J. Nig. Soc. Phys. Sci. 4 (2022) 83–87

Journal of the
Nigerian Society

of Physical
Sciences

On Some Pursuit Differential Game Problem in a Hilbert Space

Jamilu Adamua,∗, Aminu Sulaiman Hallirub, Bala Ma’aji Abdulhamidc

aDepartment of Mathematics, Federal University, Gashua, Nigeria.
bDepartment of Mathematical Sciences, Bayero University, Kano, Nigeria.

cDepartment of Mathematical Sciences, Abubakar Tafawa Balewa University, Bauchi, Nigeria.

Abstract

We study pursuit differential game problem in which a countable number of pursuers chase one evader. The problem is formulated in a Hilbert
space l2 with pursuers’ motions described by nth order differential equations and that of the evader by mth order differential equation. The control
functions of the pursuers and evader are subject to integral and geometric constraints respectively. Duration of the game is denoted by the positive
number θ. Pursuit is said to be completed if there exist strategies u j of the pursuers P j such that for any admissible control v(·) of the evader E the
inequality ‖y(θ) − x j(θ)‖ ≤ r j is satisfied for some j ∈ {1, 2, . . . }. In this paper, sufficient condition for completion of pursuit were obtained and
also strategies of the pursuers that ensure completion of pursuit are constructed.

DOI:10.46481/jnsps.2022.379

Keywords: differential game, pursuer, evader, geometric constraint, integral constraint, Hilbert space.

Article History :
Received: 03 September 2021
Received in revised form: 25 January 2022
Accepted for publication: 25 January 2022
Published: 28 February 2022

c©2022 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: T. Latunde

1. Introduction

A considerable amount of literature on differential game
problem in which finite number of pursuers chase one evader
in the Hilbert space l2, control function of players subjected
to either geometric, integral or both constraints has been
published. (See for example, [1]- [25] and some reference their
in.

In many studies of differential game problems, motions of the
two players (i.e. pursuer and evader) are explicitly stated and
are considered to be differential equations of the same order.
For example, in the papers [3, 8, 10, 13, 20, 24], motion of
each of the player is considered to obey first order differential

∗Corresponding author tel. no: +2347033885836
Email address: jamiluadamu88@gmail.com (Jamilu Adamu )

equation. In other studies such as [5, 6, 17, 22], players’
motions are described by second order differential equations.
Whereas in [1, 2, 4, 21], motion of the pursuers and evader
are described by first and second order differential equations
respectively.

Adamu et. al [1], studied pursuit-evasion differential game
problem in a Hilbert space l2, in which motions of pursuers
and evader are described by first and second order differential
equations respectively. Control functions of both the pursuers
and evader are subject to integral constrains. Under certain
condition the authors found the value of the game and con-
structed optimal strategies of the players.

In the paper [2] by Badakaya, a pursuit differential game
problem with finite number of pursuers and one evader in
the space l2 is considered. Pursuers’ motions are described

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Adamu et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 83–87 84

by a first order differential equations and that of the evader
by a second order differential equation. Control functions of
the pursuers and evader are subject to integral and geometric
constraints respectively. Duration of the game is denoted
by the positive number θ. Theorems are stated and proved
each of which provides a condition for completion of pursuit.
Consequently, strategies of the pursuers that ensure completion
of pursuit are constructed.

In the other hand, the paper [4], is about study of pursuit
differential game problem in l-catch sense with finite number
of pursuers and one evader in the space l2. In this paper, control
functions of the pursuers and evader are subject to integral
and geometric constraints respectively. Sufficient condition for
completion of pursuit were obtained.

Leong and Ibragimov [12] studied simple motion pursuit
differential game with m pursuers and one evader on a closed
convex subset of the Hilbert space l2 . 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.
Strategy of pursuers were constructed sufficient to complete
the pursuit from any initial position.

In the paper [20], simple motion differential game with many
players and geometric constraints on the control functions of
the players was studied. By using Lyapunov function method
for an auxiliary problem, they obtained sufficient conditions to
find the pursuit time in Rn.

