J. Nig. Soc. Phys. Sci. 2 (2020) 115–119

Journal of the
Nigerian Society

of Physical
Sciences

Original Research

A Pursuit Differential Game Problem on a Closed Convex Subset
of a Hilbert Space

Abbas Ja’afaru Badakayaa,∗, Bilyaminu Muhammadb

aDepartment of Mathematical Sciences, Bayero University, Kano
bDepartment of Mathematics,Federal College of Eduction(Technical), Gusau, Zamfara State

Abstract

In this paper, we study a pursuit differential game problem with finite number of pursuers and one evader on a nonempty closed convex subset of
the Hilbert space l2. Players move according to certain first order ordinary differential equations and control functions of the pursuers and evader
are subject to integral constraints. Pursuers win the game if the geometric positions of a pursuer and the evader coincide. We formulate and prove
theorems that are concerned with conditions that ensure win for the pursuers. Consequently, winning strategies of the pursuers are constructed.
Furthermore, illustrative example is given to demonstrate the result.

Keywords: Pursuit, integral constraint, closed convex set.

Article History :
Received: 01 December 2019
Received in revised form: 07 April 2020
Accepted for publication: 11 April 2020
Published: 01 August 2020

c©2020 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: T. Latunde

1. Introduction

Differential game has been an area of great interest to many
Applied Mathematicians due its numerous applications in solv-
ing many real life problems. it was birthed as a result of inter-
field research activities in game theory and optimal control..
Thus, many research articles have been devoted to this field and
a lot of results were published ( see for example, [1-16]).

In some of these research works, players move according to
the following differential equations:{

ẋ = a(t)u, x(0) = x0,
ẏ = b(t)v, y(0) = y0

(1)

∗Corresponding author tel. no:
Email addresses: ajbadakaya.mth@buk.edu.ng (Abbas Ja’afaru

Badakaya ), bilyamamman@gmail.com (Bilyaminu Muhammad)

where u(t) and v(t) are control functions of the players which
are either subject to integral or geometrical constraints and a(·), b(·)
are scalar measurable functions.

The problems considered in [1, 7, 12, 13, 18, 19] involve
players’ motion described by the differential equations (1), where
a(t) = b(t) = 1. Whereas, in the problems considered in [6, 8, 9,
10, 14, 17], players move according to the differential equations
(1), where a(t) = b(t) , 1. Problem in which players move ac-
cording to (1), with a(t) , b(t) are investigated in [2]. In this
work, control functions of the players are subject to integral
constraints. Optimal strategies of the players are constructed
and value of the game is found.

In all of the above cited works, only in [1, 8, 10, 18] con-
straints on the state variables are considered. The paper [1] re-
ports study of pursuit problem on a closed convex subset of Rn

with Control functions of players subject to coordinate-wise in-

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Abbas Ja’afaru Badakaya & Bilyaminu Muhammad / J. Nig. Soc. Phys. Sci. 2 (2020) 115–119 116

tegral constraints. Some conditions under which pursuit can be
completed from any position of the players in the given set are
obtained. Moreover, strategies for the pursuers are constructed.

The work [8] is concerned with pursuit problem in which
players control functions are subject to integral constraints. Play-
ers are not allowed to move out of a closed convex subset of Rn.
Optimal time of pursuit is found and optimal strategies for the
players are constructed. Ibragimov and Satimov in [10] studied
pursuit differential game problem on a nonempty convex subset
of Rn. In the game, both pursuers and evaders are not allowed to
leave the given set and their control functions subject to integral
constraints. Sufficient conditions for completion of pursuit are
obtained.

A differential pursuit problem of an evader by finite number
of pursuers on a closed convex set of l2, was studied by Leong
and Ibragimov in [18]. The authors showed that an evading
player cannot avoid an exact contact with any finite number of
pursuing players whose individual resources are less than that
of this evading players.

