J. Nig. Soc. Phys. Sci. 3 (2021) 12–16

Journal of the
Nigerian Society

of Physical
Sciences

Simple Motion Pursuit Differential Game

J. Adamua,∗, B. M. Abdulhamidb, D. T. Gbandec, A. S. Halliruc

aDepartment of Mathematics, Federal University Gashua, Yobe, Nigeria.
bDepartment of Mathematical Sciences, Abubakar Tafawa Balewa University, Bauchi, Nigeria.

cDepartment of Mathematical Sciences, Bayero University, Kano, Nigeria.

Abstract

We study a simple motion pursuit differential game of many pursuers and one evader in a Hilbert space l2. The control functions of the pursuers
and evader are subject to integral and geometric constraints respectively. Duration of the game is denoted by 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(τ)−x j(τ)‖ ≤ l j
is satisfied for some j ∈ {1, 2, . . . } and some time τ. In this paper, sufficient condition for completion of pursuit were obtained. Consequently
strategies of the pursuers that ensure completion of pursuit are constructed.

DOI:10.46481/jnsps.2021.148

Keywords: Differential game, pursuer, evader, geometric constraint, integral constraint, Hilbert space

Article History :
Received: 01 January 2021
Received in revised form: 28 January 2021
Accepted for publication: 29 January 2021
Published: 27 February 2021

c©2021 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: B. J. Falaye

1. Introduction

In view of the extensive literature on differential games of
several player’s with the control functions subjected to either
geometric, integral or both constraints. The work in the papers
[1-26] and some reference their in attract the attention of many
researchers .

In many studies of differential game, motion of each player
are explicitly stated and considered to be a system of differen-
tial equations of the same order. In the papers [2,5,7-9,13,16],
motion of each of the player is considered to obey first order dif-
ferential equation. In other studies such as Refs. [4,6,12,15,19],

∗Corresponding author tel. no: +2347033885836.
Email addresses: jamiluadamu88@gmail.com (J. Adamu ),

mbabdulhamid67@atbu.edu.ng (B. M. Abdulhamid),
princedavison4@gmail.com (D. T. Gbande),
aminuhalliru1@gmail.com (A. S. Halliru)

players’ motions are described by second order differential equa-
tions. Whereas in Refs. [1,23,25] motion of the players are
described by first and second order differential equations.

In Ref, [24], Rikhsiev studied simple motion differential
game of optimal pursuit with one evader and many pursuers
on a closed convex subset of the Hilbert space l2. A sufficient
condition for optimality of pursuit time is obtained, when the
initial position of the evader belong to the interior of the convex
hull of the initial position of the pursuers.

Simple motion differential game of many players with ge-
ometric constraints on the control functions of the players is
studied in [13]. By using lyapunov function method for an aux-
iliary problem, they obtained sufficient conditions to find the
pursuit time in Rn. Vagin and Petrov in Ref. [22] Studied a
pursuit differential game problem with finite number pursuers
and one evader in the Hilbert space Rn. Motions of each player
is described by nth order differential equation. Control func-
tions of the players are subject to geometric constraints. They

12



Adamu et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 12–16 13

obtained sufficient condition for completion of pursuit.
The work in Ref. [26] Leong and Ibragimov studied sim-

ple motion pursuit differential game with m pursuers and one
evader on a closed convex subset of the Hilbert space l2 . Con-
trol 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 suffi-
cient to complete the pursuit from any initial position.

In Ref. [23] pursuit differential game problem for the so-
called boy(evader) and crocodile (pursuer) in the space Rn is
studied. Boy’s motion is described by first order differential
equations and that of the crocodile by second order differen-
tial equation. Control functions of the pursuer and evader are
subject to integral and geometric constraints respectively. They
obtained Sufficient conditions of completion of pursuit.

In this piece of research, we study pursuit differential game
problem in a Hilbert space l2, where motions of the pursuers and
evader described by first and second order differential equations
respectively. Control functions of the pursuers are subject to
integral constrains. Whereas, geometric constraint is imposed
on the control function of the evader.

2. Statement of the Problem

Consider the space l2 =

% = (%1,%2, . . . ) :
∞∑

k=1

%2k < ∞

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

〈x, y〉 =
∞∑

k=1

xkyk, ||%|| =

 ∞∑
k=1

%2k

1/2 ,
where x, y,% ∈ l2, respectively.

