systems are of increasing interest (see, e.g., [714]).

Satimov and Tukhtasinov [10, 11] studied pursuit and evasion problems described by the parabolic equation 𝑧𝑡𝐴𝑧=𝑢+𝑣,𝑧|𝑡=0=𝑧0(𝑥),𝑧|𝑆𝑇=0,(1.1) where 𝑧=𝑧(𝑡,𝑥) is unknown function; 𝑥=(𝑥1,𝑥2,,𝑥𝑛)Ω𝑅𝑛,𝑛1 is parameter in a bounded domain Ω; 𝑡[0,𝑇],𝑇>0; 𝑢=𝑢(𝑡,𝑥),𝑣=𝑣(𝑡,𝑥) are control functions of the players; 𝑆𝑇={(𝑥,𝑡)|𝑥𝜕Ω,𝑡(0,𝑇)}, 𝐴 is a differential operator defined in the space 𝐿2(Ω). In this problem, the authors used the fact in [15] that under certain conditions, the problem (1.1) has a unique generalized (in the sense of distribution) solution of the form 𝑧(𝑡,𝑥)=𝑘=1𝑧𝑘(𝑡)𝜓𝑘(𝑥),(1.2) where the functions 𝑧𝑘(𝑡), 0𝑡𝑇, 𝑘=1,2,, constitute the solution of the Cauchy problem for the following infinite system of differential equations and initial conditions: ̇𝑧𝑘+𝜆𝑘𝑧𝑘=𝑢𝑘(𝑡)+𝑣𝑘(𝑡),𝑧𝑘(0)=𝑧0𝑘,𝑘=1,2,,(1.3) and 𝜆𝑘, 𝑘=1,2,, satisfying the condition that 0<𝜆1𝜆2, are eigenvalues of the operator 𝐴, the functions 𝜓𝑘(𝑥),𝑘=1,2,, constitute an orthonormal and complete system of eigenfunctions of the operator 𝐴, 𝑢𝑘(𝑡),𝑣𝑘(𝑡), and 𝑧0𝑘 are the Fourier coefficients in the expansion of 𝑢(𝑡,𝑥),𝑣(𝑡,𝑥), and 𝑧0(𝑥), respectively, in the system {𝜓𝑘(𝑥)}.

The work above shows the significant relationship between differential game problems described by (1.1) in one side and those described by (1.3) in the other side. Therefore, it is logical to study the latter in an independent frame work (see, e.g., [16, 17]).

In the present paper, we solve pursuit and evasion problems described by system (1.3), with function coefficients 𝜆𝑘,𝑘=1,2, instead of constants coefficients considered in the previous study. Different forms of constraints on the controls of the players are considered.

2. Statement of the Problem

Let 𝑙2=𝛼𝛼=1,𝛼2,𝑘=1𝛼2𝑘,<(2.1) with inner product and norm 𝛼,𝛽=𝑘=1𝛼𝑘𝛽𝑘,𝛼,𝛽𝑙2,𝛼=𝑘=1𝛼2𝑘1/2,𝑤()𝐿2(0,𝑇,𝑙2)=𝑘=1𝑇0𝑤2𝑘(𝑠)𝑑𝑠1/2.(2.2)

Let 𝐿20,𝑇,𝑙2=𝑤𝑤(𝑡)=1(𝑡),𝑤2(𝑡),𝑤()𝐿2(0,𝑇,𝑙2)<,𝑤𝑘()𝐿2,(0,𝑇)(2.3) where 𝑇,𝑇>0, is a given number.

