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Differential Equations and Dynamical Systems
https://doi.org/10.1007/s12591-020-00545-5
1 3
ORIGINAL RESEARCH
A Dierential Game Problem ofMany Pursuers andOne
Evader intheHilbert Space
𝓁2
JewaiduRilwan1,4· PoomKumam2 · GafurjanIbragimov3·
AbbasJa’afaruBadakaya4· IdrisAhmed1
© Foundation for Scientific Research and Technological Innovation 2020
Abstract
In this paper, we investigate a differential game problem of multiple number of pursuers
and a single evader with motions governed by a certain system of first-order differential
equations. The problem is formulated in the Hilbert space
𝓁2,
with control functions of
players subject to integral constraints. Avoidance of contact is guaranteed if the geometric
position of the evader and that of any of the pursuers fails to coincide for all time t. On the
other hand, pursuit is said to be completed if the geometric position of at least one of the
pursuers coincides with that of the evader. We obtain sufficient conditions that guarantees
avoidance of contact and construct evader’s strategy. Moreover, we prove completion of
pursuit subject to some sufficient conditions. Finally, we demonstrate our results with some
illustrative examples.
Keywords Differential game· Integral constraint· Hilbert space· Avoidance of contact·
Pursuit
Mathematics Subject Classication Primary 91A23· Secondary 49N75
* Poom Kumam
poom.kum@kmutt.ac.th
Jewaidu Rilwan
jrilwan.mth@buk.edu.ng
1 KMUTTFixed Point Research Laboratory, Department ofMathematics, Room SCL 802
Fixed Point Laboratory, Science Laboratory Building, Faculty ofScience, King Mongkut’s
University ofTechnology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru,
Bangkok10140, Thailand
2 Center ofExcellence inTheoretical andComputational Science (TaCS-CoE), Science Laboratory
Building, Faculty ofScience, King Mongkut’s University ofTechnology Thonburi (KMUTT), 126
Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok10140, Thailand
3 Institute forMathematical Research andDepartment ofMathematics, Faculty ofScience (FS),
Universiti Putra Malaysia, Serdang, Selangor43400, Malaysia
4 Department ofMathematical Sciences, Bayero University, Kano, Nigeria
Differential Equations and Dynamical Systems
1 3
Introduction
In the mid twentieth century, the need for solving conflict problems brought forth an
area of research in mathematics known as Differential game. It was pioneered by Isaacs
[27]. Thereafter, fundamental results have been obtained and published in books such as
Berkovitz [8], Friedman [14], Krasovskii and Subbotin [29, 30], Lewis [33], Petrosyan
[39] and Pontryagin [40].
Evasion differential game problem is a class of differential game problem that
involves finding sufficient conditions for avoidance of contacts of one (or more)
dynamic object(s) from many other objects. This class of game problem amongst others
has received attention from researchers and many results were obtained (see, e.g. [4, 6,
15, 34, 41, 42] and some references therein).
Pontryagin and Mischenko in [41], studied problem of evasion in linear differential
game and obtained interesting results for the case of finite dimensional state spaces on
an infinite time interval with geometric constraints. Azimov [6] and Mezentsev [34]
investigated and extended this result to the case of integral constraints. Thereafter, Gam-
krelidze and Kharatashvili [15] developed a new method of solving quasilinear evasion
differential game problems.
Among the works dedicated to differential games of several players, equation of
motions described by
integral or geometric constraints on players control parameters, where a(t) is a scalar func-
tion defined on some intervals, were investigated in [1–3, 9, 12, 13, 16, 20–25, 28, 31, 35,
43, 44].
Alias et. al [3] investigated evasion differential game problem of countably many
pursuers and countably many evaders in the Hilbert space
𝓁2
with
a(t)=1
and integral
constraints imposed on players control functions. They proved that evasion is possible
under the assumption that the total resource of evaders exceeds (or equals) that of the
pursuers and initial positions of all the evaders are not limit points for initial positions
of the pursuers.
In [22] and [23], pursuit-evasion differential game described by (1) was studied in
the Hilbert space
𝓁2
with geometric and integral constraints on control functions of
the players respectively, where
a(t)=𝜃−t,
and
𝜃
is the duration of the game. In both
papers, sufficient conditions for completion of pursuit as well as value of the game were
obtained.
Ibragimov and Satimov [25] considered differential game problem of many pursuers
and many evaders described by (1) on a nonempty convex subset of
ℝn
where all players
are confined within the convex set. In the paper, a(t) is a scalar measurable function satis-
fying some conditions and the control functions of players are subjected to integral con-
straints. It was proven that pursuit can be completed if the total resources of the pursuers is
greater than that of the evaders.
The work in [43] deals with the case when all players are endowed with equal dynamic
capabilities with geometric constraints, where
a(t)=1
. The pursuit problem was solved
with the assumption that the evader’s initial position must lie in the convex hull of that of
the pursuers, otherwise evasion is possible. The results in [43] was later adopted in devel-
oping an efficient method of resolving functions for a linear group pursuit problem in [44].
(1)
{
̇xj(t)=a(t)uj(t),xj(0)=x
0
j,j=1, 2,
…
̇y(t)=a(t)v(t),y(0)=y0,
Differential Equations and Dynamical Systems
1 3
Reducing control problems for parabolic and hyperbolic partial differential equations to
infinite system of differential equations
by using decomposition method based on Fourier expansion was proposed by
Chernous’ ko [11], where
zk
,𝜔
k
∈ℝ
1
,𝜔
k
,k=1, 2,
…
are control parameters,
𝜆k,0<𝜆
1≤𝜆2≤⋯≤𝜆k≤⋯
→
∞
are generalized eigenvalues of the elliptic operator
The concept has been applied in studying differential game problems for systems described
by parabolic equations but reduced to the system (2) in Ibragimov [18, 19], Ichikawa [26],
Osipov [36], Satimov and Tukhtasinov [45].
Ibragimov [18] adopted the proposed decomposition method [11] in solving an optimal
pursuit problem described by (2) with integral constraints on controls of players. A detailed
analysis and solution of the problem is presented in explicit form.