We consider a differential game described by the following
equations:{

P j : ẋ j = u j(t), x j(0) = x j0, j ∈ J,
E : ÿ = v(t), ẏ(0) = y1, y(0) = y0,

(1)

where x j, x j0, ui, y, y0, y1, v ∈ l2, u j = (u j1, u j2, . . . ) is a control
parameter of the pursuer P j and v = (v1, v2, . . . ) is that of the
evader E. Here and below J = {1, 2, . . . }, ||x j0 − y0|| > l j, where
l j ≥ 0 are given numbers.

In the space l2, we define a ball (respectively, sphere) of
radius r and center at x0 by B(x0, r) = {x ∈ l2 : ||x − x0|| ≤ r} (
respectively, by S (x0, r) = {x ∈ l2 : ||x − x0|| = r}).

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 ≤ ρ2j, (2)

where ρ 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)|| ≤ σ, t ≥ 0, (3)

is called admissible control of the evader.

If the pursuers P j and evader E chose their admissible controls
u j(·) and v(·) respectively, the solutions to the dynamic equa-
tions (1) are given by:

x j(t) = x j0 +
∫ t

0
u j(s)d s, (4)

y(t) = y0 + ty1 +
∫ t

0

∫ r
0

v(s)d sdr. (5)

It is not difficult to see that∫ t
0

∫ r
0

v(s)d sdr =
∫ t

0
(t − s)v(s)d s (6)

Therefore equation (5) becomes

y(t) = y0 + ty1 +
∫ t

0
(t − s)v(s)d s, (7)

One can readily see that x j(·), y(·) ∈ C(0,θ; l2) , where C(, 0,θ; l2)
is the space of functions

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

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

With this instead of differential game described by (1) we
can consider an equivalent differential game with the same con-
trol functions described by:{

P : ẋ j = u(t), x j(0) = x j0
E : ẏ = (θ− t)v(t), y(0) = y1θ + y0 = y0.

(8)

Indeed, if the evader uses an admissible control v(t) = (v1(t), v2(t), . . . ),
then according to (1), we have

y(θ) = y0+y1θ+
∫ θ

0

∫ r
0

v(s)d sdr = y0+y1θ+
∫ θ

0
(θ−t)v(t)dt,(9)

and the same result can be obtained by (8)

y(θ) = y0 +
∫ θ

0
(θ−t)v(t)dt = y0 +y1θ+

∫ θ
0

(θ−t)v(t)dt,(10)

Also, the same argument can be made for the pursuer P j, there-
fore in the distance ‖y(θ)−x j(θ)‖ we can take either the solution
of (1) or the solution of (8).

The attainability domain of the pursuer Pi from the initial
position xi0 up to the time θ is the ball B(x j0,ρi

√
θ). Indeed, by

Cauchy-Schwartz inequality we have∣∣∣∣∣∣x j(θ) − x j0∣∣∣∣∣∣
13



Adamu et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 12–16 14

=

∣∣∣∣∣∣
∣∣∣∣∣∣x j0 +

∫ θ
0

u j(s)d s − x j0

∣∣∣∣∣∣
∣∣∣∣∣∣

=

∣∣∣∣∣∣
∣∣∣∣∣∣
∫ θ

0
u j(s)d s

∣∣∣∣∣∣
∣∣∣∣∣∣

≤

∫ θ
0
||u j(s)||d s

≤

(∫ θ
0

12d s
) 1

2
(∫ θ

0
||u j(s)d s||

2
) 1

2

≤ ρ j
√
θ

On the other hand, let x̄ ∈ B(x j0,ρ j
√
θ). If the pursuer P j uses

the control u j(s) =
x̄−x j0
θ
, 0 ≤ s ≤ θ, then we have

x j(θ) = x j0+
∫ θ

0
u j(s)d s = x j0+

∫ θ
0

x̄ − x j0
θ

d s = x j0+x̄−x j0 = x̄.

In a similar fashion we can show that the attainability domain
of the evader E from the initial position y0 up to the time θ is
the ball B(y0,σ

θ2

2 ).

Definition 2.3. A strategy of the jth pursuer is a function U j(t, x j, y, v),
U j : [0,∞) × l2 × l2 × l2 → l2, such that the system{

ẋ j = U j(t, x, y, v(t)), x j(0) = x j0,
ÿ = v(t), ẏ(0) = y1, y(0) = y0

has a unique solution (x j(·), y(·)), and that x j(·), y(·) ∈ C(0,θ; l2),
for an arbitrary admissible control v = v(t), 0 ≤ t ≤ θ, of the
evader E. A strategy U j is said to be admissible if each control
formed by this strategy is admissible.