In the present paper, pursuit differential game problem with
finite number of pursuers and one evader on a nonempty closed
convex subset of l2 is investigated. Control functions of the
pursuers and evader are subject to integral constraints. During
the game players are to stay within a closed convex subset of l2.
Players move according to (1), where a(t) , b(t).

2. Statement of the Problem

Consider the Hilbert space

l2 =

α = (α1,α2, ...,αk..., ) |
∞∑

k=1

α2k < ∞

 ,
with the inner product

(α,β) =
∞∑

k=1

αkβk

and norm ‖α‖ =
(∑
∞
k=1 α

2
k

)1/2
. Let N be closed convex subset

of l2 and define a ball in l2, with center at x0 and radius r by
H(x0, r) := {x ∈ l2 : ‖x − x0‖ ≤ r}. Let C(0,θ; l2) denotes
space of continuous functions f (t) = ( f1(t), f2(t), . . . ) defined
on the interval [0,θ] with absolutely continuous coordinates.
The following definitions are important in the paper:

Definition 1. Let (X,µ1) and (Y,µ2), where µ1 and µ2 are σ-
algebra on X and Y respectively, be a measurable spaces . The
function f : (X,µ1) → (Y,µ2) is Borel measurable if f −1(E) ∈
µ1 for all E ∈ µ2 [20].

Definition 2. Let A be a subset of a Hilbert space X , if every
point of the Hilbert space has exactly one projection onto A,
then A is a Chebyshev set [3].

Definition 3. Let X and Y be normed linear spaces and let T :
X → Y be a linear map, then T is said to be Lipschitz, if there
exists a constant K > 0 such that, for each x ∈ X ‖T x‖ ≤ K‖x‖
[4].

We define a pursuit differential game problem in which count-
able but finite number of pursuers P j, j ∈ J = {1, 2, . . . , m} and
evader E move according to the following equations:{

P j : ẋ j = a(t)u j, x j(0) = x j0,
E : ẏ = b(t)v, y(0) = y0,

(2)

where x j, x j0, u j, y, y0, v ∈ l2, u j = (u j1, u j2, . . . ) is a control pa-
rameter of the pursuer P j and v = (v1, v2, . . . ) is that of the
evader E. Additionally, a(t) and b(t) are scalar measurable func-
tions such that 1 ≤ b(t) ≤ a(t) for all t ∈ [0,θ]. The positive
number θ is denoting duration of the game.

Definition 4. A Borel measurable function u j(·); u j : [0,θ] →
H(0,ρ j) such that(∫

∞

0
||u j(t)||

2dt
)1/2
≤ ρ j, (3)

is called admissible control of the pursuer P j.

Definition 5. A Borel measurable function v(·); v : [0,θ] →
H(0,σ) such that(∫

∞

0
‖v(t)‖2dt

)1/2
≤ σ, (4)

is called admissible control of the evader.

Definition 6. A function U(t, x j, y, v), U j : [0,θ) × l2 × l2 ×
H(0,σ) → H(0,ρ j), such that the system{

ẋ j = U j(t), x j(0) = x0 j,
ẏ = v(t), y(0) = y0,

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

In what follows, we refer the game described by (2) in which the
control functions u(·) and v(·) satisfying (3) and (4) respectively,
as game G1.

Definition 7. Pursuers win the game G1, if there exist pur-
suer’s strategy U j that ensure the equality x j(θ) = y(θ) for some
j ∈ J.

When the pursuer P j and evader E use admissible controls u j(t) =
(u j1(t), u j2(t), . . . ) and v(t) = (v1(t), v2(t), . . . ) respectively, then
from (2) their corresponding motions is given by

x j(t) = (x j1(t), x j2(t), . . . ), y(t) = (y1(t), y2(t), . . . ),
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Abbas Ja’afaru Badakaya & Bilyaminu Muhammad / J. Nig. Soc. Phys. Sci. 2 (2020) 115–119 117

where

x j(t) = (x j1(t), x j2(t), . . . , x jk(t), . . . ), x jk(t) = x j0k+
∫ t

0
a(t)u jk(s)d s;

y(t) = (y1(t), y2(t), . . . , yk(t), . . . ), yk(t) = yk0 +
∫ t

0
b(t)vk(s)d s.