The paper [21] deal with the study of a pursuit differential
game problem for the so-called ”boy and crocodile” game
in the space Rn. Boy’s motion is described by a first order
differential equations and that of the crocodile by a second
order differential equation. Control functions of the pursuer
and evader are subject to integral and geometric constraints
respectively. They obtained sufficient conditions of completion
of pursuit.

Vagin and Petrov [25] studied a pursuit differential game prob-
lem with finite number pursuers and evader in the space Rn.
Motions of the players are described by nth order differential
equations. Control functions of the players are subject to
geometric constraints. Sufficient condition for completion of
pursuit was obtained.

In the present paper, motivated by the above development, we
proposed to study pursuit differential game problem with count-
able number of pursuers and one evader in the Hilbert space l2.
The motions of the pursuers and evader are described by nth

and mth order differential equations respectively. Control func-
tions of the pursuers are subject to integral constrains. Whereas,
geometric constraint is imposed on the control function of the
evader. We found the sufficient condition of completion of pur-
suit in l-catch sense.

2. Differential game formulation

Let l2 =

ξ = (ξ1,ξ2, . . . ) :
∞∑
j=1

ξ2j < ∞

 , with inner product
〈·, ·〉 : l2 × l2 → R and norm ‖ · ‖ : l2 → [0, +∞), defined
as follows:

〈ξ, y〉 =
∞∑
j=1

ξ jy j, ‖z‖ =

 ∞∑
j=1

z2j


1/2

,

where ξ, y, z ∈ l2.
We consider a differential game described by the following ini-
tial value problem:

P j :
dn x j
dtn = u j(t),

E : d
m y

dtm = v(t),
(1)

subject to:
x j(0) = x0j,

d x j
dt (0) = x

1
j,

d2 x j
dt2 (0) = x

2
j, . . . ,

dn−1 x j
dtn−1 (0) = x

n−1
j , j ∈ J,

y(0) = y0, dydt (0) = y
1,

d2 y
dt2 (0) = y

2, . . . ,
dm−1 y
dtm−1 (0) = y

m−1

(2)

where x j, x0j, x
1
j, x

2
j, . . . , x

n−1
j , u j, y, y

0, y1, y2, . . . , ym−1, v ∈ l2,
u j and v are controls parameter of the pursuer P j and evader
E respectively. Here and below n ≤ m for all n, m ∈ N,
J = {1, 2, . . . } and ‖y0 − x j0‖ > r j, where r j ≥ 0.

Let C(0,θ; l2) be the space of functions

h(t) = (h1(t), h2(t), · · · , h j(t), · · · ) ∈ l2, t ≥ 0, (3)

such that the following conditions hold:
(a) h j(t), 0 ≤ t ≤ θ, j = 1, 2, · · · , are absolutely continuous
functions;
(b) h(t), 0 ≤ t ≤ θ, is a continuous function in the norm of l2.

Definition 2.1. A function u j(t) = (u j1(t), u j2(t), . . . ) with Borel
measurable coordinates such that∫ θ

0
||u j(t)||

2dt ≤ β2j, (4)

where β j is given positive number, is called an admissible con-
trol of the jth pursuer.

Definition 2.2. A function v(t) = (v1(t), v2(t), . . . ) with Borel
measurable coordinates such that

||v(t)|| ≤ γ, t ≥ 0 (5)

where γ is given positive number, is called admissible control
of the evader E.

Definition 2.3. A function U j(t, x j, y, v), U j : [0,θ] × l2 × l2 ×
l2 → l2, is called a strategy of the jth pursuer if, for any ad-
missible control v(·) of the evader E, the system (1)-(2) has a
unique solution (x j(·), y(·)), with x j(·), y(·) ∈ C(0,θ, l2). A strat-
egy U j is admissible if each control involved in the formation of
this strategy is admissible.

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Adamu et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 83–87 85

Definition 2.4. The system described by (1)-(2) in which the
controls u j(·) and v(·) satisfy the inequalities (4) and (5) re-
spectively is called game G1.

Definition 2.5. Pursuit is said to be completed in l-catch sense
in the game G1 if there exist strategies u j of the pursuers P j
such that for any admissible control v(·) of the evader E the
inequality ‖y(θ)− x j(θ)‖ ≤ r j is satisfied for some j ∈ {1, 2, . . . }.