We examine a pursuit and evasion differential game problems described by the following infinite system of differential equations ̇𝑧𝑘(𝑡)+𝜆𝑘(𝑡)𝑧𝑘(𝑡)=𝑢𝑘(𝑡)+𝑣𝑘(𝑡),𝑧𝑘(0)=𝑧𝑘0,𝑘=1,2,,(2.4) where 𝑧𝑘,𝑢𝑘,𝑣𝑘𝑅1,𝑘=1,2,,𝑧0=(𝑧10,𝑧20,)𝑙2,𝑢𝑘,𝑣𝑘,, are control parameters of pursuer and evader respectively, 𝜆𝑘(𝑡),𝑘=1,2,, are bounded, non-negative continuous functions on the interval [0,𝑇] such that 𝜆𝑘(0)=0,𝑘=1,2,.

Definition 2.1. A function 𝑤(),𝑤[0,𝑇]𝑙2, with measurable coordinates 𝑤𝑘(𝑡),0𝑡𝑇,𝑘=1,2,, subject to 𝑘=1𝑇0𝑤2𝑘(𝑠)𝑑𝑠𝜌2𝑘=1𝑤2𝑘(𝑡)𝜌2[],,𝑡0,𝑇(2.5) where 𝜌 is a positive number, is referred to as an admissible control subject to integral constraint (resp., geometric constraint).

We denote the set of all admissible controls with respect to integral constraint by 𝑆1(𝜌) and with respect to geometric constraint by 𝑆2(𝜌).

The control 𝑢()=(𝑢1(),𝑢2(),) of the pursuer and 𝑣()=(𝑣1(),𝑣2(),) of the evader are said to be admissible if they satisfy one of the following conditions 𝑘=1𝑇0𝑢2𝑘(𝑠)𝑑𝑠1/2𝜌,𝑘=1𝑇0𝑣2𝑘(𝑠)𝑑𝑠1/2𝜎,(2.6)𝑘=1𝑢2𝑘(𝑡)1/2[],𝜌,𝑡0,𝑇𝑘=1𝑣2𝑘(𝑡)1/2[],𝜎,𝑡0,𝑇(2.7)𝑘=1𝑇0𝑢2𝑘(𝑠)𝑑𝑠1/2𝜌,𝑘=1𝑣2𝑘(𝑡)1/2[],𝜎,𝑡0,𝑇(2.8)𝑘=1𝑢2𝑘(𝑡)1/2[],𝜌,𝑡0,𝑇𝑘=1𝑇0𝑣2𝑘(𝑠)𝑑𝑠1/2𝜎,(2.9) where 𝜌 and 𝜎 are positive constants. We will call the system (2.4) in which 𝑢() and 𝑣() satisfy inequalities (2.6) (resp., (2.7), (2.8), and (2.9)), game 𝐺1 (resp., 𝐺2,𝐺3,𝐺4).

Definition 2.2. A function 𝑧(𝑡)=(𝑧1(𝑡),𝑧2(𝑡),),0𝑡𝑇, is called the solution of the system (2.4) if each coordinate 𝑧𝑘(𝑡)(i)is absolutely continuous and almost everywhere on [0,𝑇] satisfies (2.4),(ii)𝑧()𝐶(0,𝑇;𝑙2).

Definition 2.3. A function []𝑈(𝑡,𝑣),𝑈0,𝑇×𝑙2𝑙2(2.10) is referred to as the strategy of the pursuer with respect to integral constraint if:(1)for any admissible control of the evader 𝑣=𝑣(𝑡),𝑡[0,𝑇], the system (2.4) has a unique solution at 𝑢=𝑢(𝑡,𝑣1(𝑡),𝑣2(𝑡),),(2)𝑈(,𝑣())𝑆1(𝜌).

In a similar way, we define strategy of the pursuer with respect to geometric constraint.

Definition 2.4. One will say that pursuit can be completed in the game 𝐺1 (resp., 𝐺2,𝐺3) from an initial position 𝑧0, if there exists a strategy of the pursuer to ensure that 𝑧(𝑡)=0 for some 𝑡[0,𝑇] and for any admissible control of the evader 𝑣(), where 𝑧(𝑡) is the solution to (2.4).