It can easily be shown that xi(·), y(·) ∈ C(0,θ; l2).

Problem. What are the sufficient conditions for pursuer to
win the game G1?

3. Results

To present our result, we need the following notations: ρ2 :=
m∑

j=1

ρ2j and σ j :=
σ
ρ
ρ j, f or j ∈ J. Then, it is easy to see that

σ2j < ρ
2
j and σ

2 =
∑m

j=1 σ
2
j.

The following Lemma is useful in the presentation of our results

Lemma 8. Let 1 ≤ p ≤ ∞. If f, g ∈ Lp(X,µ) then f + g ∈
Lp(X,µ) and ‖ f + g‖ ≤ ‖ f‖p + ‖g‖p [5].

The theorem below gives sufficient conditions for pursuers to
win the game when the players move freely without state con-
straint in the space l2.

Theorem 1. . If ρ2j < σ
2 for all j ∈ J and

m∑
j=1

ρ2j > σ
2 , then

for any initial positions of the players, pursuers win the game
G1.

Proof:
If y0 = x j0, for some j ∈ J, then the proof is trivial. Therefore,
let y0 , x j0, for all j ∈ J. We construct the strategies of the
pursuers as follows:

In the first phase of the game, we allow only one pursuer
to move using a define strategy. Without loss of generality, we
allow the first pursuer to move and others to stay static. That is,
the pursuers to use the strategies define byu1(t) =

y0−x10
a(t)θ1

+
b(t)
a(t) v(t), j = 1

u j(t) = 0, j , 1,
(5)

where x10 and y0 is the initial position of the first pursuer and
the evader respectively; θ1 =

(
||y0−x10||
ρ1−σ1

)2
. Now we show that the

strategy (5) is admissible whenever
∫ θ1

0
||v(t)||2dt ≤ σ1. Indeed,

(∫ θ1
0
‖u1(t)‖

2dt
)1/2

=

(∫ θ1
0

∥∥∥∥∥ y0 − x10a(t)θ1 + b(t)a(t) v(t)
∥∥∥∥∥2 dt

)1/2
≤

(∫ θ1
0

∥∥∥∥∥ y0 − x10a(t)θ1
∥∥∥∥∥2 dt

)1/2
+

(∫ θ1
0

∥∥∥∥∥ b(t)a(t) v(t)
∥∥∥∥∥2 dt

)1/2

≤

(∫ θ1
0

‖y0 − x10‖
2

a2(t)θ12
dt

)1/2
+

(∫ θ1
0
‖v(t)‖2dt

)1/2
≤
‖y0 − x10‖

θ1

(∫ θ1
0

1
a2(t)

dt
)1/2

+ σ1

≤
‖y0 − x10‖

θ1

(∫ θ1
0

dt
)1/2

+ σ1

=
‖y0 − x10‖

θ
1/2
1

+ σ1

=
‖y0 − x10‖((
||y0−x10||
ρ1−σ1

)2)1/2 + σ1
≤ ρ1 −σ1 + σ1 = ρ1.

Now if the pursuer P1 uses the strategy (5) then,

x1(θ1) − y(θ1) = x10 − y0 +
∫ θ1

0
a(t)u1(t)dt −

∫ θ1
0

b(t)v(t)dt

= x10 − y0 +
∫ θ1

0
a(t)

(
y0 − x10
a(t)θ1

+
b(t)
a(t)

v(t)
)

dt

−

∫ θ1
0

b(t)v(t)dt

= x10 − y0 +
∫ θ1

0

y0 − x10
θ1

dt +
∫ θ1

0
b(t)v(t)dt

−

∫ θ1
0

b(t)v(t)dt

= x10 − y0 +
y0 − x10
θ1

θ1

= x10 − y0 + y0 − x10 = 0.

