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

Authors: Abbas Badakaya Ja'afaru; Gafurjan Ibragimov
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We study pursuit and evasion differential game problems described by infinite number of first-order differential equations with function coefficients in Hilbert spacel2. 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.


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## Body

## 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 inRn and partial differential equations. In particular, pursuit and evasion differential game problems involving distributed parameter systems are of increasing interest (see, e.g., [7–14]).Satimov and Tukhtasinov [10, 11] studied pursuit and evasion problems described by the parabolic equation
(1.1)zt-Az=-u+v,z|t=0=z0(x),z|ST=0,
where z=z(t,x) is unknown function; x=(x1,x2,…,xn)∈Ω⊂Rn,n≥1 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
(1.2)z(t,x)=∑k=1∞‍zk(t)ψk(x),
where the functions zk(t), 0≤t≤T, k=1,2,…, constitute the solution of the Cauchy problem for the following infinite system of differential equations and initial conditions:
(1.3)z˙k+λkzk=-uk(t)+vk(t),zk(0)=z0k,k=1,2,…,
and λk, k=1,2,…, satisfying the condition that 0<λ1≤λ2≤⋯→∞, 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(2.1)l2={α=(α1,α2,…):∑k=1∞‍αk2<∞},
with inner product and norm
(2.2)〈α,β〉=∑k=1∞‍αkβk,α,β∈l2,‖α‖=(∑k=1∞‍αk2)1/2,∥w(·)∥L2(0,T,l2)=(∑k=1∞‍∫0T‍wk2(s)ds)1/2.Let(2.3)L2(0,T,l2)={w(t)=(w1(t),w2(t),…):‖w(⋅)‖L2(0,T,l2)<∞,wk(⋅)∈L2(0,T)},
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(2.4)z˙k(t)+λk(t)zk(t)=-uk(t)+vk(t),zk(0)=zk0,k=1,2,…,
where zk,uk,vk∈R1,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 functionw(·),w:[0,T]→l2, with measurable coordinates wk(t),0≤t≤T,k=1,2,…, subject to
(2.5)∑k=1∞‍∫0T‍wk2(s)ds≤ρ2(∑k=1∞‍wk2(t)≤ρ2,t∈[0,T]),
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 byS1(ρ) and with respect to geometric constraint by S2(ρ).The controlu(·)=(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
(2.6)(∑k=1∞‍∫0T‍uk2(s)ds)1/2≤ρ,(∑k=1∞‍∫0T‍vk2(s)ds)1/2≤σ,(2.7)(∑k=1∞‍uk2(t))1/2≤ρ,t∈[0,T],(∑k=1∞‍vk2(t))1/2≤σ,t∈[0,T],(2.8)(∑k=1∞‍∫0T‍uk2(s)ds)1/2≤ρ,(∑k=1∞‍vk2(t))1/2≤σ,t∈[0,T],(2.9)(∑k=1∞‍uk2(t))1/2≤ρ,t∈[0,T],(∑k=1∞‍∫0T‍vk2(s)ds)1/2≤σ,
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 functionz(t)=(z1(t),z2(t),…),0≤t≤T, 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(2.10)U(t,v),U:[0,T]×l2→l2
is referred to as the strategy of the pursuer with respect to integral constraint if:(1)
for any admissible control of the evaderv=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 gameG1 (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 gameG4 from an initial position z0, if for arbitrary ɛ>0, there exists a strategy of the pursuer to ensure that ∥z(t)∥≤ɛ 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 gameG1 (resp., G2,G3,G4) from the initial position z0≠0, 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 statez0 for which pursuit can be completed for a finite time;(2)
conditions for which evasion is possible from any initial positionz0≠0  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

Thekth equation in (2.4) has a unique solution of the form
(3.1)zk(t)=e-αk(t)(zk0-∫0t‍uk(s)eαk(s)ds+∫0t‍vk(s)eαk(s)ds),
where αk(t)=∫0t‍λk(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(3.2)Y={z0=(z10,z20,…)∣∃k=j:zj02≤ρ24,Aj(t)≥1},Y1(T)={z0=(z10,z20,…)∣∑k=1∞‍zk02Ak(T)≤(ρ-σ)2},Y2(T)={z0=(z10,z20,…)∣∑k=1∞‍zk02Bk2(T)≤(ρ-σ)2},Y3(T)={z0=(z10,z20,…)∣∥z0∥+σ2ɛsupkAk(T)≤ρT,ɛ>0},
where Ak(T)=∫0T‍e2αk(s)ds and Bk(T)=∫0T‍eαk(s)ds.