Definition 2.5. One will say that pursuit can be completed in the game 𝐺4 from an initial position 𝑧0, if for arbitrary 𝜀>0, there exists a strategy of the pursuer to ensure that 𝑧(𝑡)𝜀 for some 𝑡[0,𝑇] and for any admissible control of the evader 𝑣(), where 𝑧(𝑡) is the solution to (2.4).

Definition 2.6. One will say that evasion is possible in the game 𝐺1 (resp., 𝐺2,𝐺3,𝐺4) from the initial position 𝑧00, if there exists a function 𝑣(𝑡)𝑆1(𝜎) (𝑣(𝑡)𝑆2(𝜎),𝑣(𝑡)𝑆2(𝜎),𝑣(𝑡)𝑆1(𝜎)) such that, for arbitrary function 𝑢0(𝑡)𝑆1(𝜌) (𝑢0(𝑡)𝑆2(𝜌),𝑢0(𝑡)𝑆1(𝜌),𝑢0(𝑡)𝑆2(𝜌)), the solution 𝑧(𝑡) of (2.4) does not vanish, that is, 𝑧(𝑡)0 for any 𝑡[0,𝑇].
The problem is to find (1)conditions on the initial state 𝑧0 for which pursuit can be completed for a finite time; (2)conditions for which evasion is possible from any initial position 𝑧00 in the differential game 𝐺𝑖, for 𝑖=1,2,3,4.In problems 1 and 2, different forms of constraints on the controls of the players are to be considered.

3. Differential Game Problem

The kth equation in (2.4) has a unique solution of the form 𝑧𝑘(𝑡)=𝑒𝛼𝑘(𝑡)𝑧𝑘0𝑡0𝑢𝑘(𝑠)𝑒𝛼𝑘(𝑠)𝑑𝑠+𝑡0𝑣𝑘(𝑠)𝑒𝛼𝑘(𝑠),𝑑𝑠(3.1) where 𝛼𝑘(𝑡)=𝑡0𝜆𝑘(𝑠)𝑑𝑠.

It has been proven in [18] that the solution 𝑧(𝑡)=(𝑧1(𝑡),𝑧2(𝑡),) of (2.4), where 𝑧𝑘,𝑘=1,2, defined by (3.1), belongs to the space 𝐶(0,𝑇;𝑙2).

Let 𝑧𝑌=0=𝑧10,𝑧20,𝑘=𝑗𝑧2𝑗0𝜌24,𝐴𝑗,𝑌(𝑡)11𝑧(𝑇)=0=𝑧10,𝑧20,𝑘=1𝑧2𝑘0𝐴𝑘(𝑇)(𝜌𝜎)2,𝑌2z(𝑇)=0=𝑧10,𝑧20,𝑘=1𝑧2𝑘0𝐵2𝑘(𝑇)(𝜌𝜎)2,𝑌3(𝑧𝑇)=0=𝑧10,𝑧20𝑧,0+𝜎2𝜀sup𝑘𝐴𝑘(,𝑇)𝜌𝑇,𝜀>0(3.2) where 𝐴𝑘(𝑇)=𝑇0𝑒2𝛼𝑘(𝑠)𝑑𝑠 and 𝐵𝑘(𝑇)=𝑇0𝑒𝛼𝑘(𝑠)𝑑𝑠.

3.1. Pursuit Differential Game

Theorem 3.1. If 𝜌𝜎 then from the initial position 𝑧0𝑌1(𝑇), pursuit can be completed in the game 𝐺1.