Research problem: In the game G1, find sufficient condition
for completion of pursuit.

3. MAIN RESULT

Once the players’ admissible controls u j(·) and v(·) are chosen,
the solution to the equations of motion for the jth pursuer and
evader in (1) with initial condition (2) are respectively given by

x j(θ) = x j0 +
∫ θ

0

∫ t1
0

∫ t2
0
· · ·

∫ tn−1
0

u j(t)dt dtn−1

. . . dt2 dt1, (6)

and

y(θ)

= y0 +
∫ θ

0

∫ t1
0

∫ t2
0
· · ·

∫ tm−1
0

v(t)dt dtm−1 . . . dt1, (7)

where

x j0 = x
0
j + θx

1
j +

θ2

2!
x2j + · · · +

θn−1

(n − 1)!
xn−1j (8)

y0 = y
0

+ θy1 +
θ2

2!
y2 + · · · +

θm−1

(m − 1)!
ym−1 (9)

The expression with the multiple integrals in (6) and (7) can
be reduced to that with single integral using the formula below
(see [27])∫ θ

0

∫ t1
0

∫ t2
0
· · ·

∫ tn−1
0

u(t) dtn−1 . . . dt2 dt1

=

∫ θ
0

(θ− t)n−1

(n − 1)!
u(t)dt. (10)

Therefore, equation (6) and (7) become

x j(θ) = x j0 +
∫ θ

0

(θ− t)n−1

(n − 1)!
u j(t)dt, (11)

y(θ) = y0 +
∫ θ

0

(θ− t)m−1

(m − 1)!
v(t)dt, (12)

Alternatively, in place of the players’ dynamic equations (1),
we can consider an equivalent differential game described by
the following first order differential equations:

P j :
d x j
dt =

(θ−t)n−1

(n−1)! u j(t), x j(0) = x j0,

E : dydt =
(θ−t)m−1

(m−1)! v(t), y(0) = y0,
(13)

Let π1 = ω0(β2j − γ
2ω1) + ||y0||

2
−

∣∣∣∣∣∣x j0∣∣∣∣∣∣2 and
π2 = ω0(β2j − θγ

2) + ||y0||
2
−

∣∣∣∣∣∣x j0∣∣∣∣∣∣2, where ω0 = θ2n−1(2n−1)((n−1)!)2
and ω1 =

((n−1)!)2θ2(m−n)+1

(2(m−n)+1)((m−1)!)2 .

We define the sets

X :=


⋃
j∈J

{
z ∈ l2 : 2〈y0 − x j0, z〉 ≤ π1

}
, n < m.⋃

j∈J

{
z ∈ l2 : 2〈y0 − x j0, z〉 ≤ π2

}
, n = m.

(14)

The following statement gives sufficient condition for comple-
tion of pursuit.

Theorem 3.1. . If y(θ) ∈ X then the pursuit can be completed
in the game G1.

Proof. To prove this theorem, we first introduce dummy pur-
suer with state variable z whose motion obey the following ini-
tial value problem.

dz
dt

=
(θ− t)n−1

(n − 1)!
Ū(t), z(0) = z0 = x j0, (15)

and that the control function Ū(·) is such that(∫ θ
0

∣∣∣∣∣∣Ū(t)∣∣∣∣∣∣2 dt) 12 ≤ β̄ = β j + r j√
ω0

(16)

Clearly, β̄ > β j and (β̄−β j)
√
ω0 = r j for all j ∈ J.

For all j ∈ J and 0 ≤ t ≤ θ, we construct a dummy pursuer’s
strategy as follows: If n = m, we set

Ū(t) =
(2n − 1)(n − 1)!
θ2n−1(θ− t)1−n

(y0 − z0) + v(t). (17)

If n < m, we set

Ū(t) =
(2n − 1)(n − 1)!
θ2n−1(θ− t)1−n

(y0−z0)+
(n − 1)!v(t)

(m − 1)!(θ− t)n−m
.(18)

For the case n = m, let the dummy pursuer z use the strategy
(17), we show that this strategy is admissible and ensure the
equality z(θ) = y(θ). Indeed,

z(θ) = z0 +
∫ θ

0

(θ− t)n−1

(n − 1)!