Ibragimov and Risman [21] considered pursuit and evasion differential game described
by (2) in a certain Hilbert space they introduced as
𝓁2
r
with integral constraints on control
functions of the players, where the space
with inner product and norm defined by
respectively, for a given fixed number r and monotonically increasing sequence of positive
numbers
{𝜆k}k∈ℕ
. The evasion problem was studied on some disjoint subset (octants) of
𝓁2
r
and solved with the assumption that the total resources of the pursuers is less than the
evader’s while the pursuit problem was solved in contrary to this assumption.
There are several other techniques of analyzing pursuit-evasion scenarios involving
multiple pursuers and one evader proposed in the literature, for example; situations involv-
ing attackers, defenders and one evader are analyzed via the linear quadratic differential
game approach in [10, 37], scenarios where the multiple pursuers and evader are restricted
within a bounded domain are investigated and analyzed via the geometric approach based
on dynamic Voronoi diagram in [7, 17, 46].
Motivated by the results in [21], we study the same problem for a differential game
described by (1) in the Hilbert space
𝓁2
(coinciding with 𝓁
2
r
and r=0
)
; which amounts to
solving the pursuit version of the problem in [3] and evasion version of the problem con-
sidered in [25]. More precisely, we investigate a differential game problem of avoidance
of contacts (evasion) and completion of pursuit described by (1), but reduced to (2) in the
Hilbert space
𝓁2
. With all players control parameters subject to integral constraint, we will
show that if the total energy resources of the pursuers is less than that of the evader, then
avoidance of contact is guaranteed. Furthermore, we obtain sufficient condition for com-
pletion of pursuit. It should be noted that in this work, a scalar non-negative function a(t) is
(2)
̇zk(t)+𝜆kzk=𝜔k,k=1, 2, …
Az =−
n
∑
i,j=1
𝜕
𝜕xi
(
aij(x)𝜕z
𝜕xj
).
𝓁
2
r∶=
{
𝛼=(𝛼1,𝛼2,…) ∶
∞
∑
k=1
𝜆r
k𝛼2
k<∞
},
𝛼,𝛽
r=
∞
k=1
𝜆r
k𝛼k𝛽k,
𝛼
=
∞
k=1
𝜆r
k𝛼2
k
1∕2
Differential Equations and Dynamical Systems
1 3
introduced into the equation of motion of the players (similar to the problem considered in
Ibragimov and Satimov [25] in a finite dimensional space but in this paper, we will adopt
a different solution concept with a mild condition on a(t)). Also, the case considered in [3,
21] is a special case, where
a(t)=1.
This paper is organized as follows. Statement of the problem, notations and some basic
definitions are given in Sect.2. Section3 constitutes the main results of this paper. It com-
prises three subsections, that is, Sects.3.1,3.2 and3.3. In the first two subsections, we pro-
vide sufficient conditions for avoidance of contacts (Theorem1) and completion of pursuit
(Theorem 2), respectively. In addition to Sect.3.1, we include an interesting subsection
where estimation of distances between the evader and the pursuers is obtained. We illus-
trate our results with some examples in Sect.3.3. Section4 concludes the paper.
Statement oftheProblem
Consider the Hilbert space
whose inner product and norm are respectively defined by
Let the motions of countably many pursuers
Pj
,
j=1, 2, …
and an evader E be described
by the equations
where xj(t),x
0
j
,uj(t),y(t),y
0
,
v(t)∈𝓁2
;
uj(t)=(uj1(t),uj2(t),…)
and
v(t)=(v1(t),v2(t),…)
are control parameters of the pursuer
Pj
and evader E respectively. Without any loss of
generality we assume that the scalar measurable function
a(t)≥0
is not identically equal to
zero on any interval
t1<t<t2
with
t1≥0
, and moreover a(t) is such that for each
n=1, 2,
the integral
Suppose that
uj(⋅),v(⋅)∈L2(t0,T;𝓁2)
, where
T>0
is the termination instant of the game
(not necessarily fixed with respect to pursuit problem).
Denition 1 A function
uj(⋅)=(uj1(⋅),uj2(⋅),…)
with Borel measurable coordinates
ujk(⋅),k=1, 2, …,
satisfying the inequality
𝓁
2=
{
𝛼=(𝛼1,𝛼2,…) ∶
∞
∑
k=1
𝛼2
k<∞
},
𝛼,𝛽
=
∞
k
=1
𝛼k𝛽k,
𝛼
=
∞
k
=1
𝛼2
k
1
∕
2
,𝛼,𝛽∈𝓁2
.
(3)
{
Pj∶̇xj(t)=a(t)uj(t),xj(t0)=x
0
j,
E∶̇y(t)=a(t)v(t),y(t0)=y0,x0
j
≠y0
,
∫t
2
t1
an(t)dt →∞as t2→∞
.
Differential Equations and Dynamical Systems
1 3
where
𝜌j,j=1, 2, …
are given positive numbers, is called admissible control of the
jth
pursuer.
Denition 2 A function
v(⋅)=(v1(⋅),v2(⋅),…)
with Borel measurable coordinates
vk(⋅),
satisfying the inequality
where
𝜎
is a given positive number, is called admissible control of the evader.
Remark 1 Note that the players dynamics (3) together with the integral constraints (4) (or
(5)) allows the pursuer
Pj
(evader E, respectively) to move arbitrary far from its initial posi-
tion. That is, if the control resource constraint constant is
𝜌j
, then applying a constant con-
trol
uj
with the magnitude
‖
u
j‖≡
𝜌
j
∕
√
T during a time interval with length T, the pursuer
Pj
spends its control resource totally and moves for distance
𝜌j√
T
.
Thus, the distance can
be made arbitrary large.
Whenever the players’ admissible controls
uj(⋅)
and
v(⋅)
are chosen, the corresponding
motions of the
jth
pursuer and evader (solutions of equation (3)) are given by
respectively.