### 3.1. Pursuit Differential Game

Theorem 3.1.
Ifρ≥σ then from the initial position z0∈Y1(T), pursuit can be completed in the game G1.Proof.
Let define the pursuer's strategy as(3.3)uk(t)={zk0Ak-1(T)eαk(t)+vk(t),0≤t≤T,0,t>T.
The admissibility of this strategy follows from the relations
(3.4)(∑k=1∞‍∫0T‍uk2(s)ds)1/2=(∑k=1∞‍∫0T‍|zk0Ak-1(T)eαk(s)+vk(s)|2ds)1/2≤(∑k=1∞‍∫0T‍(|zk0Ak-1(T)eαk(s)|+|vk(s)|)2ds)1/2≤(∑k=1∞‍∫0T‍|zk0Ak-1(T)eαk(s)|2ds)1/2+(∑k=1∞‍∫0T‍|vk(s)|2ds)1/2≤(∑k=1∞‍zk02Ak-1(T))1/2+σ=ρ-σ+σ=ρ,
here we used the Minkowski inequality and the fact that z0∈Y1(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,
(3.5)zk(T)=e-αk(T)(zk0-∫0T‍zk0Ak-1(T)e2αk(s)ds)=e-αk(T)(zk0-zk0)=0.
Therefore, pursuit can be completed in the game G1. This ends the proof of the theorem.Theorem 3.2.
Ifρ≥σ then from the initial position z0∈Y2(T), pursuit can be completed in the game G2.Proof.
We define the pursuer's strategy as(3.6)uk(t)={zk0Bk-1(T)+vk(t),0≤t≤T,0,t>T.
The inclusion u(·)∈S2(ρ) follows from the relations
(3.7)(∑k=1∞‍uk2(t))1/2=(∑k=1∞‍|zk0Bk-1(T)+vk(t)|2)1/2≤(∑k=1∞‍(|zk0Bk-1(T)|+|vk(t)|)2)1/2≤(∑k=1∞‍|zk0Bk-1(T)|2)1/2+(∑k=1∞‍|vk(t)|2)1/2=ρ-σ+σ=ρ,
here we used the Minkowski inequality and the fact that z0∈Y2(T).
Suppose that the pursuer uses the strategy (3.6). One can easily see that zk(T)=0,k=1,2,…, that is,
(3.8)zk(T)=e-αk(T)(zk0-∫0T‍zk0Bk-1(T)eαk(s)ds)=e-αk(T)(zk0-zk0)=0.
Therefore, pursuit can be completed in the game G2. This completes the proof of the theorem.Theorem 3.3.
Ifρ≥σ and z0∈Y1(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, thatz0∈Y1(T), T∈(0,1] and let v0(t) be an arbitrary admissible control of the evader.
Let the pursuer use the strategyu(t)=(u1(t),u2(t),…) defined by
(3.9)uk(t)={zk0Ak-1(T)eαk(t)+v0k(t),0≤t≤T,0,t>T.
Then, using (3.1), we have
(3.10)z(T)=e-αk(T)(zk0-∫0T‍zk0Ak-1(T)e2αk(t)ds)=e-αk(T)(zk0-zk0)=0.
We now show the admissibility of the strategy used by the pursuer. From the inclusion v0(t)∈S2(σ) we can deduce that
(3.11)(∑k=1∞‍∫0T‍v0k2(s)ds)1/2≤σT,(3.12)(∑k=1∞‍∫0T‍uk2(s)ds)1/2=(∑k=1∞‍∫0T‍|zk0Ak-1(T)eαk(s)+v0k(s)|2ds)1/2≤(∑k=1∞‍∫0T‍(|zk0Ak-1(T)eαk(s)|+|vk(s)|)2ds)1/2≤(∑k=1∞‍∫0T‍|zk0Ak-1(T)eαk(s)|2ds)1/2+(∑k=1∞‍∫0T‍|v0k(s)|2ds)1/2≤(∑k=1∞‍zk02Ak-1(T))1/2+σT=ρ-σ+σT≤ρ,