×

(
(2n − 1)(n − 1)!
θ2n−1(θ− t)1−n

(y0 − x j0) + v(t)
)

dt

= z0 +
(2n − 1)(y0 − x j0)

θ2n−1

∫ θ
0

(θ− t)2n−2dt

+

∫ θ
0

(θ− t)n−1

(n − 1)!
v(t)dt

= z0 + y0 − x j0 +
∫ θ

0

(θ− t)n−1

(n − 1)!
v(t)dt

= y0 +
∫ θ

0

(θ− t)m−1

(m − 1)!
v(t)dt = y(θ).

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Adamu et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 83–87 86

It is left to show that the strategy (17) is admissible. By as-
sumption y(θ) ∈ X and if n = m, we have,

2〈y0 − x j0, y(θ)〉 ≤ ω0
(
β2j − θγ

2
)

+ ||y0||
2
− ||x j0||

2 (19)

In accordance with inequality (19) and using state equation (12)
of the evader, we have:

2
〈
y0 − x j0,

∫ θ
0

(θ− t)m−1

(m − 1)!
v(t)dt

〉
≤ ω0

(
β2j − θγ

2
)
− ||y0 − x j0||

2. (20)

Using the inequality (20), we obtained∫ θ
0
||Ū(t)||2dt

=

∫ θ
0

∣∣∣∣∣
∣∣∣∣∣ (2n − 1)(n − 1)!θ2n−1(θ− t)1−n (y0 − z0) + v(t)

∣∣∣∣∣
∣∣∣∣∣2 dt

=

∫ θ
0

∣∣∣∣∣
∣∣∣∣∣ (2n − 1)(n − 1)!θ2n−1(θ− t)1−n (y0 − x j0)

∣∣∣∣∣
∣∣∣∣∣2 dt

+ 2
∫ θ

0

〈
(2n − 1)(n − 1)!
θ2n−1(θ− t)1−n

(y0 − x j0), v(t)
〉

dt

+

∫ θ
0
||v(t)||2 dt

≤
||y0 − x j0||2

ω0
+

2
ω0

〈
y0 − x j0,

∫ θ
0

(θ− t)m−1

(m − 1)!
v(t)dt

〉
+ θγ2

≤
||y0 − x j0||2

ω0
+

1
ω0

(
ω0

(
β2j − θγ

2
)
− ||y0 − x j0||

2
)

+ θγ2

= β2j < β̄
2

For the case n < m and if the dummy pursuer z uses the strategy
(18), we have z(θ) = y(θ). Indeed, clearly

z(θ) = x j0 +
(2n − 1)(y0 − x j0)

θ2n−1

∫ θ
0

(θ− t)2n−2dt

+

∫ θ
0

(θ− t)m−1

(m − 1)!
v(t)dt

= x j0 +
(2n − 1)(y0 − x j0)

θ2n−1
θ2n−1

2n − 1

+

∫ θ
0

(θ− t)m−1

(m − 1)!
v(t)dt

= x j0 + y0 − x j0 +
∫ θ

0

(θ− t)m−1

(m − 1)!
v(t)dt

= y0 +
∫ θ

0

(θ− t)m−1

(m − 1)!
v(t)dt = y(θ).

The admissibility of the strategy (18) can be shown as follows.
The inclusion y(θ) ∈ X, implies that,

2〈y0 − x j0, y(θ)〉 ≤ ω0
(
β2j −γ

2ω1
)

+ ||y0||
2
− ||x j0||

2 (21)

In accordance with inequality (21) and using state equation (12)
of the evader , we have:

2
〈
y0 − x j0,

∫ θ
0

(θ− t)m−1

(m − 1)!
v(t)dt

〉
≤ ω0

(
β2j −γ

2ω1
)
− ||y0 − x j0||

2.

