Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 748096, 10 pages
doi:10.1155/2012/748096
Research Article
Simple Motion Evasion Differential Game of Many
Pursuers and One Evader with Integral Constraints
on Control Functions of Players
Gafurjan Ibragimov and Yusra Salleh
INSPEM and Department of Mathematics, UPM, 43400 Serdang, Malaysia
Correspondence should be addressed to Gafurjan Ibragimov, gafur@science.upm.edu.my
Received 7 May 2012; Revised 15 September 2012; Accepted 15 September 2012
Academic Editor: Alexander Timokha
Copyright q 2012 G. Ibragimov and Y. Salleh. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We consider an evasion differential game of many pursuers and one evader with integral
constraints in the plane. The game is described by simple equations. Each component of the control
functions of players is subjected to integral constraint. Evasion is said to be possible if the state
of the evader does not coincide with that of any pursuer. Strategy of the evader is constructed
based on controls of the pursuers with lag. A sufficient condition of evasion from many pursuers
is obtained and an illustrative example is provided.
1. Introduction
1.1. Related Works
Among a large number of papers devoted to differential games with integral constraints
see, e.g., 1–14, a few works are dedicated to evasion games with integral constraints see,
e.g., 1, 4, 7, 12. Note that integral constraints arise if we consider constraints on energy,
resources, and so forth. In 1, an evasion differential game was considered. In the paper 1,
the dynamics of the object zt is given by the equation
ż Az − u v,
z0 z0 ,
1.1
where z ∈ Rn , A is a constant matrix, and u and v are control parameters of the
pursuer and evader, respectively. The control functions utand vt of the pursuer and
evader,respectively, are defined as measurable functions satisfying conditions
∞
0
2
|ut| dt ≤ ρ ,
2
∞
0
|vt|2 dt ≤ σ 2 .
1.2
2
Journal of Applied Mathematics
/ M, t ≥ 0. In
The terminal set M is a subspace of Rn . By definition evasion is possible if zt ∈
this work Pontryagin’s method was extended to the evasion game with integral constraints
and sufficient conditions of evasion from any initial position were obtained.
In 4, a differential game described by 1.1 was studied. The control functions ut
and vt satisfy conditions 1.2. Evasion is said to be possible if zt ∈
/ M, where M ∪m
i1 Mi ,
n
Mi are subspaces of R . The pursuers’ purpose is to take the point zt onto the terminal set
M; the evader’s one, to guarantee evasion from the set M. By definition, l-escape is possible if
|zt| ≥ lt > 0, t ≥ 0 for some function lt. Sufficient conditions for l-escape were obtained.
In the book 12 chapter 11, linear evasion differential game with integral constraints
was examined. The game is described by the equation
ż Az − Bu Cv,
1.3
z0 z0 ,
where z ∈ Rn , A, B, and C are constant matrices and u and v are control parameters of
the pursuer and evader, respectively. The control functions ut and vt of the pursuer and
evader are subjected to the conditions 1.2. The terminal set M is a subspace of Rn . By
definition evasion is possible if zt ∈
/ M, t ≥ 0. Some sufficient conditions of evasion were
obtained.
In 5, a differential game of countably many pursuers and one evader is described by
the following infinite system of differential equations:
żik −λk zik − uik vk ,
zik t0 z0ik ,
1.4
k 1, 2, . . . ,
2
, where
which was studied in the Hilbert space lr1
zik , uik , vk , t0 , z0ik ∈ R,
2
z0i z0i1 , z0i2 , . . . ∈ lr1
,
z0i /
0,
1.5
ui ui1 , ui2 , . . . is the control parameter of ith pursuer, i 1, 2, . . ., and v v1 , v2 , . . . is that
of the evader. The control functions of the players are subject to integral constraints
T
2
ui t dt ≤
t0
ρi2 ,
T
2
vt dt ≤ σ ,
2
vt t0
∞
λrk vk2 t
1/2
,
T > 0.