This means that x1(θ1) = y(θ1) and implies that pursuers win the
game. However, if x1(t) , y(t) for all t ∈ [0,θ1], then energy
expanded by the evader in the time interval [0,θ1] is more than
the energy of the first pursuer. That is∫ θ1

0
‖v(t)‖2dt > ρ21.

But from the fact that ρ21 > σ
2
1, then we have∫ θ1

0
‖v(t)‖2dt > σ21. (6)

If the first pursuer cannot win the game for the pursuers then
the game will proceed into the second phase. In the second
phase, the second pursuer will move using the strategy similar
to (5) and the remaining pursuers P j, j = 1, 3, 4, . . . , m remain
static. In general, during the kth phase of the game, pursuers
P j, j = 1, 2, ...k − 1, k + 1, . . . , m, remain static and only the kth

pursuer moves. That is, the strategies of the pursuers in the kth

phase of the game are given byuk(t) =
y(τk−1 )−xk0

a(t)θk
+

b(t)
a(t) v(t) j = k

u j(t) = 0, j , k;
(7)

117



Abbas Ja’afaru Badakaya & Bilyaminu Muhammad / J. Nig. Soc. Phys. Sci. 2 (2020) 115–119 118

where t ∈ [τk−1,τk], k = 1, 2, . . . , m,τ0 = 0 ; τk =
k∑

j=1

θ j, and

θk =
(
||y(τk−1 )−xk0||

ρk−σk

)2
. Again, for any admissible control of the

evader v(·) the constructed strategy (5) is admissible and pur-
suers win the game.

Now, if in each phase of the game the equation x j(t) = y(t)
does not hold for some t ∈ [0,τm], then the following inequali-
ties must hold∫ τ1

0
‖v(t)‖2dt >σ21,

∫ τ2
τ1

‖v(t)‖2dt > σ22 , ...,

+

∫ τm
τm−1

‖v(t)‖2dt > σ2m.

Consequently,∫
∞

0
‖v(t)‖2dt ≥

∫ τ1
0
‖v(t)‖2dt +

∫ τ2
τ1

‖v(t)‖2dt+

. . . +

∫ τm
τm−1

‖v(t)‖2dt > σ21 + σ
2
2 + . . . + σ

2
m = σ

2.

This contradicts (4). Therefore, we must have the equality x j(t) =
y(t) holding for some j ∈ J and for some t ∈ [0,τm]. This
completes the proof. Now, consider the game G1 in which
x j, x0, y, y0 ∈ N ⊂ l2 and all player cannot move out of the
closed convex set N. Then we have the following theorem:

Theorem 2. If ρ2j < σ
2 for all j ∈ J and

∑m
j=1 ρ

2
j > σ

2, then for
any initial positions of the players, pursuers win the game G1.

Proof: For purpose of the proof of this theorem, we in-
troduce m number of dummy pursuers P̄ j, j = 1, ..., m which
moves according to the following their equations

P̄ j : ẇ = a(t)ū j(t), j = 1, ..., m; w j(0) = w j0. (8)

where the controls ū j is such that∫
∞

0
||ū j(t)||

2dt ≤ ρ2j.

The dummy pursuers has no restriction in their movements.
That is, they can move outside the set N. Therefore, dummy
pursuers can win the game if we define the strategy of each of
the dummy pursuer P̄ j, j ∈ I (same with the pursuers’ strategies
define in the proof of theorem (1) as follows:

ū j(t) =
y(τ j−1) − z j0

a(t)θ j
+

b(t)
a(t)

v(t), τ j−1 ≤ t ≤ τ j;

ūk(t) = 0,∀k = 1, 2, ..., j − 1, j + 1, ..., m.

Define by FN (x) the projection of a point x ∈ l2 onto N :

|x − FN (x)| = min
y∈N
|x − y|.