recall that T∈(0,1] and (3.11). This completes the proof.Theorem 3.4.
For arbitraryρ>0,σ>0 and initial position z0∈Y3(T), pursuit can be completed in the game G4.Proof.
Letv0 be an arbitrary admissible control function of the evader. When the pursuer uses the admissible control function
(3.13)uk(t)=zk0T1-1e-αk(t),k=1,2,…,0≤t≤T1,
for time T1=∥z0∥ρ-1, the solution (3.1) of (2.4) becomes
(3.14)zk(T1)=e-αk(T1)∫0T1‍v0k(s)eαk(s)ds.
Then for arbitrary positive numberɛ, it is obvious that either(1)
∥z(T1)∥≤ɛ, or(2)
∥z(T1)∥>ɛ.If (1) is true then the proof is complete. Obviously T1≤T.
Suppose that (1) is not true then (2) must hold. We now assume thatz0=z(T1) and repeat previous argument by setting
(3.15)uk(t)=zk0(T1)T2-1e-αk(t),k=1,2,…,0≤t≤T2,
with time T2=∥z(T1)∥ρ-1 (we will later prove that the sum of Ti is less than or equal to T). For this step the solution (3.1) becomes
(3.16)zk(T1+T2)=e-αk(T2)∫0T2‍v0k(T1+s)eαk(s)ds.
Yet again, we have either of the following cases holding:(1)
∥z(T1+T2)∥≤ɛ, or(2)
∥z(T1+T2)∥>ɛ.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 beforenth finite step, where
(3.17)n=⌈σ2sup⁡kAk(T)ɛ2⌉.
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 fornth step. In this case, we must have
(3.18)∑i=1n‍σi2≤σ2.
But in the first instance, we have
(3.19)|zk(T1)|2≤e-2αk(T1)∫0T1‍v0k2(s)ds∫0T1‍e2αk(s)ds≤sup⁡kAk(T)∫0T1‍v0k2(s)ds,
here we used (3.14) and Cauchy-Schwarz inequality.
Therefore,(3.20)‖z(T1)‖2≤sup⁡kAk(T)∑k=1∞‍∫0T1‍v0k2(s)ds=sup⁡kAk(T)σ12,
and by using the assumption that ∥z(T1)∥>ɛ, we have
(3.21)σ12>ɛ2sup⁡kAk(T).
Since the right hand side of this inequality is independent of  n, we can conclude that
(3.22)σn2>ɛ2sup⁡kAk(T).
Using this inequality and definition of  n, we have
(3.23)∑i=1n‍σi2>nɛ2sup⁡kAk(T)>σ2,
contradicting (3.18). Hence, pursuit must be completed for the initial position z0∈Y3(T) before the nth step. Furthermore, the pursuit time is given by T(z0)=T1+T2+⋯+Tn-1, and the inclusion T(z0)∈[0,T] is satisfied. Indeed (see (3.20), definition of n and that z0∈Y3(T)),
(3.24)T(z0)=‖z0‖ρ+‖z(T1)‖ρ+⋯+‖z(Tn-2)‖ρ≤1ρ(‖z0‖+sup⁡kAk(T)∑i=1n-2‍σi)≤1ρ(‖z0‖+σ(n-2)sup⁡kAk(T))≤1ρ(‖z0‖+σ2ɛsup⁡kAk(T))≤T.
This proves the theorem.