1.6
k1
Avoidance of contact is said to be possible in the game with the initial position z0 2
, if z t zi1 t, zi2 t, . . . /
0, for all t ∈ 0, T and i 1, 2, . . .. It
{z01 , z02 , . . . , z0m , . . .}, z0i ∈ lr1
∞ 2 i
2
was proven that if σ > k1 ρk , then avoidance of contact is possible in the game from any
2
, z0i /
0, i 1, 2, . . . .
initial position z0 {z01 , . . . , z0m , . . .}, z0i ∈ lr1
The work 7 deals with the evasion differential game of m pursuers x1 , . . . , xm and
one evader y described by equations
ẋi ui ,
xi 0 xi0 ,
ẏ v,
y0 y0 ,
xi0 /
y0 ,
i 1, . . . , m.
1.7
Journal of Applied Mathematics
3
The game occurs in the plane and the following conditions:
∞
2
|ui s| ds ≤
0
∞
i 1, . . . , m,
ρi2 ,
|vs|2 ds ≤ σ 2
1.8
0
are imposed on control functions of the players. The main result of the paper is as follow: if
2
ρ12 ρ22 · · · ρm
≤ σ 2 , then evasion is possible. The important point to note here is the fact
2
2
2
> σ 2 , then pursuit can be completed 13.
that if ρ1 ρ2 · · · ρm
It should be noted that in all of these works, an integral constraint was imposed on the
whole control function of each player, such as
∞
|vs|2 ds ≤ σ 2 .
1.9
0
However, in contrast to other works, in the present paper, we study a differential game
with integral constraints imposed on each coordinate of control functions see the constraints
1.12-1.13.
2
2
2
At first glance, the inequalities ρ1i
ρ2i
· · ·ρmi
> σi2 , i 1, 2 seem to be the conditions
for completing pursuit. It turns out, as shown in the example, evasion is possible from some
initial positions even though these inequalities hold.
1.2. Statement of the Problem
In R2 , we consider an evasion differential game of many pursuers x1 , . . . , xm and one evader
y described by the equations
ẋi ui ,
i 1, . . . , m,
xi 0 xi0 ,
ẏ v,
y0 y0 ,
1.10
1.11
0
0
and assume that xi0 /
, xi2
, i 1, . . . , m, y0 y10 , y20 .
y0 , xi0 xi1
Definition 1.1. A measurable function ui t ui1 t, ui2 t, t ≥ 0 is called an admissible
control of the pursuer xi if
∞
0
2
,
|ui1 s|2 ds ≤ ρi1
∞
0
2
,
|ui2 s|2 ds ≤ ρi2
1.12
where ρi1 , ρi2 , i 1, . . . , m are given positive numbers.
Definition 1.2. A measurable function vt v1 t, v2 t, t ≥ 0 is called an admissible control
of the evader y if
∞
0
|v1 s|2 ds ≤ σ12 ,
where σ1 and σ2 are given positive numbers.
∞
0
|v2 s|2 ds ≤ σ22 ,
1.13
4
Journal of Applied Mathematics
Definition 1.3. A function of the form
0,
0 ≤ t ≤ ε,
V t fu1 t − ε, . . . , um t − ε, t > ε
1.14
is called a strategy of the evader, where ε is a positive number, f : R2m → R2 is a continuous
function, and u1 t, . . . , um t, t ≥ 0 are admissible controls of the pursuers.
0
Definition 1.4. We say that evasion is possible from the initial position x10 , x20 , . . . , xm
, y0 in
yt, t ≥ 0,
the game 1.10–1.13 if there exists a strategy of the evader V such that xi t /
i 1, . . . , m for any admissible controls of the pursuers.
We can now state the evasion problem.
0
Problem 1. Find conditions for all initial positions x10 , x20 , . . . , xm
, y0 and parameters σ1 , σ2 ,
ρi1 , ρi2 , i 1, . . . , m which guarantee evasion in the game 1.10–1.13.