Denition 3 Suppose that
A function
z(t)=(z1(t),z2(t),…),t0≤t≤T,
is called the solution of the system of
equations
if:
(1) each coordinate
zk(t)
is absolutely continuous function and almost everywhere on
[t0,T]
satisfies (7);
(2)
z(⋅)∈C(t0,T∶𝓁2)
where
C(t0,T∶𝓁2)
is the space of continuous functions
z(t)=(z1(t),z2(t),…),t0≤t≤T
with values in
𝓁2.
(4)
�
T
t
0
∥uj(t)∥
2dt
≤
𝜌j
2,
uj(t)
=
∞
k=1
u2
jk(t)
1∕2
,
(5)
�
T
t0
∥v(t)∥
2dt
≤
𝜎2,
v(t)
=
∞
k=1
v2
k(t)
1∕2
,
(6)
x
j(t)=(xj1(t),xj2(t),…),xjk(t)=x0
jk(t)+∫
t
t0
a(s)ujk(s)ds
,
y(t)=(y1(t),y2(t),…),yk(t)=y0
k(t)+∫t
t0
a(s)vk(s)ds
,
w
(⋅)=(w
1
(⋅),w
2
(⋅),…) ∈ L
2
(t
0
,T∶𝓁
2
),(z
0
1
,z
0
2
,…) ∈ 𝓁
2.
(7)
̇
zk
=a(t)w
k
(t),z
k
(t
0
)=z
0
k
,k=1, 2, …
,
Differential Equations and Dynamical Systems
1 3
Denition 4 A function
is called strategy of the evader if for any admissible controls of the pursuers
uj(⋅)
the evad-
er’s control
v(t)=V(u1(t),u2(t),…),[t0,T],
is admissible and system (3) has a unique
solution after substitution of the players’ controls into it.
Denition 5 Avoidance of contact is said to be guaranteed in the game described by (3)-
(5) with initial positions
{
x
0
1
,x
0
2
,…,x
0
m
,…,y
0
},x
0
j
,y
0
∈𝓁
2
, if there exists a strategy V of
the evader such that for all admissible controls of the pursuers
uj(⋅),j=1, 2, …
, the rela-
tion
xj(t)≠y(t)
holds, for all
t∈[t0,T]
.
Before stating the research problem, let introduce a dummy variable
zj(t)=y(t)−xj(t)
so that the players dynamics (3) reduces to
Therefore, the system (3) is represented by the infinite system of differential equations of
the form
where
w
j(t)=(wj1(t),wj2(t),…),z
0
j
=(z
0
j1
,z
0
j2
,…)
≠
0, wji =vi−uji,z
0
ji
=y
0
i
−x
0
ji
,i,j=1, 2,
…
.
By this notation, the solution to the system (8) becomes
with
zj(t)
satisfying the conditions in Definition (3),
uj(t)
and v(t) satisfying (1) and (5),
respectively, for all
j=1, 2, ….
We also denote that
Remark 2 In view of the properties of the function
a(⋅)
, it is worth noting that the integral
An(t0,t)>0
if
t>t0
and
An(t0,t)<0
if
t<t0
for
n=1, 2
.
Remark 3 The problem of finding sufficient conditions for completion of pursuit in a
pursuit problem described by the system (2) often requires conditions on the parameters
𝜆k,k=1, 2, …
. However, by considering the system (8) with a generalized scalar func-
tion
a(⋅)
, these conditions have been circumvented, except the condition
𝜌>𝜎
. That is, we
examine the case where
The fact that (10) implies the existence of
m>0,
such that
V(u1,u2,…),V∶𝓁2×𝓁2×⋯
→
𝓁2,
̇
z
j(t)=a(t)(v(t)−uj(t)),zj(t
0
)=y
0
−x
0
j.
(8)
̇
z
j(t)=a(t)wj(t),zj(t0)=z
0
j
,j=1, 2, …
,
(9)
z
j(t)=(zj1(t),zj2(t),…),zji(t)=z0
ji +∫
t
t
0
a(s)wji(s)ds
,
An(t0,t)=�
t
t0
an(s)ds;t
≠
t0,n=1, 2 ; 𝜌2=𝛴∞
j=1𝜌2
j<∞
.
(10)
∞
∑
j=1
𝜌2
j>𝜎
2
.
Differential Equations and Dynamical Systems
1 3
motivated the following definitions:
Denition 6 A function
uj
(t,z
j
(t),𝜎(t),v(t)),j={1, 2, …,m},u
j
∶[t0,T]×𝓁2×[0, 𝜎
2]
×𝓁2
→
𝓁2,
is called a strategy of the
jth
pursuer
j∈{1, 2, …,m}
if there exists a unique
absolutely continuous vector-function
(𝜎(⋅),zj(⋅)),zj(⋅)∈C(t0,T∶𝓁2),t∈[to,T],
satisfy-
ing the system
at
uj=uj(t,zj(t),𝜎(t),v(t))
and
v=v(t)
almost everywhere on [0,T], where
v(t),t0≤t≤T,
is an arbitrary admissible control of the evader. The strategy of the
jth
pursuer is called
admissible if each control formed by this strategy is admissible.
To guarantee the existence and uniqueness of the solution to the system (11), we assume
Lipschitzian (or even linear) dependence of the pursuers’ strategies
uj(t,zj(t),𝜎(t),v(t))
on
the phase coordinate
zjk(t),k=1, 2, …
.
Note that we have only defined strategies of the first m pursuers, for
j>m
, we set
ujk(t)=0, k=1, 2, …
.
Denition 7 Pursuit is said to be completed in the game described by (8) not later than
time
t(z0)≥t0
, from the initial positions
if there exists admissible strategies
u1=u1(t,z1,𝜎,v),u2=u2(t,z2,𝜎,v),…,um=um(t,zm,𝜎,v),
of pursuers
Pj,j=1, 2, …,m,
such that for any admissible control
v=v(t)
of
the evader at some
j∈{1, 2, …,m},
the solution
zj(t)
of the system (8) with
uj
=u
j
(t,z
j
,𝜎,v),v=v(t),t
0≤
t
≤
t(z
0)
satisfies the equality
zj(𝜏)=0
, for some
𝜏∈[t0,t(z0)].