Definition 2.4. The system described by (1) in which the con-
trols u j(·) and v(·) satisfy the inequalities (2) and (3) respec-
tively is called game G.

Definition 2.5. Pursuit is said to be completed in l-catch sense
in the game G if there exist strategies u j of the pursuer P j such
that for any admissible control v(·) of the evader E the inequal-
ity ‖y(τ) − x j(τ)‖ ≤ l j is satisfied for some j ∈ {1, 2, . . . } and
some time τ ∈ [0,θ].

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

We define the half space

Φ j =

{
α ∈ l2 : 2〈y0 − x j0,α〉 ≤ θ

(
ρ2j −σ

2 θ
3

3

)
+ ||y0||

2
−

∣∣∣∣∣∣x j0∣∣∣∣∣∣2} .
3. MAIN RESULT

In this section, we present the main result of the paper.

Theorem 3.1. If y(θ) ∈ Φ j, then pursuit can be completed in
the game G.

Proof To prove this theorem, we first introduce dummy pur-
suer with state variable z and motion described by the following
equation.

ż(t) = w(t), z(0) = x j0,

where the control function w(t) is such that

(∫ θ
0
||w(t)||2 dt

) 1
2

≤ ρ̄ = ρ j +
l j
√
θ
.

Clearly, ρ̄ > ρ j and (ρ̄−ρ j)
√
θ = l j for all j ∈ J.

We construct the strategy of the dummy pursuer as follows:

w(t) =
{ y0−x j0

θ
+ (θ− t)v(t), 0 ≤ t ≤ θ,

0, t > θ.
(11)

To show that the strategy (11) is admissible, we use the fact
that y(θ) ∈ Φ j. This means

2〈y0 − x j0, y(θ)〉 ≤ θ
(
ρ2j −σ

2 θ
3

3

)
+ ||y0||

2
− ||x j0||

2 (12)

In accordance with the inequality (12) and using the state equa-
tion of the evader (10), we have:

2
〈
y0 − x j0,

∫ θ
0

(θ− t)v(t)dt
〉

(13)

= 2〈y0 − x j0, y(θ) − y0〉
= 2〈y0 − x j0, y(θ)〉− 2〈y0 − x j0, y0〉
= 2〈y0 − x j0, y(θ)〉− 2||y0||

2
+ 2〈x j0, y0〉

≤ θ

(
ρ2j −σ

2 θ
3

3

)
+ ||y0||

2
− ||x j0||

2
− 2||y0||

2
+ 2〈x j0, y0〉

= θ

(
ρ2j −σ

2 θ
3

3

)
− ||y0||

2
− ||x j0||

2
+ 2〈x j0, y0〉

= θ

(
ρ2j −σ

2 θ
3

3

)
−

(
||y0||

2
+ ||x j0||

2
− 2〈x j0, y0〉

)
= θ

(
ρ2j −σ

2 θ
3

3

)
− ||y0 − x j0||

2. (14)

Using inequality (13), we have∫ θ
0
||w(t)||2dt

=

∫ θ
0

∣∣∣∣∣
∣∣∣∣∣ y0 − x j0θ + (θ− t)v(t)

∣∣∣∣∣
∣∣∣∣∣2 dt

=

∫ θ
0

(∣∣∣∣∣
∣∣∣∣∣ y0 − x j0θ

∣∣∣∣∣
∣∣∣∣∣2 + 2 〈 y0 − x j0θ , (θ− t)v(t)

〉
+ ||(θ− t)v(t)||2

)
dt

=

∫ θ
0

||y0 − x j0||2

θ2
d s + 2

∫ θ
0

〈 y0 − x j0
θ

, (θ− t)v(t)
〉

dt

+

∫ θ
0

(θ− t)2||v(t)||2dt

≤
||y0 − x j0||2

θ
+

2
θ

〈
y0 − x j0,

∫ θ
0

(θ− t)v(t)dt
〉

+ σ2
∫ θ

0
(θ− t)2dt

≤
||y0 − x j0||2

θ
+

1
θ

(
θ

(
ρ2j −σ

2 θ
3

3

)
− ||y0 − x j0||

2
)

+ σ2
θ3

3

= ρ2j < ρ̄

Therefore the strategy (11) is admissible.