This is the main problem to be investigated in the present paper. It should be noted
that, in the evasion game, the pursuers use arbitrary admissible controls u1 t, . . . , um t, t ≥
0, and the evader uses a strategy. By Definition 1.3 this strategy is constructed based on values
u1 t − ε, . . . , um t − ε.
After the proof of the main result Theorem 2.1 we will give illustrative examples.
For the initial positions, which do not satisfy hypotheses of the theorem, we will show that
pursuit can be completed. Therefore, at first we have to give a definition for “pursuit can be
completed”. To this end we need to define strategies of the pursuers.
Definition 1.5. A Borel measurable function Ui v Ui1 v, Ui2 v, Ui : R2 → R2 is called
a strategy of the pursuer xi if for any control of the evader vt, t ≥ 0 and the inequalities
∞
0
2
,
|Ui1 vs|2 ds ≤ ρi1
∞
0
2
|Ui2 vs|2 ds ≤ ρi2
1.15
hold.
Definition 1.6. We say that pursuit can be completed from the initial position
0
, y0 in the game 1.10–1.13 if there exist strategies of the pursuers Ui ,
x10 , x20 , . . . , xm
i 1, . . . , m such that for any admissible control of the evader the equality xi τ yτ holds
at some i ∈ {1, . . . , m} and τ ≥ 0.
Note that in the pursuit game, the pursuers use strategies and the evader uses any
admissible control vt see Definition 1.6. According to Definition 1.5 at current time t the
pursuers use vt to construct their strategies.
2. Main Result
The main result of the paper is formulated as follows.
Journal of Applied Mathematics
5
Theorem 2.1. Assume the following conditions hold.
1 There exists a subset I of the set {1, . . . , m} such that
2
ρi2
≤ σ22 ,
2
ρi1
≤ σ12 ,
i∈I
J {1, . . . , m} \ I.
2.1
i∈J
/ x1 , x1 , y20 ∈
/ x2 , x2 , where
2 y10 ∈
0
x1 min xi1
,
i∈I
0
x1 max xi1
,
i∈I
0
x2 min xi2
,
i∈J
0
x2 max xi2
.
i∈J
2.2
Then evasion is possible in the game 1.10–1.13.
2 2 1/2
Proof. Let ρ m
, and let ε, ε < 1/4ρ2 min{y10 −x1 2 , y20 −x2 2 }, be a positive
j1 ρij i1
number. We construct a strategy for the evader as follows:
0,
0 ≤ t ≤ ε,
v1 t 0
1/2
2
y − x0 , t > ε,
i∈I ui1 t − ε
1
k1
⎧
0 ⎨0,
0 ≤ t ≤ ε,
y0 − xl2
1/2
v2 t 20
y − x0 ⎩ u2 t − ε
, t > ε,
i∈J i2
2
l2
0
y10 − xk1
2.3
2.4
where u1 t, . . . , um t, t ≥ 0 are any admissible controls of the pursuers, and k ∈ I and
l ∈ J are arbitrary numbers. The pursuers use any admissible controls and the evader uses
the strategy 2.3 and 2.4. We’ll prove that evasion is possible. We examine the case y10 >
x1 , y20 > x2 . Other cases such as
y10 > x1 ,
y20 < x2 ;
y10 < x1 ,
y20 > x2 ;
y10 < x1 ,
y20 < x2
2.5
0
,
can also be considered similarly. It follows from the inequalities y10 > x1 , y20 > x2 that y10 > xk1
0
0
y2 > xl2 , and so
0
y0 − xk1
10
1,
y − x0 1
k1
0
y0 − xl2
20
1.
y − x0 2
2.6
l2
Then 2.3 and 2.4 take the following form:
0,
v1 t 2
i∈I ui1 t
⎧
⎨0,
v2 t ⎩
2
i∈J ui2 t
1/2
,
− ε
− ε
1/2
0 ≤ t ≤ ε,
t > ε,
0 ≤ t ≤ ε,
,
t > ε.
