2025 -Yil
13-Fevral
RAQAMLI DUNYO: MATEMATIK VA INFORMATIK
YONDASHUVLAR
Respublika ilmiy-uslubiy konferensiyasi
8
LINEAR DIFFERENTIAL PURSUIT PROBLEM WITH MANY PLAYERS UNDER
INTEGRAL AND GEOMETRIC RESTRICTIONS
Soyibboev Ulmasjon
teacher at the department of Mathematics,
Namangan State University, PhD,
+99899-975-98-38
Abdurazzokova Marjona
Bachelor student, Namangan State University,
abdurazzaqovamarjona046@gmail.com,
+99897-427-50-57
Turgunpolatova Odina
Bachelor student, Namangan State University,
odinaturgunpulatova952@gmail.com,
+99897-827-12-08
Abdusalomova Feruzakhon
Bachelor student, Namangan State University
, abdusalomovaferuza15@gmail.com,
+99888-686-15-03
https://doi.org/10.5281/zenodo.14845001
Abstract.
We have considered a simple motion differential game of
𝑚
pursuers with
integral restriction and one evader with geometric restriction in
ℝ
𝑛
. To solve a pursuit problem,
the attainability domain of each pursuer is constructed. Sufficient condition of pursuit is obtained
by intersection of the attainability domains. Lifeline game is solved for the advantage of the
pursuers.
Keywords:
Differential game, pursuer, evader, geometric restriction, integral restriction,
attainability domain, lifeline.
Consider the differential game when the Pursuers
𝑋
𝑖
,
𝑖 = 1,2, … , 𝑚
, and the Evader
𝑌
having radius vectors
𝑥
𝑖
and
𝑦
respectively move in
ℝ
𝑛
. If their velocity vectors are
𝑢
𝑖
and
𝑣
then
the game will be described by the equations
𝑋
𝑖
: 𝑥
𝑖
̇ + 𝑎𝑥
𝑖
= 𝑢
𝑖
,
𝑥
𝑖
(0) = 𝑥
𝑖0
,
(1)
𝑌:
𝑦̇ + 𝑎𝑦 = 𝑣
,
𝑦(0) = 𝑦
0
, (2)
where
𝑥
𝑖
, 𝑦, 𝑢
𝑖
, 𝑣 ∈ ℝ
𝑛
, 𝑛 ≥ 2
;
𝑥
𝑖0
, 𝑦
0
are the initial positions of the objects
𝑋
𝑖
and
𝑌
and
it is assumed
𝑥
𝑖0
≠ 𝑦
0
. Here the temporal variation of
𝑢
𝑖
must be a measurable function
𝑢
𝑖
(∙): [0, ∞) → ℝ
𝑛
, on which is imposed the constraint
∫ |𝑢
𝑖
(𝑠)|
2
𝑑𝑠 ≤ 𝜌
𝑖0
𝑡
0
for almost every
𝑡 ≥ 0
, (3)
where
𝑖 = 1,2, … , 𝑚
, and
𝜌
𝑖0
are the positive numbers. From the physical point of view,
the right-hand of (3) corresponds to the given resource of the Pursuer
𝑋
𝑖
. We call the inequality
(3) as integral restriction (briefly,
𝐼 −
constraint) and denote by
𝑈
𝐼
𝑖
the class of admissible controls
satisfying (3).
Similar, the temporal variation of
𝑣
should be a measurable function
𝑣(∙): [0, ∞) → ℝ
𝑛
and
on this function, we impose geometrical restriction (briefly,
𝐺 −
constraint)
|𝑣(𝑡)| ≤ 𝛽
for almost every
𝑡 ≥ 0
, (4)
where
𝛽
is a positve number which means the maximal velocity of the Evader. We denote
by
𝑉
𝐺
the class of the Evader’s admissible controls satisfying (4).
2025 -Yil
13-Fevral
RAQAMLI DUNYO: MATEMATIK VA INFORMATIK
YONDASHUVLAR
Respublika ilmiy-uslubiy konferensiyasi
9
We are going to study mainly the game with phase restrictions for the Evader being given
by a subset
𝐴
of which is called the “Life-line” [1].
Definition 1.
For each triple
(𝜌
𝑖0
, 𝑢
𝑖
(∙)), 𝑢
𝑖
(∙) ∈ 𝑈
𝐼
𝑖
,
𝑖 = 1,2, … , 𝑚
, the scalar function
𝜌
𝑖
(𝑡) = 𝜌
𝑖0
− ∫ |𝑢
𝑖
(𝑠)|
2
𝑑𝑠, 𝜌
𝑖
(0) = 𝜌
𝑖0
,
𝑡
0
𝑡 ≥ 0
, is called the
residual resource
of the Pursuer
𝑋
𝑖
at the time
𝑡
.