Proof. Let define the pursuer's strategy as 𝑢𝑘𝑧(𝑡)=𝑘0A𝑘1(𝑇)𝑒𝛼𝑘(𝑡)+𝑣𝑘(𝑡),0𝑡𝑇,0,𝑡>𝑇.(3.3) The admissibility of this strategy follows from the relations 𝑘=1𝑇0𝑢2𝑘(𝑠)𝑑𝑠1/2=𝑘=1𝑇0||𝑧𝑘0𝐴𝑘1(𝑇)𝑒𝛼𝑘(𝑠)+𝑣𝑘||(𝑠)2𝑑𝑠1/2𝑘=1𝑇0||𝑧𝑘0𝐴𝑘1(𝑇)𝑒𝛼𝑘(𝑠)||+||𝑣𝑘||(𝑠)2𝑑𝑠1/2𝑘=1𝑇0||𝑧𝑘0𝐴𝑘1(𝑇)𝑒𝛼𝑘(𝑠)||2𝑑𝑠1/2+𝑘=1𝑇0||𝑣𝑘||(𝑠)2𝑑𝑠1/2𝑘=1𝑧2𝑘0𝐴𝑘1(𝑇)1/2+𝜎=𝜌𝜎+𝜎=𝜌,(3.4) here we used the Minkowski inequality and the fact that 𝑧0𝑌1(𝑇).
Suppose that the pursuer uses the strategy (3.3), one can easily see that for any admissible control of the evader 𝑧𝑘(𝑇)=0,𝑘=1,2,, that is, 𝑧𝑘(𝑇)=𝑒𝛼𝑘(𝑇)𝑧𝑘0𝑇0𝑧𝑘0𝐴𝑘1(𝑇)𝑒2𝛼𝑘(𝑠)𝑑𝑠=𝑒𝛼𝑘(𝑇)𝑧𝑘0𝑧𝑘0=0.(3.5) Therefore, pursuit can be completed in the game 𝐺1. This ends the proof of the theorem.

Theorem 3.2. If 𝜌𝜎 then from the initial position 𝑧0𝑌2(𝑇), pursuit can be completed in the game 𝐺2.

Proof. We define the pursuer's strategy as 𝑢𝑘𝑧(𝑡)=𝑘0𝐵𝑘1(𝑇)+𝑣𝑘(𝑡),0𝑡𝑇,0,𝑡>𝑇.(3.6) The inclusion 𝑢()𝑆2(𝜌) follows from the relations 𝑘=1𝑢2𝑘(𝑡)1/2=𝑘=1||𝑧𝑘0𝐵𝑘1(𝑇)+𝑣𝑘||(𝑡)21/2𝑘=1||𝑧𝑘0𝐵𝑘1||+||𝑣(𝑇)𝑘||(𝑡)21/2𝑘=1||𝑧𝑘0𝐵𝑘1||(𝑇)21/2+𝑘=1||𝑣𝑘||(𝑡)21/2=𝜌𝜎+𝜎=𝜌,(3.7) here we used the Minkowski inequality and the fact that 𝑧0𝑌2(𝑇).
Suppose that the pursuer uses the strategy (3.6). One can easily see that 𝑧𝑘(𝑇)=0,𝑘=1,2,, that is, 𝑧𝑘(𝑇)=𝑒𝛼𝑘(𝑇)𝑧𝑘0𝑇0𝑧𝑘0𝐵𝑘1(𝑇)𝑒𝛼𝑘(𝑠)𝑑𝑠=𝑒𝛼𝑘(𝑇)𝑧𝑘0𝑧𝑘0=0.(3.8) Therefore, pursuit can be completed in the game 𝐺2. This completes the proof of the theorem.

Theorem 3.3. If 𝜌𝜎 and 𝑧0𝑌1(𝑇) at some 𝑇(0,𝑇], then pursuit can be completed in the game 𝐺3.