Remark 4 In view of Remark1, we further emphasize here that, if the pursuer
Pj
has con-
trol resource equal
𝜌j
, and that of the evader E equal to
𝜎
, and
𝜌j>𝜎
, then under discrimi-
nation of the evader (it declares its instantaneous control to the pursuer), the pursuer
Pj
can complete pursuit of the evader E for any of their initial mutual location by the parallel
approach strategy. The parallel approach strategy (also called
𝜋
-strategy and characterized
with the property that straight lines through positions of the pursuer and the evader are par-
allel) is an effective strategy that has been applied in solving simple motion (i.e.
a(t)=1
)
pursuit-evasion differential game problems of many pursuers one evader (see e.g. [5, 32,
38])
m
∑
j=1
𝜌2
j>𝜎
2
(11)
�
̇𝜎 (t)=−
‖
v(t)
‖2
,𝜎(t0)=𝜎
2
,
̇zjk (t)= a(t)wk(t),zjk(t0)=z0
jk,k=1, 2,
…
z0
={z
0
1
,z
0
2
,…},(z
0
j
=z
0
j1
,z
0
j2
,…),z
0
j
∈𝓁2,j=1, 2, …,m
,
Differential Equations and Dynamical Systems
1 3
Research Questions: In the game (3)–(5), find sufficient conditions for
i avoidance of contact,
ii completion of pursuit.
Main Results
In this section, we present the main results of the research work.
Avoidance ofContact Problem
The following theorem gives sufficient conditions for avoidance of contact in the game
described by (8):
Theorem1 If
𝜎>𝜌,
then avoidance of contact is guaranteed in the game described by (8),
for any initial position
z0
={z
0
1
,z
0
2
,…,z
0
m
,…},z
0
j
∈𝓁2,j=1, 2, …
.
Proof 1. Construction of evader’s Strategy
We first define octants of the space
𝓁2
as follows:
where
I⊂ℕ
and
J=ℕ⧵I.
By this definition (12), the intersection of any two distinct
octants is null. Since the cardinality of the collection of all subsets of any set is greater than
the cardinality of the set itself, then the cardinality of the set of octants of the space
𝓁2
, that
is
|2ℕ|
, is greater than
|ℕ|
. In view of this and the fact that the set of points
z0
j
,j=1, 2, …
,
is countable, then there exists an octant that do not contain the points
z0
j
,j=1, 2,
…
. Due
to this fact and without loss of generality, we assume that the points
z0
j
,j=1, 2, …
,
are not
contained in the octant defined by:
This implies
z0
j
is a vector in
𝓁2,
with at least one nonnegative coordinate, for each
j∈ℕ.
Let the evader use the strategy
v(⋅)=(v1(⋅),v2(⋅),…)
with
where
I
i={j∶z
0
jk
<0, k=1, 2, …,i−1; z
0
ji ≥
0}
.
If
l∈Ii
then
z0
l
=(z
0
l1
,z
0
l2
,…,z
0
li
,
…)
has the coordinates
z0
lr
<
0,
for all
r=1, 2, …,i−1.
According to the strategy (14) which is similar to the parallel approach method, for each
coordinate, the evader applies a control which guarantees keeping its distance on this coor-
dinate from all pursuers, which can move towards the evader on this coordinate at the cur-
rent instant, plus some addition. This addition exploits the difference between the control
(12)
X
(I,J)=
{
z=(z
1
,z
2
,…) ∶ z∈𝓁
2
,z
i
>0, i∈I,z
k
<0, k∈J
},
(13)
X(�,ℕ)=
{
z=(z
1
,z
2
,…) ∶ z∈𝓁
2
,z
i
<0, i∈ℕ
}.
(14)
v
i(t)=
j∈Ii
u2
ji(t)+ a(t)(𝜎2−𝜌2)
2iA1(t0,T)
1∕2
,t0
≤
t
≤T
0, t>T,
Differential Equations and Dynamical Systems
1 3
resource of the evader and the total control resource of the pursuers and allows the evader
even to increase the distance.
Furthermore, the set
{
I
r
∶r=1, 2, …
}
is a collection of pairwise disjoint sets. That
is,
Im∩In=�
, for
m≠n.
We prove this claim by contradiction. Suppose
p∈Im∩In
and
m≠n
. Without loss of generality, let
m<n
. We have
The second line in (15) implies that
zpm <0
, this is in contradiction with the first line.
Hence, the claim follows. That is,
Im∩In=�
, if
m≠n.
We now show that the evader’s strategy (14) is admissible. That is, it satisfies (5) as
follows
Thus,
2. Avoidance of Contact
Lastly, we show that avoidance of contact is possible for any given initial positions of
the players
z0
={z
0
1
,z
0
2
,…,z
0
m
,…},z
0
j
∈𝓁2,j=1, 2, …
.
That is, we show that
xj(t)≠y(t)
holds, for all
t∈[t0,T],j=1, 2, ….
To achieve our goal, it suffice to show
zj(t)≠0,
for
all
t∈[t0,T],j=1, 2, ….
We take an arbitrary point
zp(t),p∈Ii,
where the index
i∈ℕ
is
chosen in such a way that
z0
p
has coordinates
z0
pk
<0, k=1, 2, 3, …,i−
1
and
zpi ≥0
. It is
also easy to see that
because
u2
pi
(s
)
is among the considered
u2
ji
(s)
.
In view of (9), (14) and (16), we have
(15)
z0
pl <0, l=1, 2, 3, …,m−1; zpm
≥0
z0
pl
<0, l=1, 2, 3, …,n−1; zpn ≥
0.
�T
t0v(s)2ds =�
T
t0
∞
i=1
v2
i(s)ds
=�T
t0
∞
i=1
j∈Ii
u2
ji(s)+ a(s)(𝜎2−𝜌2)
2iA1(t0,T)ds
≤�T
t0
∞
i=1∞
j=1
u2
ji(s)+ a(s)(𝜎2−𝜌2)
2iA1(t0,T)ds
=
∞
j=1
�T
t
0
∞
i=1
u2
ji(s)ds +(𝜎2−𝜌2)
A1(t0,T)�T
t
0
a(s)ds
∞
i=1
1
2i
.