### 3.2. Evasion Differential Game

Theorem 3.5.
Ifσ-ρ≥0 then evasion is possible in the game G1 from the initial position z0≠0.Proof.
Suppose that(3.25)σ-ρ≥0,
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:
(3.26)vk(t)={Aj-1/2(T)ρeαj(t),k=j,0,k≠j.
This control function belongs to S1(σ). Indeed,
(3.27)∑k=1∞‍∫0T‍vk2(s)ds=Aj-1(T)ρ2Aj(T)≤σ2
we have used (3.26) and (3.25).
Our goal now is to show thatzj(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
(3.28)zj(t)≥e-αj(t)(zj0+ρAj(t)-ρAj(t))=zj0e-αj(t)>0,
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 z0≠0. The proof of the theorem is complete.Theorem 3.6.
Suppose thatσ≥ρ or there exists a number k=j such that z0j>0 and σ-ρAj(T)≥0. Then from the initial position z0≠0, evasion is possible in the game G2.Proof.
Suppose thatσ≥ρ and that z0≠0. The later condition means that zk0≠0 for some k=j. We construct the control function of the evader as follows:
(3.29)vk(t)={ρ,k=j,0,k≠j.
It is obvious that this control belongs to the set S2(σ).
To be definite, letzj0>0. Using (3.29) and (3.1), we have
(3.30)zj(t)>e-αj(t)(zj0+ρ∫0t‍eαj(s)ds-ρ∫0t‍eαj(s)ds)=e-αj(t)zj0>0.
This means that evasion is possible from the initial position z0≠0 in the game G2.
We now prove the theorem with the alternative condition. Suppose that there exists a numberk=j such that z0j>0 and σ-ρAj(T)≥0. Let the control of the evader be as follows:
(3.31)vk(t)={TAj(T)t+e+Tρe-αj(t),k=j,0,k≠j.
We show that this control satisfies the geometric constraint:
(3.32)∑k=1∞‍vk2(t)=TAj(T)(t+e+T)2ρ2e-2αj(t)≤ρ2Aj(T)≤σ2.
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))
(3.33)zj(t)≥e-αj(t)(zj0+ρTAj(T)ln⁡⁡(t+e+T)-ρTAj(T))>0,
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.
Ifσ-ρT≥0 then evasion is possible from the initial position z0≠0 in the game G4.Proof.
Suppose thatz0≠0 and that σ-ρT≥0. We construct the control function of the evader as follows:
(3.34)vk(t)={ρTAj(T)eαj(t),k=j,0,k≠j.
We now show that this control satisfies the integral constraint
(3.35)∑k=1∞‍∫0T‍vk2(s)ds=ρ2TAj(T)∫0T‍e2αj(s)ds≤ρ2T≤σ2.
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 letzj0>0. Substituting (3.34) into (3.1), we have
(3.36)zj(t)≥e-αj(t)(zj0+ρTAj(T)Aj(T)-ρTAj(T))=e-αj(t)zj0>0.
Therefore, z(t)≠0,t∈[0,T]. This means that evasion is possible from initial position z0≠0 in game G4. This ends the proof of the theorem.Theorem 3.8.
Ifz0∈Y and σ≥2ρeαj(T) for some k=j, then from the initial position z0≠0 evasion is possible in the game G3.Proof.
Suppose thatz0∈Y and that there exists k=j such that
(3.37)σ≥2ρeαj(T).
We construct the control function of the evader as follows:
(3.38)vk(t)={(2zj0+ρ)eαj(t),k=j,0,k≠j.
The inclusion v(·)∈S2(σ) follows from the following
(3.39)∑k=1∞‍vk2(t)=[(2zj0+ρ)eαj(t)]2≤8zj02e2αj(t)+2ρ2e2αj(t)≤2ρ2e2αj(t)+2ρ2e2αj(t)≤4ρ2e2αj(T)≤σ2.
we used (3.34); (3.37) and the inequality (a+b)2≤2a2+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
(3.40)zj(t)≥e-αj(t)(zj0-ρAj(t)+2zj0Aj(t)+ρAj(t))=e-αj(t)(zj0+2zj0Aj(t))>0.
Therefore, we have zj(t)>0,0≤t≤T, that is, evasion is possible in the game G3. This ends the proof of the theorem.