2.7
2.8
6
Journal of Applied Mathematics
First, we show that evasion is possible on the time interval 0, ε. Let t ∈ 0, ε. Then for any
i ∈ I we have:
y1 t − xi1 t ≥
y10
y10
t
v1 sds −
0
x1
−
0
xi1
−
t
ui1 sds
0
t
√
− |ui1 s|ds ≥ y10 − x1 − ρ t
2.9
0
√
1 0
≥ y10 − x1 − ρ ε ≥
y1 − x1 > 0,
2
√
since by choice of ε, ρ ε < 1/2y10 − x1 .
We now show that evasion is possible on ε, ∞. If t ∈ ε, ∞, then according to 2.7
for any i ∈ I we have:
y1 t − xi1 t y10 t
ε
0
v1 sds − xi1
−
≥ y10 − x1 ≥
y10
−
x1
ui1 sds
0
t t
u2i1 s − εds − ui1 sds
ε
0
i∈I
t−ε u2i1 sds −
≥ y10 − x1 −
≥
t
0
t
t−ε
i∈I
t−ε
t 0
|ui1 s|ds ≥ y10 − x1 −
|ui1 s|ds
2.10
t−ε
√
ερ
1 0
y1 − x1 > 0.
2
Thus, evasion from the pursuers xi , i ∈ I, is possible. Similarly, according to 2.8 for any
j ∈ J we obtain that
y2 t − xi2 t ≥ y20 − x2 −
This completes the proof of the theorem.
√
ερ ≥
1 0
y2 − x2 > 0,
2
t ≥ 0.
2.11
Journal of Applied Mathematics
7
Example 2.2. We consider two differential games of two pursuers and one evader described
by the equations
ẋi ui ,
ẏ v,
xi 0 xi0 ,
y0 y0 ,
∞
0
∞
0
u2ij sds ≤ ρij2 ,
vj2 sds ≤ σj2 ,
2.12
i, j 1, 2.
2.13
2
2
2
2
A In the first differential game, ρ11
4, ρ12
1, ρ21
1, ρ22
4, σ12 σ22 1, and
hence the sets I {2}, J {1} satisfy the first hypothesis of the theorem. Initial
y0 , x20 /
y0 , are assumed to be
positions of the players x10 , x20 , and y0 , where x10 /
any points in the plane. It is not difficult to verify that the second hypothesis of the
0
0
0
0
0
0
theorem holds, if x21
/ y1 , and x12 / y2 since x1 x1 x21 , x2 x2 x12 . Therefore,
for this case conclusion of the theorem is true and hence from such initial positions
0
0
y10 , x12
y20 is that
evasion is possible. Note that the meaning of the condition x21
/
/
0
the initial position of the evader y doesn’t lie on the horizontal and vertical lines
passing through the points x10 and x20 , respectively.
We now construct the strategy for the evader, which guarantees the evasion. Let
u1 t, u2 t, t ≥ 0 be any admissible controls of the pursuers. According to 2.3 and 2.4
the strategy of the evader takes the following form:
0,
0 ≤ t ≤ ε,
v1 t ξ|u21 t − ε|, t > ε,
0,
v2 t η|u12 t − ε|,
0 ≤ t ≤ ε,
t > ε,
2.14
where
0
y0 − x21
,
ξ 10
y − x0 1
21
0
y0 − x12
.
η 20
y − x0 2
2.15
12
2
2
2
2 1/2
For this example, ρ ρ11
ρ12
ρ21
ρ22
and ε satisfies the following conditions:
2 2 1
0
0
0
0
ε < 2 min y1 − x21 , y2 − x12
.
4ρ
2.16
0
0
According to the theorem if x21
y10 , x12
y20 , then the strategy of the evader 2.14 guarantees
/
/
the evasion.