�T
t0
‖
v(s)
‖
2ds
≤∞
�
j=1
𝜌2
j+𝜎2−𝜌2=𝜎2
.
(16)
∑
j∈Ii
u
2
ji(s)
≥
u
2
pi(s)
,
Differential Equations and Dynamical Systems
1 3
Thus,
zpi(t)>0
for all
t≥t0
. Consequently,
xpi(t)<yi(t),t0≤t≤T.
This implies that the
distance
‖zj(t)‖≠0
for all
j=1, 2, …,t0≤t≤T
. Estimation of this distance is further
discussed in the subsequent subsection.
3. Estimation of distances between the evader and the pursuers
By virtue of the evader’s strategy (14), we estimate the distance
‖zj(t)‖
of the evader
from the pursuers
Pj,j=1, 2, …
on the time interval
[t0,T]
as follows:
Since
zj(t)≠0
for all
j=1, 2, …,
let
xji(t)<yi(t)
for all
i,j=1, 2, …,t0≤t≤T
and
‖
z
0
j‖
denotes the initial distance of the evader from the
jth
pursuer.
Then, by Cauchy–Schwartz inequality
On the other hand, since
‖zj(t)‖≥�zji(t)�
for any fixed
j=1, 2, …
and
then
Set
(17)
z
pi(t)=z0
pi +�
t
t0
a(s)wpi(s)ds
=z0
pi +�t
t0
a(s)(vi(s)−upi(s))ds
≥�t
t0
a(s)
j∈Ii
u2
ji(s)+ a(s)(𝜎2−𝜌2)
2iA1(t0,T)1∕2
−upi(s)
ds
≥�t
t0
a(s)u2
pi(s)+ a(s)(𝜎2−𝜌2)
2iA1(t0,T)1∕2
−upi(s)ds
>�t
t0
a(s)
u2
pi(s)
1∕2
−upi(s)
ds =0.
(18)
‖
zj(t)‖=
�
�
�
�
�
z0
j+�
t
t0
a(s)wj(s)ds
�
�
�
�
�
≥‖z0
j‖−�
�
�
�
��t
t0
a(s)v(s)ds�
�
�
�
�
−�
�
�
�
��t
t0
a(s)uj(s)ds
�
�
�
�
�
≥
‖
z0
j
‖
−𝜎√A2(t0,T)−𝜌√A2(t0,T)
≥
‖
z0
j‖
−2𝜎
√
A2(t0,T).
z
ji(t)=z0
ji +�
t
t
0
a(s)wji(s)ds =z0
ji +�
t
t
0
a(s)(vi(s)−uji(s))ds
≥
z0
ji
,
(19)
‖
zj(t)
‖≥�
z
0
ji�.
(20)
d
j=
z0
js
,if
z0
j
≤
2𝜎
A2(t0,T),
min{
z0
js
,
z0
j
−2𝜎
A2(t0,T)}, if
z0
j
>2𝜎
A2(t0,T)
,
Differential Equations and Dynamical Systems
1 3
for some
s=1, 2, …
. Then in view of (18) and (19), we have
‖zj(t)‖≥dj,t0≤t≤T.
That
is, the value
dj
is the smallest distance the evader can preserve from the
jth
pursuer in the
time interval
[t0,T].
This completes the proof of the theorem.
◻
Pursuit Problem
Here, we provide a theorem which constitutes sufficient conditions for completion of pur-
suit as well as its proof.
Theorem2 Suppose that
∑m
j=1
𝜌
2
j
>𝜎
2
and
A2(t0,t)
→
+∞
as
t
→
+∞,
then pursuit can
be completed for the finite time
t=t(z0)
from any initial positions of players in the game
described by (8).
Proof Let
𝜎
j=
𝜎𝜌j
𝜌
, where
𝜌
=(𝜌
2
1
+…+𝜌
2
m
)
1∕2
and
Roughly put, the value
Fj(𝜏,t)
(which actually depends also on a point
zj
) is the square of
the resource necessary to guide an object with dynamics (3) from the position
(𝜏,zj)
to the
position (t,0). That is, it denotes the amount of control resources to be spent by the
jth
pur-
suer to capture the evader at any given time instant from some of their initial positions if
the evader will stay in its place.
For a fixed positive real number
𝜏
, we claim that the function (21) is endowed with the
following properties:
(i)
Fj(𝜏,t),t>𝜏,
is a decreasing function of t;
(ii)
Fj(𝜏,t)
→
+∞
as
t→𝜏+
(that is, t approaches
𝜏
from the right);
(iii)
Fj(𝜏,t)
→
0
as
t→+∞.
Property (i) follows directly from the fact that each term of the series
Fj(𝜏,t)
is a decreas-
ing function for all
t>𝜏
. To prove the second property, recall that
zj(⋅)∈C(t0,t∶𝓁2)
,
which implies, the series
∑∞
k=1z
2
jk
(𝜏)<∞
.
Then, in view of the fact that
A−1
2
(𝜏,t)→
+∞
as
t→𝜏+
, we must have
Fj(𝜏,t)
→
+∞
as
t→𝜏+
. Using similar arguments in the proof
of property (ii) along with hypothesis of the theorem, we have
Fj(𝜏,t)
→
0
as
t→+∞
, i.e.,
property (iii) holds.
Hence, the equation
with
𝜌1>𝜎
1
, has a unique solution
t=𝜃1
. Moreover, since
F1(t0,t)<0
for
t<t0
(in view
of Remark 1), then the root
𝜃1
exists only in the semiaxis
t>t0
, where
𝜃1
is the instant
when the first pursuer will capture the evader if the evader will spent control resource
𝜎1
for
avoidance from the first pursuer.
(21)
F
j(𝜏,t)=
∞
∑
k=1
z
2
jk(𝜏)
A
2
(𝜏,t),t>𝜏,j=1, 2, …,m
.