14



Adamu et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 12–16 15

Suppose that the dummy pursuer z uses the strategy (11).
One can easily see that w(θ) = y(θ). Indeed,

z(θ) = x j0 +
∫ θ

0

( y0 − x j0
θ

+ (θ− t)v(t)
)

d s

= x j0 +
∫ θ

0

( y0 − x j0
θ

)
dt +

∫ θ
0

(θ− t)v(t)dt

= x j0 + y0 − x j0 +
∫ θ

0
(θ− t)v(t)dt

= y(θ).

Using the strategy of the dummy pursuer, we define the strate-
gies of the real pursuers P j, j ∈ J as follows:

u j(t) =
ρ j

ρ̄
w(t). (15)

The admissibility of this strategies follows from the fact that the
control w(·) is admissible. Therefore it is left to show that∣∣∣∣∣∣y(θ) − x j(θ)∣∣∣∣∣∣ ≤ l j.
Indeed, using Cauchy-Schwartz inequality we have∣∣∣∣∣∣y(θ) − x j(θ)∣∣∣∣∣∣ = ∣∣∣∣∣∣z(θ) − x j(θ)∣∣∣∣∣∣

=

∣∣∣∣∣∣
∣∣∣∣∣∣x j0 +

∫ θ
0

w(t)dt − x j0 −
∫ θ

0
u j(t)dt

∣∣∣∣∣∣
∣∣∣∣∣∣

=

∣∣∣∣∣∣
∣∣∣∣∣∣
∫ θ

0
w(t)dt −

∫ θ
0

ρ j

ρ̄
w(t)dt

∣∣∣∣∣∣
∣∣∣∣∣∣

≤

∫ θ
0

∣∣∣∣∣∣
∣∣∣∣∣∣
(
1 −

ρ j

ρ̄

)
w(t)

∣∣∣∣∣∣
∣∣∣∣∣∣ dt

=

(
ρ̄−ρ j

ρ̄

) ∫ θ
0
||w(t)||dt

≤

(
ρ̄−ρ j

ρ̄

) 
(∫ θ

0
12dt

) 1
2
(∫ θ

0
||w(t)||2 dt

) 1
2


≤

(
ρ̄−ρ j

ρ̄

)
ρ̄
√
θ

= (ρ̄−ρ j)
√
θ = l j.

This complete the prove of the theorem.

Illustrative Example

Let ρ j = 5, σ = 8, θ = 1 in the game G. We consider
the following initial positions x j0 = (0, 0, . . . , 3, 0, . . . ), y0 =
(0, 0, . . . ) of the players, where the number 3 is jth coordinate
of the point x j. Observe that ρ j

√
θ = 5, σθ

2

2 = 4,
∣∣∣∣∣∣x j0 − y0∣∣∣∣∣∣ =(

02 + 02 + · · · + 32 + 02 + . . .
) 1

2
= 3 > 0. We show that y(θ) ∈

Φ j. It is suffices to show that the inclusion

B(0,σ
θ2

2
) ⊂

⋃
j∈J

B(x j0,ρ j
√
θ),

holds, where 0 is the origin. Indeed, let z = (z1, z2, . . . ) be
arbitrary nonnegative point of the ball B(0, 4) :

∑
∞
j=1 z

2
j ≤ 16.

Then

∣∣∣∣∣∣z − x j0∣∣∣∣∣∣ = (z21 + · · · + z2j−1 + (3 − z j)2 + z2j+1 + . . . ) 12
=

 ∞∑
j=1

z2j + 9 − 6z j


1
2

≤
(
16 + 9 − 6z j

) 1
2

=
(
25 − 6z j

) 1
2

≤ 5.

This means that hypothesis of our theorem is satisfied, therefore
pursuit can be completed in the game G.

4. Conclusion

We have studied a simple motion pursuit differential game
problem in which countable 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 respec-
tively. Pursuers’ motions are described by 1st order differential
equations and that of the evader by 2nd order differential equa-
tion. In this piece of research the strategies of the pursuers are
constructed and sufficient condition for completion of pursuit
were obtained. For further research, value of the game and op-
timality of the pursuit times can also be investigated.

Acknowledgements

The authors would like to thank the reviewers for giving
useful comments and suggestions for improvement of this pa-
per.

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