Proof. Suppose, as contained in the hypothesis of the theorem, that 𝑧0𝑌1(𝑇), 𝑇(0,1] and let 𝑣0(𝑡) be an arbitrary admissible control of the evader.
Let the pursuer use the strategy 𝑢(𝑡)=(𝑢1(𝑡),𝑢2(𝑡),) defined by 𝑢𝑘𝑧(𝑡)=𝑘0𝐴𝑘1(𝑇)𝑒𝛼𝑘(𝑡)+𝑣0𝑘(𝑡),0𝑡𝑇,0,𝑡>𝑇.(3.9) Then, using (3.1), we have 𝑧(𝑇)=𝑒𝛼𝑘(𝑇)𝑧𝑘0𝑇0𝑧𝑘0𝐴𝑘1(𝑇)𝑒2𝛼𝑘(𝑡)𝑑𝑠=𝑒𝛼𝑘(𝑇)𝑧𝑘0𝑧𝑘0=0.(3.10) We now show the admissibility of the strategy used by the pursuer. From the inclusion 𝑣0(𝑡)𝑆2(𝜎) we can deduce that 𝑘=1𝑇0𝑣20𝑘(𝑠)𝑑𝑠1/2𝜎𝑇,(3.11)𝑘=1𝑇0𝑢2𝑘(𝑠)𝑑𝑠1/2=𝑘=1𝑇0||𝑧𝑘0𝐴𝑘1(𝑇)𝑒𝛼𝑘(𝑠)+𝑣0𝑘||(𝑠)2𝑑𝑠1/2𝑘=1𝑇0||𝑧𝑘0𝐴𝑘1(𝑇)𝑒𝛼𝑘(𝑠)||+||𝑣𝑘||(𝑠)2𝑑𝑠1/2𝑘=1𝑇0||𝑧𝑘0𝐴𝑘1(𝑇)𝑒𝛼𝑘(𝑠)||2𝑑𝑠1/2+𝑘=1𝑇0||𝑣0𝑘||(𝑠)2𝑑𝑠1/2𝑘=1𝑧2𝑘0𝐴𝑘1(𝑇)1/2+𝜎𝑇=𝜌𝜎+𝜎𝑇𝜌,(3.12) recall that 𝑇(0,1] and (3.11). This completes the proof.

Theorem 3.4. For arbitrary 𝜌>0,𝜎>0 and initial position 𝑧0𝑌3(𝑇), pursuit can be completed in the game 𝐺4.