F1∕2
1
(t
0
,t)=𝜌
1
−𝜎
1
Differential Equations and Dynamical Systems
1 3
1. Construction of the pursuers Strategies
The pursuers strategies we construct here suggest that the pursuers act one after another
in the order of their enumeration. That is, each pursuer in its turn applies the parallel
approach control discriminating the evader (it declares its instantaneous control value).
And the excess of the pursuer’s control resource is spent in such a way that the total
resource will be exhausted at some preliminarily computed instant if the evader spends
some prescribed portion of its control resource. If the evader spends more than its planned,
then the pursuer will spend totally its control resource earlier than the precomputed instant.
And if the evader will spend not greater than this planned portion, then the evader will be
caught already by this pursuer without involving the ones with greater numbers.
To construct the pursuers strategies, we introduce the following notations.
In the game described by (8), let
𝜏j,j=1, 2, …,m
denotes the instant when the evader will spend totally the portion of
its control resource planned for the
jth
pursuer. Such a time
𝜏j
is finite (if it exist) since the
evader’s control resources is limited. On the other hand, if
𝜏j
fails to exist, then the evader
never had the chance to spend all its control resources planned on avoiding the
jth
pursuer
(that is, the evader gets caught by the
jth
pursuer). From the moment the evader gets caught
by pursuer
Pj
, its left over control resource eventually becomes time invariant (thus, mak-
ing it reasonable to put
𝜏j=∞
in this case). Also let
tj,j=1, 2, …,m
denotes the mini-
mum time planned by the
jth
pursuer to complete pursuit. At this same time
tj
, the pursuer
Pj
expects the evader to have spent its total control resource on avoiding contact from it.
We now define the strategy of the pursuers
Pj,j=1, 2, …,m
as follows.
Set
ujk(t,z1(t),𝜎(t),v(t)) = 0, t0≤t≤t1,k=1, 2, …;j=2, …,m,
where time
and
𝜏1
is the first time when
∫𝜏
1
t0
‖
v(s)
‖2
ds =𝜎
2
1
which may or may not exists.
Consequently
If such time
𝜏1
fails to exists, then we set
𝜏1= +∞.
The case
𝜏1= +∞
yields the inequality
𝜎
(t)>𝜎
2
−𝜎
2
1,
for all
t≥t0
which follows from
Hence, we have
(22)
u
1k(t,z1(t),𝜎(t),v(t)) =
⎧
⎪
⎨
⎪
⎩
a(t)
A2(t0,𝜃1)z1k(t0)+vk(t),t0
≤
t
≤
t1
,
0, t>t1,
t
1=
{
𝜃1,𝜏1
≥
𝜃1
,
𝜏
1
,𝜏
1
<𝜃
1,
𝜎
(𝜏1)=𝜎2−
∫𝜏
1
t0
‖
v(s)
‖
2ds =𝜎2−𝜎2
1
.
𝜎
(t)=𝜎2−�
t
t0
‖
v(s)
‖
2ds >𝜎
2−�
∞
t0
‖
v(s)
‖
2ds =𝜎2−𝜎2
1,t
≥
t0
.
�𝜏
1
t
0
‖
v(s)
‖
2ds =
�∞
t
0
‖
v(s)
‖
2ds
≤
𝜎2
1
.
Differential Equations and Dynamical Systems
1 3
But if
𝜏1≥𝜃1,
that is,
t1=𝜃1,
then we claim pursuit can be completed at time
t1.
Observe
that, for this case,
2. Completion of pursuit
To show completion of pursuit, we first establish the admissibility the pursuers strategy
(22), that is,
where
u1(t)=(u11(t),…,u1k(t),…)
. Hence, in accordance with Definition (1), we con-
clude that the strategy (22) is admissible for the case
𝜏1≥𝜃1
. And if the pursuers adopt the
strategy (22), then for
k=1, 2, …
, we have
This implies that pursuit can be completed in the game (3)–(5) at time
t1=𝜃1
. Thus, if the
pursuers apply the admissible strategies (22), then either the evader spend the resource of
control less than or equal to
𝜎2
1
or pursuit will be completed.
◻
If we assume that pursuit fails to be completed on the time interval
[t0,t1],
then we
must have
𝜏1<𝜃
1
which implies
[t0,t1]=[t0,𝜏1]
.
Using mathematical induction, we define the numbers
𝜃j,
𝜏
j;j=1, 2, …,m
as
follows:
Let the number
𝜃1,
𝜏
1,
𝜃
2,
𝜏
2,…,
𝜃
j,
𝜏
j,(j=1, 2, …)
be defined subject to the following
conditions
(i)
Fj(𝜏j−1,t),
where
𝜏0=t0,
has a unique solution at
t=𝜃j
;
(ii)
𝜏j
is the first instant when
∫𝜏
j
𝜏
j−1
‖
v(s)
‖2
ds =𝜎
2
j
,
for all
j≥1
. That is,
𝜎
(𝜏
j
)=𝜎2−
∑j
i=2
𝜎
2
i
, such time may exist or not. For the latter case, we let
𝜏j= +∞
;
(iii)
𝜏k<𝜃
k,k=1, 2, …,j,
and pursuit fails to be completed on
[t0,𝜏j].
In particular,
𝜏k<∞,k=1, 2, …,j.
�𝜃
1
t0
‖
v(s)
‖
2ds =�
𝜏
1
t0
‖
v(s)
‖
2ds
≤
𝜎2
1
.
(23)
�t1
t0u1(s)2ds
1
2
=
�𝜃1
t0u1(s)2ds
1
2
≤�𝜃1
t0
a(s)
A2(t0,𝜃1)z1(t0)
2
ds1
2
+�𝜃1
t0v(s)2ds
1
2
≤∞
k=1
z2
1k(t0)
A2
2(t0,𝜃1)�𝜃1
t0
a2(s)ds+𝜎1
=
∞
k=1
z2
1k(t0)
A2(t0,𝜃1)
+𝜎1=F
1
2
1(t0,𝜃1)+𝜎1=𝜌1,
z
1k(t1)=z1k(t0)+∫
𝜃
1
t0
a(t)(vk(t)−u1k(t))dt
=z1k(t0)−∫𝜃1
t0
a(t)(a(t)
A2(t0,𝜃1)z1k(t0))
dt
=z1k(t0)− z1k(t0)
A
2
(t
0
,𝜃
1
)∫𝜃1
t
0
a2(t)dt =0.
