International Journal of Pure and Applied …evasion problem. AMS Subject Classi cation: 49N70, 49N75, 91A23 Key Words: pursuit, evasion, strategy, integral constraint 1. Introduction

International Journal of Pure and Applied Mathematics————————————————————————–Volume 70 No. 7 2011, 927-938

DIFFERENTIAL GAMES DESCRIBED BY INFINITE SYSTEM

OF DIFFERENTIAL EQUATIONS OF SECOND ORDER.

THE CASE OF NEGATIVE COEFFICIENTS

Abbas Badakaya Ja’afaru1 §, Gafurjan I. Ibragimov2

1,2Department of MathematicsFaculty of Science

Institute for Mathematical ResearchUniversiti Putra

Malaysia, 43400 UPM Serdang, Selangor, MALAYSIA

Abstract: We study pursuit and evasion differential game problems for aninfinite system of differential equations of second order. Control functions aresubject to integral constraints.The pursuit is completed if z(τ) = z(τ) = 0 atsome τ > 0, where z(t) is the state of the system. The pursuer tries to completethe pursuit and the evader exactly tries to avoid this. A sufficient conditionis obtained for completing the pursuit in the differential game when controlrecourse of the pursuer greater than that of the evader. In the case where thecontrol recourse of the evader not less than that of the pursuer we study anevasion problem.

AMS Subject Classification: 49N70, 49N75, 91A23Key Words: pursuit, evasion, strategy, integral constraint

1. Introduction

The study of two person zero-sum differential games was initiated by Isaac [10].Since then many works with various approaches have been done in developingthe theory of differential games (see, for example, [5], [10], [11], [12], [14], and[15]). Control problems in systems with distributed parameters were studiedextensively in the literature, because of its considerable meaning both in theoryand applications (see e.g. [1]- [4], [9] and [13]). As to the differential gameproblems in systems with distributed parameters, there is only a few literature

Received: March 9, 2011 c© 2011 Academic Publications, Ltd.§Correspondence author

928 A.B. Ja’afaru, G.I. Ibragimov

(see e.g. [6]-[8] and [16] - [18]).In the paper [17], differential game problem defined by the second-order

time evolution equation:

z(t) + Az(t) = −u(t) + v(t), z(0) = z0, z(t), 0 < t ≤ T,

was reduced to the one described by the following system

zk(t) + µkzk(t) = −uk(t) + vk(t), 0 < t ≤ T, k = 1, 2, . . . , (1)

where the numbers µk are generalized eigenvalues of an elliptic operator of theform

Az = −n∑

i,j=1

∂

∂xi

(

aij(x)∂z

∂xj

)

that satisfy the conditions

0 < µ1 ≤ µ2 ≤ · · · → ∞,

uk, k = 1, 2, . . . , are control parameters of the pursuer, and vk, k = 1, 2, . . . , arethat of the evader.

The work [7] proposed studying the differential game problems describedby the infinite system of differential equations (1) in one frame separately fromthat described by partial differential equations. In that work, the numbers µk

are assumed to be any positive numbers, and the control functions of the playersare subject to integral constraints.

The paper [8] concerned with the differential games described by infinitesystems of first order differential equations. Control functions are subjectedto integral constraints. To solve pursuit and evasion differential games someconditions were obtained.

The main purpose of the present paper is to investigate pursuit and eva-sion differential game problems described by the system (1) in case of negativecoefficients µk.

2. Statement of the Problem

Let λ1, λ2, . . . be a bounded sequence of negative numbers and r be a realnumber. We introduce the space

l2r = {α = (α1, α2, . . . ) :

∞∑

k=1

|λk|rα2k < ∞},

DIFFERENTIAL GAMES DESCRIBED BY INFINITE SYSTEM... 929

with inner product and norm

〈α, β〉r =

∞∑

k=1

|λk|rαkβk, α, β ∈ l2r , ||α||r =

(

∞∑

k=1

|λk|rα2k

)1/2

.

Let

L2(t0, T, l2r) = {w(t) = (w1(t), w2(t), . . . ) :∞∑

k=1

|λk|r∫ T

t0

w2k(t)dt < ∞, wk(·) ∈ L2(t0, T )

}

,

‖w(·)‖L2(0, T, l2r) =

(

∞∑

k=1

|λk|r∫ T

t0

w2k(s)ds

)1/2

,

where T, T > t0, is a given number.

We examine the differential game problems described by the following infi-nite system of differential equations

zk(t) + λkz(t) = −uk(t) + vk(t), zk(0) = z0k, zk(0) = z1

k, k = 1, 2, . . . , (2)

where

zk, uk, vk ∈ R1, k = 1, 2, . . . , z0 = (z01 , z0

2 , . . . ) ∈ l2r+1, z1 = (z11 , z1

2 , . . . ) ∈ l2r ,

u = (u1, u2, . . . ) is the control parameter of the Pursuer, v = (v1, v2, . . . ) is thatof the Evader.