Since N is closed convex subset of l2, it follows that N is a
Chebyshev set and the inequality

|FN (x) − FN (y)| ≤ |x − y| (9)

holds for any x, y ∈ l2, that is FN (·) is Lipschitz continuous
with 1. The operator FN (·) associates each of the absolute con-
tinuous function w j(t), 0 ≤ t ≤ τm, to an absolute continuous
function x j(t). That is, FN : l2 → N, such that

FN (w j(t)) =

x j(t) i f w j(t) , N,w j(t) = x j(t), i f w j(t) ∈ N, (10)
where t ∈ [0,τm]. We define the strategies of the real pursuers
by

u j(t) =

ū j(t), i f w j(t) ∈ N1
a(t) Ḟ(w j(t)), i f w j(t) < N.

(11)

We now show that this strategy ensure win for the pursuers and
is admissible. By theorem (1) we can infer that the equality
wi(t∗) = y(t∗) holds for some time t∗ ∈ [0,τm], for an index i ∈ I.
Since y(t) ∈ N, then using (10) yields xi(t∗) = FN (wi(t∗)) =
wi(t∗) = y(t∗).

Lastly, we show the admissibility of the strategy (11). In-
deed, if w j(t) ∈ N then∫

∞

0
‖u j(t)‖

2dt =
∫ τ j
τ j−1

‖ū j(t)‖
2dt ≤ ρ2j.

In the other hand, we use the fact that FN (w j(·)) = x j(·) ∈ N; N
is closed convex and the inequality (9) to deduce that

‖FN (w j(t + h)) − FN (w j(t)‖ ≤ ‖w j(t + h)) − w j(t)‖.

In view of this inequality and for w j(t) < N, we have∫
∞

0
‖u j(t)‖

2dt =
∫ τ j
τ j−1

1
a2(t)

‖Ḟ(w j(t))‖
2dt

=

∫ τ j
τ j−1

1
a2(t)

∥∥∥∥∥∥limh→0 FN (w j(t + h)) − FN (w j(t)h
∥∥∥∥∥∥

2

dt

=

∫ τ j
τ j−1

1
a2(t)

lim
h→0

‖FN (w j(t + h)) − FN (w j(t)‖2

h2
dt

≤

∫ τ j
τ j−1

1
a2(t)

∥∥∥∥∥∥limh→0 w j(t + h) − w j(t)h
∥∥∥∥∥∥

2

dt

=

∫ τ j
τ j−1

1
a2(t)

‖ẇ j(t))‖
2dt =

∫ τ j
τ j−1

1
a2(t)

‖a(t)ū j(t)‖
2dt

=

∫ τ j
τ j−1

‖ū j(t)‖
2dt ≤ ρ2j.

118



Abbas Ja’afaru Badakaya & Bilyaminu Muhammad / J. Nig. Soc. Phys. Sci. 2 (2020) 115–119 119

4. Illustrative Example

Consider the game in which five dynamical object(pursuers)
versus one dynamical object(evader) confined in a circular arena
with radius 3 and whose dynamics described by{

P j : ẋ j = (2 + t)u j, x j(0) = x j0, j = 1, 2, 3.
E : ẏ = (1 + t)v, y(0) = y0,

where x10 = (1, 0, 0, . . . ); x20 = (0, 1, 0. . . . ); x30 = (0, 0, 1, . . . );
y0 = (0, 0, 0, . . .); ρ1 = 3, ρ2 = 4, ρ3 = 5,σ = 7. Observe that
1 < (1 + t) < (2 + t); ρ2j < σ

2, ∀ j = 1, ..., 3 and
∑3

j=1 ρ
2
j > σ

2.

This means that hypothesis of our theorems is satisfied, there-
fore pursuers win this game.

5. Conclusion

We studied pursuit problem in a closed convex subset of a
Hilbert space in which energy of each of pursuer is less than
that of the evader. We were able to show that pursuers win the
game when the total energy of the pursuers is greater than that
of the single evader. Furthermore, pursuers’ wining time is at
least min

j∈J
θ j and at most τm.

Acknowledgments

The valuable comments and observations of the reviewers
of this article are acknowledged. Indeed, their contributions
added quality to the article.

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