Let
𝑧
𝑖
(𝑡) = 𝑥
𝑖
(𝑡) − 𝑦(𝑡), 𝑧
𝑖0
= 𝑥
𝑖0
− 𝑦
0
,
𝜇
𝑖0
= 𝜌
𝑖0
/|𝑧
𝑖0
|
.
Suppose that the pair
(𝜇
𝑖0
, 𝛽)
is a parametric state of the game (1)–(4) and denote it by
𝑝
𝑖
.
We introduce the following nonempty connected set of such states
𝑝
𝑖
:
𝑃
𝐼𝐺
𝑖
= {𝑝
𝑖
: 𝜇
𝑖0
≥ 4𝛽, 𝛽 >
0}
.
Definition 2.
The function
𝒖
𝑖
(𝑣) = 𝑣 − 𝜆
𝑖
(𝑣)𝜉
𝑖0
(5)
is called the
parallel pursuit strategy
(
П
𝐼𝐺
𝑖
-
strategy
) for the Pursuer
𝑋
𝑖
, where
𝜆
𝑖
(𝑣) =
𝜇
𝑖0
2
+ 〈𝑣, 𝜉
𝑖0
〉 + √(
𝜇
𝑖0
2
+ 〈𝑣, 𝜉
𝑖0
〉)
2
− |𝑣|
2
,
𝜉
𝑖0
=
𝑧
𝑖0
|𝑧
𝑖0
|
.
Consider the set
𝑊
𝐼𝐺
𝑖
(𝑡) = {𝑤: |𝑤 − 𝑥
𝑖
(𝑡)|
2
= 𝑒
−𝑎𝑡
𝛬
𝑖
(𝑡, 𝑣(∙))(𝑝
𝑖
(𝑡)/𝛽)|𝑤 − 𝑦(𝑡)|,
0 ≤ 𝑡 ≤ 𝑡
∗
}
,
where
𝑡
∗
= 𝑚𝑖𝑛{𝑡: |𝑧
𝑖
(𝑡)| = 0}
and
𝑤
is a point where the Pursuers
𝑋
𝑖
should meet the
Evader
𝑌
.
Theorem 1.
Let
𝑝
𝑖
∈ 𝑃
𝐼𝐺
𝑖
be valid in the game (1)–(4). Then the relation
𝑊
𝐼𝐺
𝑖
(𝑡
2
) ⊂
𝑊
𝐼𝐺
𝑖
(𝑡
1
)
is satisfied for any
𝑡
1
< 𝑡
2
, 𝑡
1
, 𝑡
2
∈ [0, 𝑡
∗
]
.
Proposition 1.
If
𝑝
𝑖
∈ 𝑃
𝐼𝐺
𝑖
, then inclusion
𝑦(𝑡) ∈ 𝑊
𝐼𝐺
𝑖
(0)
is valid on
[0, 𝑡
∗
]
.
Theorem 2.
If in the game (1)–(4),
𝑝
𝑖
∈ 𝑃
𝐼𝐺
𝑖
holds for some
𝑖 = 1,2, … , 𝑚
, then
𝑦(𝑡) ∈
𝑊
𝐼𝐺
𝑖
is met on the time interval
[0, 𝑇
𝐼𝐺
]
, where
𝑊
𝐼𝐺
= ⋂
𝑊
𝐼𝐺
𝑖
(0)
𝑚
𝑖=1
,
𝑇
𝐼𝐺
= 𝑑/𝛽
,
𝑑 =
max {|𝑤
1
− 𝑤
2
|: 𝑤
1
, 𝑤
2
∈ 𝑊
𝐼𝐺
}
.
Theorem 3.
If
𝑊
𝐼𝐺
∩ 𝐴 = ∅
, then the Pursuers
𝑋
𝑖
win on the time interval
[0, 𝑇
𝐼𝐺
]
in the
“Life-line” game (1)-(4).
References
1.
Isaacs R. Differential games. John Wiley and Sons, New York. 1965.
2.
Petrosjan L.A. Differenetial games of pursuit. Series on optimization. Vol. 2. World
Scientific Publishing, Singapore. 1993.
3.
Azamov A.A. On the quality problem for simple pursuit games with constraint. Serdica
Bulgariacae math., Publ. Sofia, 1986, Vol. 12, No. 1, pp. 38-43.
4.
Samatov B.T. Problems of group pursuit with integral constraints on controls of the players.
Cybernetics and Systems Analysis, 2013, Vol. 49, No. 6, pp. 907-921.
5.
Samatov B.T. The Pursuit-Evasion Problem under Integral-Geometric constraints on
Pursuer controls. Automation and Remote Control, Pleiades Publishing, Ltd. New York.
2013, Vol. 74, No. 7, pp. 1072–1081.
6.
Soyibboev U.B. П-strategy in a differential game with the same type of
7.
dynamics of negative viscous coefficients. Uzbek Mathematical Journal, 2023, Issue 3, pp.
172–180.