Let ρ0, ρ and σ be given positive numbers.

Definition 1. A function w(·), w : [0, T ] −→ l2r , with measurable coordi-nates wk(t), 0 ≤ t ≤ T, k = 1, 2, . . . , subject to

∞∑

k=1

|λk|r∫ T

0w2

k(s)ds ≤ ρ20

is referred to as the admissible control.

We denote the set of all admissible controls by S(ρ0).

Definition 2. A function u(·) ∈ S(ρ) (respectively v(·) ∈ S(σ)) is referredto as the admissible control of the Pursuer (the Evader).


We call the numbers ρ and σ resources of controls of the Pursuer and theEvader, respectively. It should be noted that the system (2) is considered onthe interval [0, T ], since the existence-uniqueness theorem for this system wasproved for the finite interval. At the same time T can be any fixed positivenumber.

Definition 3. Let u(·) and v(·) are admissible controls of the pursuersand the evader, respectively. A function z(t) = (z1(t), z2(t), . . . ), 0 ≤ t ≤ T, iscalled the solution of the equation (2) if each coordinate zk(t) of that:

1) is continuously differentiable on (0, T ), and satisfies the initial conditionszk(0) = z0

k, zk = z1k;

2) has the second derivative zk(t) almost everywhere on (0, T ) that satisfiesthe equation

zk(t) + λkzk(t) = −uk(t) + vk(t), k = 1, 2, . . . ,

almost everywhere on (0, T ).

Definition 4. A function u(t, v) = (u1(t, v), u2(t, v), . . . ) , u : [0, T ] ×l2r −→ l2r , with components of the form uk(t, v) = wk(t) + vk(t), k = 1, 2, . . . ,where w(·) = (w1(·), w2(·), . . . ) ∈ S(ρ − σ), and v(·) = (v1(·), v2(·), . . . ) is anyadmissible control of the Evader, is called a strategy of the Pursuer.

Definition 5. If there exists a strategy u(·) of the Pursuer such that forany admissible control of the Evader z(τ) = 0, z(τ) = 0 at some τ, 0 ≤ τ ≤ θ,

then we say that pursuit can be completed for the time θ in the differentialgame (2) (of course θ ≤ T ).

Definition 6. A function of the form

v(t) =

{

0, 0 ≤ t ≤ ε,

u(t − ε), ε < t ≤ T.

where ε is a positive number, u(t), 0 ≤ t ≤ T, is an admissible control of thePursuer, is called a strategy of the Evader.

Definition 7. We say that evasion is possible in the game (2) if there isan ε > 0 such that for any admissible control of the Pursuer and correspondingto it strategy of the Evader

||z(t)||l2r+1

+ ||z(t)||l2r

6= 0, 0 ≤ t ≤ T.

The problems are to find conditions1) for completing pursuit for a finite time in the differential game (2),2) for evasion to be possible in the game (2).


3. Problem Analysis

First we consider the following system

zk(t) + λkz(t) = wk(t), zk(0) = z0k, zk(0) = z1

k, k = 1, 2, . . . , (3)

where w(·) = (w1(·), w2(·), . . . ) ∈ L2(0, T ; l2r) is a control.

It is not difficult to verify that the kth equation in (3) has the unique solution

zk(t) = z0k cosh(αkt) + z1

k

sinh(αkt)

αk+

∫ t

0wk(τ)

sinh(αk(t − s))

αkds, (4)

where αk =√−λk. It’s derivative is

zk(t) = αkz0k sinh(αkt) + z1

k cosh(αkt) +

∫ t

0wk(τ) cosh(αk(t − s))ds. (5)

Let C(t0, T, l2r) be the space of continuous functions z(t) = (z1(t), z2(t), . . . ) ,

0 ≤ t ≤ T, with values in l2r . The following assertion is true [1]

Assertion 1. If λk < 0, k = 1, 2, . . . , is a bounded below sequence, thenthe functions z(t) = (z1(t), z2(t), . . . ) , and z(t) = (z1(t), z2(t), . . . ) defined by(4) and (5) belong to the spaces C(0, T ; l2r+1) and C(0, T ; l2r), respectively.

In (4) and (5), letting

xk(t) = αkzk(t), xk0 = αkz0k, yk(t) = zk(t), yk0 = z1

k (6)

we obtain

{

xk(t).= xk0 cosh(αkt) + yk0 sinh(αkt) +

∫ t0 sinh(αk(t − s))wk(s)ds,

yk(t).= xk0 sinh(αkt) + yk0 cosh(αkt) +

∫ t0 cosh(αk(t − s))wk(s)ds.