0
0
y10 or x12
y20 since the
However, the theorem gives no information if either x21
second hypothesis of the theorem is not satisfied for these cases. We’ll show that in each of
8
Journal of Applied Mathematics
these cases pursuit can be completed in the game 2.12 and 2.13 see Definitions 1.5 and
0
y10 , define the strategies of the pursuers as follows:
1.6. In the case x21
u11 t 0,
u12 t 0,
t ≥ 0,
u21 t v1 t,
⎧
⎨ 1 y0 − x0 v t, 0 ≤ t ≤ θ,
2
22
u22 t θ 2
⎩0,
t > θ,
2.17
2.18
2
0
where θ |y20 − x22
| .
0
y20 , define the strategies of the pursuers as follows:
In the case x12
u21 t 0, u22 t 0, t ≥ 0,
⎧ ⎨ 1 y0 − x0 v t,
1
11
u11 t θ1 1
⎩
0,
0 ≤ t ≤ θ1 ,
2.19
t > θ1 ,
u12 t v2 t,
2
0
|.
where θ1 |y10 − x11
0
0
y10 . The case x12
y20 can be analyzed in a similar fashion.
We consider the case x21
Admissibility of the strategy 2.17 and 2.18 follows from the following relations:
∞
2
|u22 t| dt 0
θ
0
≤
≤
2
|u22 t| dt ∞
|u22 t|2 dt
θ
θ 2
1 0
0
y − x22 v2 t dt
θ 2
0
θ 2 2 1
0
2
0 0
0
y
v
−
x
−
x
dt
t
t|
|v
y
2
2
22
22
θ 2
θ2 2
0
θ
2 2 θ 1 0
0 0
0
y − x22 v2 tdt |v2 t|2 dt
y − x22 θ 2
θ 0 2
0
θ
2 0 1 y20 − x22
· |v2 t|dt σ22
θ
0
θ
θ
2 0
0 2 y2 − x22
12 dt
|v2 t|2 dt
·
θ
0
0
2σ 2 0 ≤ 2 √ 2 y20 − x22
4.
θ
2.20
Journal of Applied Mathematics
9
Here, we used the Cauchy-Schwartz inequality. Next, we show that x2 θ yθ, that is,
pursuit will be completed by the pursuer x2 at the time θ. We have
0
x21 t x21
t
0
u21 sds y10 t
v1 sds y1 t
2.21
0
for all t ≥ 0. In particular, x21 θ y1 θ. In addition,
0
x22 θ x22
y20
θ
0
θ
0
u22 sds x22
θ 1 0
0
v2 s ds
y2 − x22
θ
0
2.22
v2 sds y2 θ.
0
Hence, x22 θ y2 θ. Thus x2 θ yθ.
B Consider an example of a differential game described by 2.12 and 2.13, for which
2
2
2
2, ρ12
1, ρ21
the second hypothesis of the theorem is not satisfied. Let ρ11
0
0
2
2
2
0
2, ρ22 1, σ1 σ2 2, x1 0, −1, x2 0, 1, y 0, a, where −1 < a < 1.
Let the strategies of the pursuers be defined by formulas
u11 t u21 t v1 t,
u12 t 1,
u22 t −1,
t ≥ 0.
2.23
We show that pursuit is completed by the time t 1. Indeed, for all t ≥ 0 we have
xi1 t 0
xi1
t
ui1 sds 0
t
v1 sds y1 t,
i 1, 2.
2.24
0
What is left is to show that xi2 τ y2 τ for some i ∈ {1, 2} and τ, 0 < τ ≤ 1.