## 3.1. Pursuit Differential Game

Theorem 3.1.
Ifρ≥σ then from the initial position z0∈Y1(T), pursuit can be completed in the game G1.Proof.
Let define the pursuer's strategy as(3.3)uk(t)={zk0Ak-1(T)eαk(t)+vk(t),0≤t≤T,0,t>T.
The admissibility of this strategy follows from the relations
(3.4)(∑k=1∞‍∫0T‍uk2(s)ds)1/2=(∑k=1∞‍∫0T‍|zk0Ak-1(T)eαk(s)+vk(s)|2ds)1/2≤(∑k=1∞‍∫0T‍(|zk0Ak-1(T)eαk(s)|+|vk(s)|)2ds)1/2≤(∑k=1∞‍∫0T‍|zk0Ak-1(T)eαk(s)|2ds)1/2+(∑k=1∞‍∫0T‍|vk(s)|2ds)1/2≤(∑k=1∞‍zk02Ak-1(T))1/2+σ=ρ-σ+σ=ρ,
here we used the Minkowski inequality and the fact that z0∈Y1(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,
(3.5)zk(T)=e-αk(T)(zk0-∫0T‍zk0Ak-1(T)e2αk(s)ds)=e-αk(T)(zk0-zk0)=0.
Therefore, pursuit can be completed in the game G1. This ends the proof of the theorem.Theorem 3.2.
Ifρ≥σ then from the initial position z0∈Y2(T), pursuit can be completed in the game G2.Proof.
We define the pursuer's strategy as(3.6)uk(t)={zk0Bk-1(T)+vk(t),0≤t≤T,0,t>T.
The inclusion u(·)∈S2(ρ) follows from the relations
(3.7)(∑k=1∞‍uk2(t))1/2=(∑k=1∞‍|zk0Bk-1(T)+vk(t)|2)1/2≤(∑k=1∞‍(|zk0Bk-1(T)|+|vk(t)|)2)1/2≤(∑k=1∞‍|zk0Bk-1(T)|2)1/2+(∑k=1∞‍|vk(t)|2)1/2=ρ-σ+σ=ρ,
here we used the Minkowski inequality and the fact that z0∈Y2(T).
Suppose that the pursuer uses the strategy (3.6). One can easily see that zk(T)=0,k=1,2,…, that is,
(3.8)zk(T)=e-αk(T)(zk0-∫0T‍zk0Bk-1(T)eαk(s)ds)=e-αk(T)(zk0-zk0)=0.
Therefore, pursuit can be completed in the game G2. This completes the proof of the theorem.Theorem 3.3.
Ifρ≥σ and z0∈Y1(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, thatz0∈Y1(T), T∈(0,1] and let v0(t) be an arbitrary admissible control of the evader.
Let the pursuer use the strategyu(t)=(u1(t),u2(t),…) defined by
(3.9)uk(t)={zk0Ak-1(T)eαk(t)+v0k(t),0≤t≤T,0,t>T.
Then, using (3.1), we have
(3.10)z(T)=e-αk(T)(zk0-∫0T‍zk0Ak-1(T)e2αk(t)ds)=e-αk(T)(zk0-zk0)=0.
We now show the admissibility of the strategy used by the pursuer. From the inclusion v0(t)∈S2(σ) we can deduce that
(3.11)(∑k=1∞‍∫0T‍v0k2(s)ds)1/2≤σT,(3.12)(∑k=1∞‍∫0T‍uk2(s)ds)1/2=(∑k=1∞‍∫0T‍|zk0Ak-1(T)eαk(s)+v0k(s)|2ds)1/2≤(∑k=1∞‍∫0T‍(|zk0Ak-1(T)eαk(s)|+|vk(s)|)2ds)1/2≤(∑k=1∞‍∫0T‍|zk0Ak-1(T)eαk(s)|2ds)1/2+(∑k=1∞‍∫0T‍|v0k(s)|2ds)1/2≤(∑k=1∞‍zk02Ak-1(T))1/2+σT=ρ-σ+σT≤ρ,