Proof. Let 𝑣0 be an arbitrary admissible control function of the evader. When the pursuer uses the admissible control function 𝑢𝑘(𝑡)=𝑧𝑘0𝑇11𝑒𝛼𝑘(𝑡),𝑘=1,2,,0𝑡𝑇1,(3.13) for time 𝑇1=𝑧0𝜌1, the solution (3.1) of (2.4) becomes 𝑧𝑘𝑇1=𝑒𝛼𝑘(𝑇1)𝑇10𝑣0𝑘(𝑠)𝑒𝛼𝑘(𝑠)𝑑𝑠.(3.14)
Then for arbitrary positive number 𝜀, it is obvious that either(1)𝑧(𝑇1)𝜀, or(2)𝑧(𝑇1)>𝜀.If (1) is true then the proof is complete. Obviously 𝑇1𝑇.
Suppose that (1) is not true then (2) must hold. We now assume that 𝑧0=𝑧(𝑇1) and repeat previous argument by setting 𝑢𝑘(𝑡)=𝑧𝑘0𝑇1𝑇21𝑒𝛼𝑘(𝑡),𝑘=1,2,,0𝑡𝑇2,(3.15) with time 𝑇2=𝑧(𝑇1)𝜌1 (we will later prove that the sum of 𝑇𝑖 is less than or equal to 𝑇). For this step the solution (3.1) becomes 𝑧𝑘𝑇1+𝑇2=𝑒𝛼𝑘(𝑇2)𝑇20𝑣0𝑘𝑇1𝑒+𝑠𝛼𝑘(𝑠)𝑑𝑠.(3.16)
Yet again, we have either of the following cases holding:(1)𝑧(𝑇1+𝑇2)𝜀, or(2)𝑧(𝑇1+𝑇2)>𝜀.If (1) holds then the game is completed in the time 𝑇1+𝑇2, else we assume 𝑧0=𝑧0(𝑇1+𝑇2) and repeat the process again and so on.
We now proof a claim that the game will be completed before 𝑛th finite step, where 𝜎𝑛=2sup𝑘𝐴𝑘(𝑇)𝜀2.(3.17) Note that the existence of the supreme of the sequence 𝐴1(𝑇),𝐴2(𝑇),, follows from the fact that 𝜆1(𝑡),𝜆2(𝑡), is a bounded sequence of continuous functions and 𝑡[0,𝑇].
Suppose that it is possible that the game can continue for 𝑛th step. In this case, we must have 𝑛𝑖=1𝜎2𝑖𝜎2.(3.18) But in the first instance, we have ||𝑧𝑘𝑇1||2𝑒2𝛼𝑘(𝑇1)𝑇10𝑣20𝑘(𝑠)𝑑𝑠𝑇10𝑒2𝛼𝑘(𝑠)𝑑𝑠sup𝑘𝐴𝑘(𝑇)𝑇10𝑣20𝑘(𝑠)𝑑𝑠,(3.19) here we used (3.14) and Cauchy-Schwarz inequality.
Therefore, 𝑧(𝑇1)2sup𝑘𝐴𝑘(𝑇)𝑘=1𝑇10𝑣20𝑘(𝑠)𝑑𝑠=sup𝑘𝐴𝑘(𝑇)𝜎21,(3.20) and by using the assumption that 𝑧(𝑇1)>𝜀, we have 𝜎21>𝜀2sup𝑘𝐴𝑘.(𝑇)(3.21) Since the right hand side of this inequality is independent of 𝑛, we can conclude that 𝜎2𝑛>𝜀2sup𝑘𝐴𝑘.(𝑇)(3.22) Using this inequality and definition of 𝑛, we have 𝑛𝑖=1𝜎2𝑖>𝑛𝜀2sup𝑘𝐴𝑘(𝑇)>𝜎2,(3.23) contradicting (3.18). Hence, pursuit must be completed for the initial position 𝑧0𝑌3(𝑇) before the 𝑛th step. Furthermore, the pursuit time is given by 𝑇(𝑧0)=𝑇1+𝑇2++𝑇𝑛1, and the inclusion 𝑇(𝑧0)[0,𝑇] is satisfied. Indeed (see (3.20), definition of 𝑛 and that 𝑧0𝑌3(𝑇)), 𝑇𝑧0=𝑧0𝜌+z𝑇1𝜌𝑧𝑇++𝑛2𝜌1𝜌𝑧0+sup𝑘𝐴𝑘(𝑇)𝑛2𝑖=1𝜎𝑖1𝜌𝑧0+𝜎(𝑛2)sup𝑘𝐴𝑘1(𝑇)𝜌𝑧0+𝜎2𝜀sup𝑘𝐴𝑘(𝑇)𝑇.(3.24)
This proves the theorem.

3.2. Evasion Differential Game

Theorem 3.5. If 𝜎𝜌0 then evasion is possible in the game 𝐺1 from the initial position 𝑧00.

Proof. Suppose that 𝜎𝜌0,(3.25) and let 𝑢0(𝑡) be an arbitrary control of the pursuer subjected to integral constraint. We construct the control function of the evader as follows: 𝑣𝑘𝐴(𝑡)=𝑗1/2(𝑇)𝜌𝑒𝛼𝑗(𝑡),𝑘=𝑗,0,𝑘𝑗.(3.26) This control function belongs to 𝑆1(𝜎). Indeed, 𝑘=1𝑇0𝑣2𝑘(𝑠)𝑑𝑠=𝐴𝑗1(𝑇)𝜌2𝐴𝑗(𝑇)𝜎2(3.27) we have used (3.26) and (3.25).
Our goal now is to show that 𝑧𝑗(𝑡)0 for any 𝑡[0,𝑇] as defined by (3.1). Substituting (3.26) into (3.1) and using the Cauchy-Schwartz inequality, we have 𝑧𝑗(𝑡)𝑒𝛼𝑗(𝑡)𝑧𝑗0+𝜌𝐴𝑗(𝑡)𝜌𝐴𝑗(𝑡)=𝑧𝑗0𝑒𝛼𝑗(𝑡)>0,(3.28) for any 𝑡[0,𝑇]. It follows that 𝑧(𝑡)0 on the interval [0,𝑇]. Hence, evasion is possible in the game 𝐺1 from the given initial position 𝑧00. The proof of the theorem is complete.