It follows from the relations
x12 0 < y2 0 < x22 0,
1
1
0
u12 sds −1 1ds 0,
x12 1 x12
0
0
x22 1 x22
1
0
u22 sds 1 1
0
2.25
−1ds 0
0
that one of the equalities x12 τ yτ or x22 τ yτ holds at some τ, 0 ≤ τ ≤ 1. According
to 2.24, xi1 τ y1 τ and therefore one of the equalities x1 τ yτ or x2 τ yτ holds
and hence pursuit is completed at τ. Thus, if the initial position of the evader is in the line
segment x10 , x20 then pursuit can be completed by the time t 1.
10
Journal of Applied Mathematics
3. Conclusion
We have studied a simple motion evasion differential game with integral constraints on
control functions of the players. Unlike the traditional integral constraints, in the present
paper the integral constraint is imposed on each component of control functions of the
players. Under certain conditions, we have proven that the evasion is possible, that is, we
have obtained a sufficient condition of evasion. The problem is open for further investigation
if the hypotheses of the theorem are not satisfied. Further studies can be done to obtain
complete solution for the evasion problem 1.10–1.13.
Acknowledgment
This research was partially supported by the Research Grant RUGS of the Universiti Putra
Malaysia, no. 05-04-10-1005RU.
References
1 A. Y. Azimov, “A linear differential evasion game with integral constraints on the controls,” USSR
Computational Mathematics and Mathematical Physics, vol. 14, no. 6, pp. 56–65, 1974.
2 A. A. Azamov and B. Samatov, π-Strategy. An Elementary Introduction to the Theory of Differential Games,
NUU press, Tashkent, Uzbekistan, 2000.
3 A. A. Chikrii and A. A. Belousov, “On linear differential games with integral constraints,” Trudy
Instituta Matematiki i Mekhaniki, vol. 15, no. 4, pp. 290–301, 2009 Russian.
4 P. B. Gusiatnikov and E. Z. Mohon’ko, “On 1∞ -escape in a linear many-person differential game with
integral constraints,” Journal of Applied Mathematics and Mechanics, vol. 44, no. 4, pp. 436–440, 1980.
5 G. I. Ibragimov and R. M. Hasim, “Pursuit and evasion differential games in Hilbert space,”
International Game Theory Review, vol. 12, no. 3, pp. 239–251, 2010.
6 G. I. Ibragimov, A. A. Azamov, and M. Khakestari, “Solution of a linear pursuit-evasion game with
integral constraints,” ANZIAM Journal, vol. 52, pp. E59–E75, 2010.
7 G. I. Ibragimov, M. Salimi, and M. Amini, “Evasion from many pursuers in simple motion differential
game with integral constraints,” European Journal of Operational Research, vol. 218, no. 2, pp. 505–511,
2012.
8 N. N. Krasovskii, The Theory of Motion Control, Nauka, Moscow, Russia, 1968.
9 A. V. Mesencev, “A sufficient conditions for evasion in linear games with integral constraints,”
Doklady Akademii Nauk SSSR, vol. 218, pp. 1021–1023, 1974.
10 M. S. Nikolskii, “The direct method in linear differential games with integral constraints,” in Control
Systems, pp. 49–59, IM, IK, SO AN SSSR, 1969.
11 B. N. Pshenichnyy and Y. N. Onopchuk, “Linear differential games with integral constraints,”
Akademii Nauk SSSR, Tekhnicheskaya Kibernetika, vol. 1968, no. 1, pp. 13–22, 1968.
12 N. Y. Satimov and B. B. Rikhsiev, Methods of Solving of Evasion Problems in Mathematical Control Theory,
Fan, Tashkent, Uzbekistan, 2000.
13 N. Y. Satimov, B. B. Rikhsiev, and A. A. Khamdamov, “On a pursuit problem for n person linear
differential and discrete games with integral constraints,” Mathematics of the USSR-Sbornik, vol. 46,
no. 4, pp. 456–469, 1982.
14 V. N. Ushakov, “Extremal strategies in differential games with integral constraints,” Journal of Applied
Mathematics and Mechanics, vol. 36, no. 1, pp. 15–23, 1972.
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