Differential Equations and Dynamical Systems
1 3
We now define
t=𝜃j+1
as a unique solution of the equation
where
F
j+1(𝜏j,t)=
∑
∞
k=1
z
2
j+1,k(𝜏j)
A
2
(𝜏
j
,𝜃
j+1
)
∫
t
𝜏ja2(s)
ds
.
The strategies of the pursuers for all
t≥𝜏j,
is defined as follows:
Set
uik(t,zj+1(t),𝜎(t),v(t)) = 0, 𝜏j≤t≤tj+1,i=1, 2, …,j,j+2, …,m,
where time
and
𝜏j+1
is the first time when
∫𝜏
j+1
𝜏
j
‖
v(s)
‖2
ds =𝜎
2
j+
1
which may or may not exists.
Consequently
If
𝜏j+1
fails to exist, we let
𝜏j+1= +∞
, which yields
For the case
𝜏j+1≥𝜃j+1
(that is,
tj+1=𝜃j+1
), observe that
Admissibility of the pursuers strategy (24) can verified using similar argument in (23).
That is,
F
1
2
j+1
(𝜏j,t)=𝜌j+1−𝜎j+1
,
(24)
u
j+1,k(t,zj+1(t),𝜎(t),v(t)) =
⎧
⎪
⎨
⎪
⎩
0, t<𝜏
j,
a(t)
A2(𝜏j,𝜃j+1)zj+1,k(𝜏j)+vk(t),𝜏j≤t≤tj+1
,
0, t>t
j+1
,
t
j+1=
{
𝜃j+1,𝜏j+1
≥
𝜃j+1
,
𝜏
j+1
,𝜏
j+1
<𝜃
j+1,
𝜎
(𝜏j+1)=𝜎2−∫
𝜏
j+1
𝜏0
‖
v(s)
‖
2ds =𝜎2−(𝜎2
1+…+𝜎2
j+1)
.
�𝜏
j+1
𝜏
j
‖
v(s)
‖
2ds =𝜎(𝜏j)−𝜎(𝜏j+1)
≤
𝜎2
j+1
.
�𝜃
j+1
𝜏
j
‖
v(s)
‖
2ds
≤
�
𝜏
j+1
𝜏
j
‖
v(s)
‖
2ds
≤
𝜎2
j+1
.
�𝜃j+1
𝜏juj+1(s)2ds
1
2≤
�𝜃j+1
𝜏j
a(s)
A2(𝜏j,𝜃j+1)zj+1(𝜏j)
2
ds
1
2
+�𝜃j+1
𝜏jv(s)2ds1
2
=∞
k=1
z2
j+1,k(𝜏j)
A2(𝜏j,𝜃j+1)1
2
+𝜎j+1
=F
1
2
j
(𝜏j,𝜃j+1)+𝜎j+1=𝜌j+1.
Differential Equations and Dynamical Systems
1 3
In addition, the strategy (24) ensures
zj+1(𝜃j+1)=0
if adopted by the pursuers on the time
interval
[𝜏j,𝜏j+1].
That is, pursuit will be completed at time
𝜃j+1
and so on. And if pursuit
goes on until time
𝜏m−1
without completion, then let
𝜃m
be the unique solution of
where
Since we have
𝜎
(𝜏
m−1
)=𝜎
2
−(𝜎
2
1
+…+𝜎
2
m−1
)=𝜎
2
m,
then
Let
uik(t,zm(t),𝜎(t),v(t)) = 0, t>𝜏
m−1,k=1, 2, …;i=1, 2, …,m−1
.
Admissibility of the pursuers strategy (25) can easily be verified using similar argument
in the admissibility of (24). Thus,
It can also be verified that the strategy (25) ensures the equality
zm(𝜃m)=0
if adopted by
the pursuers on the time interval
[𝜏m−1,𝜏m].
That is, pursuit is completed at time
𝜃m
. Hence,
the conclusion of the theorem follows.
Remark 5 At first glance, the pursuers strategies (22) and (24) may seem to be independent
on
𝜎(⋅)
as stated in Definition6. But, of course, they do since they are defined by usage of
instances
𝜃j
, which depend on
𝜎(⋅)
. In addition, the question of optimality of the strategies
(22) is addressed in the recent work of Ahmed et. al. [1] wherein the authors, using the
pursuers strategies (22) estimated the value of the game described by (8). The game value is
value of the payoff function at the instant of the termination of the game and when players
are using their optimal strategies. To estimate this value, the authors [1] not only constructed
the optimal strategies (22) but also presented the tools (that is, players attainability domain
and also the strategies of some fictitious pursuers) required to construct (22). In this paper,
we present a more detailed application of the strategy in showing completion of pursuit and
moreover, we address evasion problem under the same dynamic equations of players.
Illustrative Examples
Here, we present some examples to illustrate our results.
F1∕2
m
(𝜏
m−1
,t)=𝜌
m
−𝜎
m,
F
m(𝜏m−1,t)=
∞
∑
k=1
z
2
mk(𝜏m−1)
A
2
(𝜏
m−1
,𝜃
m
)∫t
𝜏m−1
a2(s)ds
.
�𝜃
m
𝜏m−1
‖
v(s)
‖
2ds =𝜎(𝜏m−1)−𝜎(𝜏m)
≤
𝜎(𝜏m−1)=𝜎2
m
.
(25)
u
mk(t,zm(t),𝜎(t),v(t)) =
⎧
⎪
⎨
⎪
⎩
0, t<𝜏
m−1,
a(t)
A2(𝜏m−1,𝜃m)zmk(𝜏m−1)+vk(t),𝜏m−1≤t≤𝜃m
,
0, t>𝜃
m
,
�𝜃m
𝜏m−1
um(t)
2dt
1
2
≤
𝜌m
.