It is clear that zk(t) = 0, zk(t) = 0 if and only if xk(t) = 0, yk(t) = 0. Thelatter equalities can be written as

Ak(t)

[

xk0

yk0

]

+

∫ t

0Ak(t)

[

− sinh(αks)cosh(αks)

]

wk(s)ds =

[

00

]

, (7)

where

Ak(t) =

[

cosh(αkt) sinh(αkt)sinh(αkt) cosh(αkt)

]

.


Multiplying (7) by A−1k (t), yields

xk0 =∫ t0 sinh(αks)wk(s)ds,

−yk0 =∫ t0 cosh(αks)wk(s)ds.

(8)

Our task is now to find a control w(·) = (w1(·), w2(·), . . . ) that satisfy (8) forall k = 1, 2, . . . . To this end we define the following set

Xk(θ, σ) =

{

(xk, yk) : xk =

∫ θ

0sinh(αks)wk(s)ds,

yk =

∫ θ

0cosh(αks)wk(s)ds, wk(·) ∈ S(θ, σ)

}

,

where

S(θ, σ) =

{

wk(·) :

∫ θ

0w2

k(s)ds ≤ σ2, wk(·) ∈ L2(0, θ)

}

.

It is clear that (xk0, −yk0) ∈ Xk(θ; σ) if and only if there exists wk(·) ∈ S(θ, σ)to satisfy (8).

4. Pursuit Differential Game

Let

X(θ, σ) =⋃

(σ1,σ2,... ),

∞∏

k=1

Xk(θ, σk),

where∏

is cartesian product of the sets Xk(θ, σk), and the union is taken overall the sequences σ1, σ2, · · · such that

∞∑

k=1

σ2k = (ρ − σ)2, σk ≥ 0.

Theorem 1. If ρ > σ and z0 ∈ X(θ, ρ − σ), where

z0 =(

(α1z01 , −z1

1), (α2z02 , −z1

2), . . .)

,

then the differential game (2) can be completed for the time θ.


Proof. As

X(θ, ρ − σ) =⋃

(σ1, σ2,... ),

∞∏

k=1

Xk(θ, σk),

∞∑

k=1

σ2k = (ρ − σ)2, σk ≥ 0, k = 1, 2, . . . ,

then z0 ∈ X(θ, ρ − σ) implies that there exists a sequence σ1, σ2, . . . , σk ≥0, k = 1, 2, . . . ,

∑∞k=1 σ2

k = (ρ− σ)2, such that (αkz0k, −z1

k) ∈ Xk(θ, σk), k =1, 2, . . . .

This gives(xk0, −yk0) ∈ Xk(θ, σk), k = 1, 2, . . . .

This means that there exists a control w0(t) = (w01(t), w0

2(t), . . . ), w0k(·) ∈

S(θ, σk), 0 ≤ t ≤ θ, such that (see (8))

xk(θ) = xk0 −θ∫

0

sinh(αks)w0k(s)ds = 0,

yk(θ) = yk0 +θ∫

0

cosh(αks)w0k(s)ds = 0, k = 1, 2, , . . . .

(9)

We construct the strategy of the pursuer as follows:

uk(t) = −w0k(t) + vk(t), (10)

where vk(t) is a control of the evader.We show admissibility of the pursuer’s strategy, i.e., we show that

(

∞∑

k=1

|λk|r∫ θ

0|uk(t)|2dt

)1/2

≤ ρ.

We have

(

∞∑

k=1

|λk|r∫ θ

0|uk(t)|2dt

)1/2

=

(

∞∑

k=1

|λk|r∫ θ

0| − w0

k(t) + vk(t)|2dt

)1/2

≤(

∞∑

k=1

|λk|r∫ θ

0

(

|w0k(t)| + |vk(t)|

)2dt

)1/2

≤(

∞∑

k=1

|λk|r∫ θ

0|w0

k(t)|2dt

)1/2

+

(

∞∑

k=1

|λk|r∫ θ

0|vk(t)|2dt

)1/2


≤ ρ − σ + σ = ρ.

We now show that the pursuit is completed for the time θ. According to (4)and (5) equation (2) has the solution

zk(t) = z0k cosh(αkt) + z1

k

sinh(αkt)

αk+

∫ t

0(−uk(s) + vk(s))

sinh(αk(t − s))

αkds,

and its derivative is

zk(t) = αkz0k sinh(αkt) + z1

k cosh(αkt) +

∫ t

0(−uk(s) + vk(s)) cosh(αk(t − s))ds.