Using the inequality (22), we obtained∫ θ
0
||Ū(t)||2dt

=

∫ θ
0

∣∣∣∣∣
∣∣∣∣∣ (2n − 1)(n − 1)!θ2n−1(θ− t)1−n (y0 − x j0) + (n − 1)!v(t)(m − 1)!(θ− t)n−m

∣∣∣∣∣
∣∣∣∣∣2 dt

=

∫ θ
0

∣∣∣∣∣
∣∣∣∣∣ (2n − 1)(n − 1)!θ2n−1(θ− t)1−n (y0 − x j0)

∣∣∣∣∣
∣∣∣∣∣2 dt

+ 2
∫ θ

0

〈
(2n − 1)(n − 1)!
θ2n−1(θ− t)1−n

(y0 − x j0),
(n − 1)!v(t)

(m − 1)!(θ− t)n−m

〉
dt

+

∫ θ
0

∣∣∣∣∣
∣∣∣∣∣ (n − 1)!v(t)(m − 1)!(θ− t)n−m

∣∣∣∣∣
∣∣∣∣∣2 dt

≤
||y0 − x j0||2

ω0
+

2
ω0

〈
y0 − x j0,

∫ θ
0

(θ− t)m−1

(m − 1)!
v(t)dt

〉
+ γ2ω1

≤
||y0 − x j0||2

ω0
+

1
ω0

(
ω0

(
β2j −γ

2ω1
)
− ||y0 − x j0||

2
)

+ γ2ω1

= β2j < β̄
2

This shows that for each two cases we established that z(θ) =
y(θ). With this and if the real pursuers use the strategies

U j(t) =
β j

β̄
Ū(t), 0 ≤ t ≤ θ, (22)

we show that∣∣∣∣∣∣y(θ) − x j(θ)∣∣∣∣∣∣ ≤ r j (23)
Indeed, using equation (22) and Cauchy-Schwartz inequality,
we get∣∣∣∣∣∣y(θ) − x j(θ)∣∣∣∣∣∣ = ∣∣∣∣∣∣z(θ) − x j(θ)∣∣∣∣∣∣

=

∣∣∣∣∣∣
∣∣∣∣∣∣z0 +

∫ θ
0

(θ− t)n−1

(n − 1)!
Ū(t)dt − x j0 −

∫ θ
0

(θ− t)n−1

(n − 1)!
U j(t)dt

∣∣∣∣∣∣
∣∣∣∣∣∣

=

∣∣∣∣∣∣
∣∣∣∣∣∣
∫ θ

0

(θ− t)n−1

(n − 1)!

(
Ū(t) −

β j

β̄
Ū(t)

)
dt

∣∣∣∣∣∣
∣∣∣∣∣∣

≤

∫ θ
0

∣∣∣∣∣∣
∣∣∣∣∣∣ (θ− t)n−1(n − 1)!

(
1 −

β j

β̄

)
Ū(t)

∣∣∣∣∣∣
∣∣∣∣∣∣ dt

=

(
β̄−β j

β̄

) ∫ θ
0

(θ− t)n−1

(n − 1)!

∣∣∣∣∣∣Ū(t)∣∣∣∣∣∣ dt
≤

(
β̄−β j

β̄

) 
(∫ θ

0

(θ− t)2n−2

((n − 1)!)2
dt

) 1
2
(∫ θ

0

∣∣∣∣∣∣Ū(t)∣∣∣∣∣∣2 dt) 12


≤

(
β̄−β j

β̄

) (
θ2n−1

(2n − 1)((n − 1)!)2

) 1
2

β̄

≤ (β̄−β j)
√
ω0 = r j.

This complete the prove of the theorem.
86



Adamu et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 83–87 87

4. Conclusion

We studied pursuit differential game problem in which a count-
able number of pursuers chase one evader in the Hilbert space
l2. Control function of the pursuers and evader are subject
to integral and geometric constraints respectively. Pursuers’
motions are described by nth order differential equations and
that of the evader by mth order differential equation where
n ≤ m, n, m ∈ N. In this piece of research the strategies of
the pursuers are constructed and sufficient condition for com-
pletion of pursuit were obtained. The result of this paper is the
generalization of some result in the literature. For example, the
result of the paper [4] is a corrolary to this paper when we set
n = 1 and m = 2 in the players’ dynamic equation (1)-(2).

Acknowledgements

The authors would like to thank the referee for giving useful
comments and suggestions for improvement of this paper.

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