recall that T∈(0,1] and (3.11). This completes the proof.Theorem 3.4.
For arbitraryρ>0,σ>0 and initial position z0∈Y3(T), pursuit can be completed in the game G4.Proof.
Letv0 be an arbitrary admissible control function of the evader. When the pursuer uses the admissible control function
(3.13)uk(t)=zk0T1-1e-αk(t),k=1,2,…,0≤t≤T1,
for time T1=∥z0∥ρ-1, the solution (3.1) of (2.4) becomes
(3.14)zk(T1)=e-αk(T1)∫0T1‍v0k(s)eαk(s)ds.
Then for arbitrary positive numberɛ, it is obvious that either(1)
∥z(T1)∥≤ɛ, or(2)
∥z(T1)∥>ɛ.If (1) is true then the proof is complete. Obviously T1≤T.
Suppose that (1) is not true then (2) must hold. We now assume thatz0=z(T1) and repeat previous argument by setting
(3.15)uk(t)=zk0(T1)T2-1e-αk(t),k=1,2,…,0≤t≤T2,
with time T2=∥z(T1)∥ρ-1 (we will later prove that the sum of Ti is less than or equal to T). For this step the solution (3.1) becomes
(3.16)zk(T1+T2)=e-αk(T2)∫0T2‍v0k(T1+s)eαk(s)ds.
Yet again, we have either of the following cases holding:(1)
∥z(T1+T2)∥≤ɛ, or(2)
∥z(T1+T2)∥>ɛ.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 beforenth finite step, where
(3.17)n=⌈σ2sup⁡kAk(T)ɛ2⌉.
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 fornth step. In this case, we must have
(3.18)∑i=1n‍σi2≤σ2.
But in the first instance, we have
(3.19)|zk(T1)|2≤e-2αk(T1)∫0T1‍v0k2(s)ds∫0T1‍e2αk(s)ds≤sup⁡kAk(T)∫0T1‍v0k2(s)ds,
here we used (3.14) and Cauchy-Schwarz inequality.
Therefore,(3.20)‖z(T1)‖2≤sup⁡kAk(T)∑k=1∞‍∫0T1‍v0k2(s)ds=sup⁡kAk(T)σ12,
and by using the assumption that ∥z(T1)∥>ɛ, we have
(3.21)σ12>ɛ2sup⁡kAk(T).
Since the right hand side of this inequality is independent of  n, we can conclude that
(3.22)σn2>ɛ2sup⁡kAk(T).
Using this inequality and definition of  n, we have
(3.23)∑i=1n‍σi2>nɛ2sup⁡kAk(T)>σ2,
contradicting (3.18). Hence, pursuit must be completed for the initial position z0∈Y3(T) before the nth step. Furthermore, the pursuit time is given by T(z0)=T1+T2+⋯+Tn-1, and the inclusion T(z0)∈[0,T] is satisfied. Indeed (see (3.20), definition of n and that z0∈Y3(T)),
(3.24)T(z0)=‖z0‖ρ+‖z(T1)‖ρ+⋯+‖z(Tn-2)‖ρ≤1ρ(‖z0‖+sup⁡kAk(T)∑i=1n-2‍σi)≤1ρ(‖z0‖+σ(n-2)sup⁡kAk(T))≤1ρ(‖z0‖+σ2ɛsup⁡kAk(T))≤T.
This proves the theorem.