Differential Equations and Dynamical Systems
1 3
Example 3.3.1 Consider the motions of countably many pursuers
Pj,j∈I={1, 2, 3, …}
and an evader E in the space
𝓁2
governed by the equations
where all the variables are defined as in Sect.2 with control functions
uj(⋅)
and
v(⋅)
satisfy-
ing the inequalities
Given the initial position of the pursuers
Pj
and evader E as
x
0
j
=(2
−1
,2
−2
,2
−3
,…,2
−(i−1)
,−2
−i
,−2
−(i+1)
,
…)
and
y0=(0, 0, 0, …)
respectively, where
the number
−2−i
is in the
(j+1)th
coordinate of point x
0
j
. Let
T=4
, observe that
A1(0, 4)=(e4𝜆−1)∕𝜆,𝜌=
�∑
∞
j=1𝜌2
j=
√3
and
z0
j
=y0−x
0
j
is not contained in the
octant
X(�,
ℕ
)
defined in (13) for each
j∈I
. Since
𝜌<𝜎=2
, then by Theorem1, if the
evader adopt the admissible strategy
v(t)=(v1(t),v2(t),…)
where
avoidance of contact from all the pursuers
Pj
is guaranteed for all
t>0
. That is, for any
arbitrary point
zp(t)=(zp1(t),zp2(t),…),p∈Ii,
we have
Thus,
zpi(t)>0
for all
t≥0
. Consequently,
xpi(t)≠yi(t),t≥0.
Example 3.3.2 For the pursuit problem, we consider the case of a particular pursuer
Ps
for
some
s∈I
and evader E with motions govern by the equations in Example3.3.1 and initial
positions
respectively. Let the players control functions be subjected to the integral constraints
{
̇xj(t)=e𝜆tuj(t),xj(0)=x
0
j
̇y(t)=e𝜆tv(t),y(0)=y0,𝜆>
0
�∞
0
∥uj(t)∥
2dt
≤
3∕2j
,
�
∞
0
∥v(t)∥
2dt ≤4.
v
i(t)=
j∈Ii
u2
ji(t)+ 𝜆(𝜎2−𝜌2)e𝜆t
2i(e4𝜆−1)
1∕2
,0≤t≤
4
0, t>4,
z
pi(t)=z0
pi +�
t
0
e𝜆swpi(s))ds
=z0
pi +�t
0
e𝜆s(vi(s)−upi(s))ds
≥�t
0
e𝜆s
j∈Ii
u2
ji(s)+ 𝜆(𝜎2−𝜌2)e𝜆s
2i(e4𝜆−1)1∕2
−upi(s)
ds
>�t
0
e𝜆s
u2
pi(s)
1∕2
−upi(s)
ds =0.
xs(0)=(0, −ln 2, 0, −(ln 2)
2
, 0, …),y(0)=(
√ln 2, 0, √(ln 2)
3
, 0, …)
Differential Equations and Dynamical Systems
1 3
Clearly, the hypothesis of Theorem 2 is satisfied since
𝜌s
=ln 4 >ln 2 =𝜎
and
A2
(0, t)=(e
2𝜆t
−1)∕2𝜆→
∞
as
t→∞
. Given
𝜆=1∕2
and
z
0
s
=y0−x0
s
=(
√
ln 2, ln 2,
√
(ln 2)3,(ln 2)2,
…)
. Then, from the conclusion of Theorem2,
the pursuer’s admissible strategy
guarantees completion of pursuit at time
ts=1.9
as follows
Thus,
zs(1.9)=0
. Consequently,
xs(1.9)=y(1.9)
.
Conclusion
We have studied a differential game problem of avoiding contact and completing pursuit
in the Hilbert space
𝓁2
with integral constraints on all players control functions, where the
scalar function a(t) introduced is such that it is identically non-zero on the interval
(t1,t2)
.
By virtue of the players dynamics we considered, the conditions often imposed on the
parameters
𝜆k,k=1, 2, …
in the system of differential Eq. (2) is redundant. With a mild
condition on a(t), we have proved that if the total energy resources of the pursuers is less
than that of the evader, then avoidance of contact is guaranteed through out the game. In
addition, we estimated the smallest possible distance the evader can preserve from the pur-
suers through out the game. For the pursuit problem, we constructed strategies of the pur-
suers and showed that pursuit can be completed for a finite time
t=t(z0)
from any initial
positions
z0
of players. The problem studied in this paper with the coefficient a(t) replaced
by the coefficients
aj(t),j=1, 2, …
is an open problem.
Acknowledgements The authors acknowledge the financial support provided by King Mongkut’s University
of Technology Thonburi through the “KMUTT
55th
Anniversary Commemorative Fund”. The first author
was supported by the Petchra Pra Jom Klao Doctoral Scholarship Academic for Ph.D. Program at KMUTT.
This project is supported by the Theoretical and Computational Science (TaCS) Center under Computa-
tional and Applied Science for Smart Innovation (CLASSIC), Faculty of Science, KMUTT. The present
research was partially supported by the National Fundamental Research Grant Scheme FRGS of Malaysia,
01-01-17-1921FR. Moreover, Poom Kumam was supported by the Thailand Research Fund and the King
Mongkut’s University of Technology Thonburi under the TRF Research Scholar Grant No.RSA6080047.
�∞
0
∥us(t)∥
2dt
≤
(ln 4)2
,
�
∞
0
∥v(t)∥
2dt ≤(ln 2)2.
u
s(t)=
⎧
⎪
⎨
⎪
⎩
et∕2
(e1.9 −1)z0
s+v(t),0
≤
t
≤1.9,
0, t>1.9
z
s(1.9)=z0
s+∫
1.9
0
et∕2(v(t)−us(t))dt
=z0
s−
z0
s
(e
1.9
−1)∫
1.9
0
etdt =
0.
Differential Equations and Dynamical Systems
1 3
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