Theorem 3.6. Suppose that 𝜎𝜌 or there exists a number 𝑘=𝑗 such that 𝑧0𝑗>0 and 𝜎𝜌𝐴𝑗(𝑇)0. Then from the initial position 𝑧00, evasion is possible in the game 𝐺2.

Proof. Suppose that 𝜎𝜌 and that 𝑧00. The later condition means that 𝑧𝑘00 for some 𝑘=𝑗. We construct the control function of the evader as follows: 𝑣𝑘(𝑡)=𝜌,𝑘=𝑗,0,𝑘𝑗.(3.29) It is obvious that this control belongs to the set 𝑆2(𝜎).
To be definite, let 𝑧𝑗0>0. Using (3.29) and (3.1), we have 𝑧𝑗(𝑡)>𝑒𝛼𝑗(𝑡)𝑧𝑗0+𝜌𝑡0𝑒𝛼𝑗(𝑠)𝑑𝑠𝜌𝑡0𝑒𝛼𝑗(𝑠)𝑑𝑠=𝑒𝛼𝑗(𝑡)𝑧𝑗0>0.(3.30) This means that evasion is possible from the initial position 𝑧00 in the game 𝐺2.
We now prove the theorem with the alternative condition. Suppose that there exists a number 𝑘=𝑗 such that 𝑧0𝑗>0 and 𝜎𝜌𝐴𝑗(𝑇)0. Let the control of the evader be as follows: 𝑣𝑘(𝑡)=𝑇𝐴𝑗(𝑇)𝑡+𝑒+𝑇𝜌𝑒𝛼𝑗(𝑡),𝑘=𝑗,0,𝑘𝑗.(3.31) We show that this control satisfies the geometric constraint: 𝑘=1𝑣2𝑘(𝑡)=𝑇𝐴𝑗(𝑇)(𝑡+𝑒+𝑇)2𝜌2𝑒2𝛼𝑗(𝑡)𝜌2𝐴𝑗(𝑇)𝜎2.(3.32)
When the evader uses the control (3.31), the non-vanishing of 𝑧𝑗(𝑡) in the interval [0,𝑇] for any admissible control of the pursuer 𝑢𝑖0, can be seen from the following (see (3.1)) 𝑧𝑗(𝑡)𝑒𝛼𝑗(𝑡)𝑧𝑗0+𝜌𝑇𝐴𝑗(𝑇)ln(𝑡+𝑒+𝑇)𝜌𝑇𝐴𝑗(𝑇)>0,(3.33) we use the fact that ln(𝑡+𝑒+𝑇)>1 for any 𝑡[0,𝑇].
Therefore, 𝑧(𝑡)0,𝑡[0,𝑇]. This completes the proof of the theorem.

Theorem 3.7. If 𝜎𝜌𝑇0 then evasion is possible from the initial position 𝑧00 in the game 𝐺4.