As the systemzk(θ) = 0, zk(θ) = 0, k = 1, 2, . . . ,

is equivalent to the one

xk0 −θ∫

0

sinh(αks)(−uk(s) + vk(s))ds = 0,

yk0 +θ∫

0

cosh(αks)(−uk(s) + vk(s))ds = 0, k = 1, 2, · · · .

(11)

From here, in view of (10) we obtain the system (9). Therefore the strategy(10) ensures that pursuit is completed at θ. The proof is complete.

5. Evasion Differential Game

In this section, we study an evasion differential game described by infinite sys-tem of differential equations (2). We formulate the main result.

Theorem 2. . If σ ≥ ρ, then from any initial position

z0 = (z01 , z0

2 , · · · ), z1 = (z11 , z1

2 , · · · ), ||z0||l2r+1

+ ||z1||l2r

6= 0,

evasion is possible in the game (2).

Proof. As mentioned above system zk(t) = 0, zk(t) = 0, k = 1, 2, · · · , isequivalent to the system (11), where x0 = (x10, x20, . . . ) ∈ l2r , y0 = (y10, y20, . . . )∈ l2r , z0, z1 and x0, y0 are connected with (6). The condition

||z0||l2r+1

+ ||z1||l2r

6= 0,


implies that there exists a number k = m such that (xm0, ym0) 6= (0, 0), sayxm0 6= 0 (the case ym0 6= 0 is similar). We show that there exists admissiblestrategy of the evader such that xm(t) 6= 0, t ∈ [0, T ]. Let the pursuer use anadmissible control u(t), t ∈ [0, T ]. We construct the strategy of the evader asfollows

vm(t) =

{

0, 0 ≤ t ≤ ε,

um(t − ε), ε < t ≤ T,

vi(t) = 0, 0 ≤ t ≤ T, i = 1, · · · , (m − 1), (m + 1), · · · .

(12)

The admissibility of evader’s strategy follows from the relation

∞∑

k=1

|λk|r∫ T

0v2k(s)ds = |λk|r

∫ T

εu2

m(s − ε)ds = |λk|r∫ T−ε

0u2

m(s)ds

≤∞∑

k=1

|λk|r∫ T

0u2

k(s)ds ≤ ρ2 ≤ σ2.

For the possibility of evasion, we have

xm(t) = xm0 +

t∫

0

sinh(αms)um(s)ds −t∫

0

sinh(αms)vm(s)ds

= xm0 +

ε∫

0

sinh(αms)um(s)ds +

t∫

ε

sinh(αms)um(s)ds

−ε∫

0

sinh(αms)vm(s)ds −t∫

ε

sinh(αms)vm(s)ds. (13)

Substituting (12) into (13) and transforming, we have

xm(t) = xm0 +

ε∫

0

[sinh(αms) − sinh(αm(s + ε))]um(s)ds

+

t−ε∫

ε

[sinh(αms) − sinh(αm(s + ε))]um(s)ds +

t∫

t−ε

sinh(αms)um(s)ds.


If ε approaches zero, then xm(t) → xm0 6= 0, since each integral approacheszero. For example, letting

I =

ε∫

0

[sinh(αms) − sinh(αm(s + ε))]um(s)ds

then we have

I = (1 − cosh(αmε))

∫ ε

0sinh(αms)um(s)ds − sinh(αmε)

∫ ε

0cosh(αms)um(s)ds.

The integral∫ ε

0sinh(αms)um(s)ds

is bounded, since

∫ ε

0sinh(αms)um(s)ds ≤

(∫ ε

0sinh2(αms)ds

∫ ε

0u2

m(s)ds

)1/2

≤ ρ

(∫ ε

0u2

m(s)ds

)1/2

.

Also∫ ε0 cosh(αms)um(s)ds is bounded. Therefore, I approaches zero as ε ap-

proaches zero.

Consequently, one can choose ε > 0 to ensure z(t) 6= 0, t ∈ [0, T ]. Thismeans evasion is possible in the game (2). This completes the proof of thetheorem.

6. Conclusions

We have studied pursuit and evasion differential game problems for an infinitesystem of differential equations of second order, the control functions beingsubject to integral constraints. In the case where the control recourse of theevader less than that of the pursuer we have studied a pursuit problem. Ifthis is not so, then we have solved an evasion problem. In solving the pursuitproblem we used the solution of a control problem.


Acknowledgments

This research was partially supported by the Research Grant (RUGS) of theUniversity Putra Malaysia, No. 05-04-10-1005RU.

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Documents

International Journal of Pure and Applied …evasion problem. AMS Subject Classi cation: 49N70, 49N75, 91A23 Key Words: pursuit, evasion, strategy, integral constraint 1. Introduction