## 3.2. Evasion Differential Game

Theorem 3.5.
Ifσ-ρ≥0 then evasion is possible in the game G1 from the initial position z0≠0.Proof.
Suppose that(3.25)σ-ρ≥0,
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:
(3.26)vk(t)={Aj-1/2(T)ρeαj(t),k=j,0,k≠j.
This control function belongs to S1(σ). Indeed,
(3.27)∑k=1∞‍∫0T‍vk2(s)ds=Aj-1(T)ρ2Aj(T)≤σ2
we have used (3.26) and (3.25).
Our goal now is to show thatzj(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
(3.28)zj(t)≥e-αj(t)(zj0+ρAj(t)-ρAj(t))=zj0e-αj(t)>0,
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 z0≠0. The proof of the theorem is complete.Theorem 3.6.
Suppose thatσ≥ρ or there exists a number k=j such that z0j>0 and σ-ρAj(T)≥0. Then from the initial position z0≠0, evasion is possible in the game G2.Proof.
Suppose thatσ≥ρ and that z0≠0. The later condition means that zk0≠0 for some k=j. We construct the control function of the evader as follows:
(3.29)vk(t)={ρ,k=j,0,k≠j.
It is obvious that this control belongs to the set S2(σ).
To be definite, letzj0>0. Using (3.29) and (3.1), we have
(3.30)zj(t)>e-αj(t)(zj0+ρ∫0t‍eαj(s)ds-ρ∫0t‍eαj(s)ds)=e-αj(t)zj0>0.
This means that evasion is possible from the initial position z0≠0 in the game G2.
We now prove the theorem with the alternative condition. Suppose that there exists a numberk=j such that z0j>0 and σ-ρAj(T)≥0. Let the control of the evader be as follows:
(3.31)vk(t)={TAj(T)t+e+Tρe-αj(t),k=j,0,k≠j.
We show that this control satisfies the geometric constraint:
(3.32)∑k=1∞‍vk2(t)=TAj(T)(t+e+T)2ρ2e-2αj(t)≤ρ2Aj(T)≤σ2.
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))
(3.33)zj(t)≥e-αj(t)(zj0+ρTAj(T)ln⁡⁡(t+e+T)-ρTAj(T))>0,
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.
Ifσ-ρT≥0 then evasion is possible from the initial position z0≠0 in the game G4.Proof.
Suppose thatz0≠0 and that σ-ρT≥0. We construct the control function of the evader as follows:
(3.34)vk(t)={ρTAj(T)eαj(t),k=j,0,k≠j.
We now show that this control satisfies the integral constraint
(3.35)∑k=1∞‍∫0T‍vk2(s)ds=ρ2TAj(T)∫0T‍e2αj(s)ds≤ρ2T≤σ2.
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 letzj0>0. Substituting (3.34) into (3.1), we have
(3.36)zj(t)≥e-αj(t)(zj0+ρTAj(T)Aj(T)-ρTAj(T))=e-αj(t)zj0>0.
Therefore, z(t)≠0,t∈[0,T]. This means that evasion is possible from initial position z0≠0 in game G4. This ends the proof of the theorem.Theorem 3.8.
Ifz0∈Y and σ≥2ρeαj(T) for some k=j, then from the initial position z0≠0 evasion is possible in the game G3.Proof.
Suppose thatz0∈Y and that there exists k=j such that
(3.37)σ≥2ρeαj(T).
We construct the control function of the evader as follows:
(3.38)vk(t)={(2zj0+ρ)eαj(t),k=j,0,k≠j.
The inclusion v(·)∈S2(σ) follows from the following
(3.39)∑k=1∞‍vk2(t)=[(2zj0+ρ)eαj(t)]2≤8zj02e2αj(t)+2ρ2e2αj(t)≤2ρ2e2αj(t)+2ρ2e2αj(t)≤4ρ2e2αj(T)≤σ2.
we used (3.34); (3.37) and the inequality (a+b)2≤2a2+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
(3.40)zj(t)≥e-αj(t)(zj0-ρAj(t)+2zj0Aj(t)+ρAj(t))=e-αj(t)(zj0+2zj0Aj(t))>0.
Therefore, we have zj(t)>0,0≤t≤T, 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.



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*Source: 717124-2012-07-08.xml*