Proof. Suppose that 𝑧00 and that 𝜎𝜌𝑇0. We construct the control function of the evader as follows: 𝑣𝑘𝜌(𝑡)=𝑇𝐴𝑗𝑒(𝑇)𝛼𝑗(𝑡),𝑘=𝑗,0,𝑘𝑗.(3.34) We now show that this control satisfies the integral constraint 𝑘=1𝑇0𝑣2𝑘(𝑠)𝑑𝑠=𝜌2𝑇𝐴𝑗(𝑇)𝑇0𝑒2𝛼𝑗(𝑠)𝑑𝑠𝜌2𝑇𝜎2.(3.35)
When the evader uses the control (3.34), our task is to show that 𝑧𝑗(𝑡) does not vanish in the interval [0,𝑇] for any admissible control of the pursuer 𝑢𝑖0.
For definiteness let 𝑧𝑗0>0. Substituting (3.34) into (3.1), we have 𝑧𝑗(𝑡)𝑒𝛼𝑗(𝑡)𝑧𝑗0+𝜌𝑇𝐴𝑗𝐴(𝑇)𝑗(𝑇)𝜌𝑇𝐴𝑗(𝑇)=𝑒𝛼𝑗(𝑡)𝑧𝑗0>0.(3.36) Therefore, 𝑧(𝑡)0,𝑡[0,𝑇]. This means that evasion is possible from initial position 𝑧00 in game 𝐺4. This ends the proof of the theorem.

Theorem 3.8. If 𝑧0𝑌 and 𝜎2𝜌𝑒𝛼𝑗(𝑇) for some 𝑘=𝑗, then from the initial position 𝑧00 evasion is possible in the game 𝐺3.

Proof. Suppose that 𝑧0𝑌 and that there exists 𝑘=𝑗 such that 𝜎2𝜌𝑒𝛼𝑗(𝑇).(3.37) We construct the control function of the evader as follows: 𝑣𝑘(𝑡)=2𝑧𝑗0𝑒+𝜌𝛼𝑗(𝑡),𝑘=𝑗,0,𝑘𝑗.(3.38) The inclusion 𝑣()𝑆2(𝜎) follows from the following 𝑘=1𝑣2𝑘(𝑡)=2𝑧𝑗0𝑒+𝜌𝛼𝑗(𝑡)28𝑧2𝑗0𝑒2𝛼𝑗(𝑡)+2𝜌2𝑒2𝛼𝑗(𝑡)2𝜌2𝑒2𝛼𝑗(𝑡)+2𝜌2𝑒2𝛼𝑗(𝑡)4𝜌2𝑒2𝛼𝑗(𝑇)𝜎2.(3.39) we used (3.34); (3.37) and the inequality (𝑎+𝑏)22𝑎2+2𝑏2.
Let the evader use the control (3.34) and for definiteness let 𝑧𝑗0>0. Using (3.1) and the Cauchy-Schwartz inequality, we have 𝑧𝑗(𝑡)𝑒𝛼𝑗(𝑡)𝑧𝑗0𝜌𝐴𝑗(𝑡)+2𝑧𝑗0𝐴𝑗(𝑡)+𝜌𝐴𝑗(𝑡)=𝑒𝛼𝑗(𝑡)𝑧𝑗0+2𝑧𝑗0A𝑗(𝑡)>0.(3.40) Therefore, we have 𝑧𝑗(𝑡)>0,0𝑡𝑇, that is, evasion is possible in the game 𝐺3. This ends the proof of the theorem.

4. Conclusion

This paper is closely related to [10, 11]. However, the game model considered in this paper is a better generalization to the one in the last cited papers. The constant coefficients of the game model considered in the cited papers are specific to function coefficients considered in this papers. Sufficient conditions for which pursuit can be completed and for which evasion is possible with various form of constraints on the control of the players have been established.

For future works, optimal pursuit and multiplayers game problems described by the model considered in this paper can be investigated. As there are four different possible combinations of geometric and integral constraints on the control functions of the two players of the game, there would be four different problems to be studied.

Acknowledgment

The authors wish to express the deepest appreciation to the reviewers for their valuable comments and observations. This research was partially supported by the Research Grant (RUGS) of the Universiti Putra Malaysia, no. 05-04-10-1005RU.

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