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THE STRATEGY OF PARALLEL PURSUIT FOR DIFFERENTIAL GAME
OF THE SECOND ORDER
Doliyev Oybek Bahodir ugli
Namangan Institute of Engineering Technology.
Address: 7 Kosonsoy st., Namangan city, Republic of Uzbekistan
E-mail: doliyevoybek7@gmail.com Tel: +998 94 174 74 73;
https://doi.org/10.5281/zenodo.12605921
Abstract.
In this work is considered a differtial game of the second
order, when control functions of the players satisfies geometric constraints. The
proposed method substantiates the parallel approach strategy in this
differential game of the second order. The new sufficient solvability conditions
are obtained for problem of the pursuit.
Keywords.
Differential game, geometric constraint, evader, pursuer,
strategy of the parallel pursuit, acceleration.
Аннотация.
В работе рассматривается дифференциальная игра
второго порядка при геометрических ограничениях на управления
игроков. При этом предлагается стратегия параллельного преследования
для преследователя и при помощи этой стратегии решается задача
преследования.
Ключевые слова:
дифференциальная игра, геометрическое
ограничение, стратегия параллельного преследования, преследователь,
убегающий, ускорения.
Annotatsiya
.
Ushbu maqolada boshqaruvlar geometrik chegaralanishga
ega hol uchun ikkinchi tartibli differensial o’yinlar o’rganiladi. Bunda quvlovchi
uchun parallel quvish strategiyasi quriladi va uning yordamida tutish masalasi
yechiladi.
Kalit so`zlar:
differensial o’yin, geometrik chegaralanish, parallel quvish
strategiyasi, quvlovchi, qochuvchi, tezlanish.
Introduction
Let
P
and
E
objects with opposite aim be given in the space
n
R
and their
movements based on the following differential equations and initial conditions
P
:
x
u
,
1
0
0
x
kx
,
u
, (1)
E
:
y
v
,
1
0
0
y
ky
,
v
, (2)
where
, , ,
x y u v
n
R
;
x
– a position of
P
object in the space
n
R
,
0
1
(0),
(0)
x
x
x
x
– its initial position and velocity respectively at
0;
t
u
–
being a controlled acceleration of the pursuer, mapping
: 0,
u
n
R
and it is
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chosen as a measurable function with respect to time; we denote a set of all
measurable functions
u
such that satisfies the condition
u
by
U
.
y
– a
position of
E
object in the space
n
R
,
0
1
(0),
(0)
y
y
y
y
– its initial position and
velocity respectively at
0;
t
v
– being a controlled acceleration of the evader,
mapping
: 0,
v
n
R
and it is chosen as a measurable function with respect
to time; we denote a set of all measurable functions
v
such that satisfies the
condition
v
by
V
.
Research
Methods
and
the
Received
Results.
Definition 1
.
For a trio of
0
1
( , , ( )), ( )
x x u
u
U
, the solution of the equation (1),
that is,
0
1
0 0
( )
( )
t s
x t
x
x t
u
d ds
is called a trajectory of the pursuer on
interval
0
t
.
Definition 2
.
For a trio of
0
1
(
,
, ( )), ( )
y
v
v
y
V
, the solution of the
equation (2), that is,
0
1
0 0
( )
( )
t s
y t
y
y t
v
d ds
is called a trajectory of the
evader on interval
0
t
.
Definition 3.
The pursuit problem for the differential game (1) - (2) is
called to be solved if there exists such control function
*
( )
u
U
of the pursuer
for any control function
( )
v
V
of the evader and the following equality is
carried out at some finite time
*
t
*
*
( )
( )
x t
y t
. (3)
Definition 4.
For the problem (1)-(2), time
T
is called a guaranteed
pursuit time if it is equal to an upper boundary of all the finite values of pursuit
time
*
t
satisfying the equality (3).
Definition 5.
For the differential game (1) - (2), the following function is
called П-strategy of the pursuer ([3]-[4]):
0
u v
v
v
, (4)
where
0
0
0
z
z
,
2
2
2
0
0
0
( ,
)
( ,
)
( ,
)
v
v
v
v
,
0
,
v
is a scalar multiplication of vectors
v
and
0
in the space
n
R
.
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Property 1.
If
, then a function
0
,
v
is continuous, nonnegative
and defined for all
v
such that satisfies the inequality
v
.
Property 2.
If
, then the following inequality is true for the
function
0
,
v
:
0
,
v
v
v
.
Theorem.
If one of the following conditions holds for the second order
differential game (1) – (2), that is, 1.
and
0;
k
or 2.
and
k
R
,
then by virtue of strategy (4) a guaranteed pursuit time becomes as follows
2
2
0
0
0
0
(
2
) / (
),
0
,
1 / ,
0
,
2
/ (
) ,
z k
z
k
z
agar k
va
k
agar k
va
T
z
0
.
agar k
va
Conclusion
Proof.
Suppose, let the pursuer choose the strategy in the form (4) when
the evader chooses any control function
( )
v
V
. Then, according to the
equations
(1)
– (2), we have the following Caratheodory’s equation:
0
z
v t
,
0
(0)
0
z
kz
Thus the following solution will be found by the given initial conditions
0
0
0
0 0
(
1)
( ),
t s
z t
z kt
v
d ds
or
2
2
2
0
0
0
0 0
(
1)
( ),
( ( ),
)
( )
t s
z t
z
kt
v
v
v
d ds
.
According to the properties 1- 2 , we will form the following inequalities
0
0 0
(
1)
(
( ) )
t s
z t
z
kt
v
d ds
2
0
(
1)
(
) / 2
z t
z
kt
t
.
If we say
2
0
, , , ,
1
,
2
t
f t a k
a kt
a
z
(5), then we will
check its properties
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1.
Let be
.
1.1.
If
0
k
, then the function
, , , ,
1
f t a k
a kt
is always
continuous and isn’t equal to zero (Fig-1).
(Figure-1)
1.2.
If
0
k
, then the function
0
, , , ,
f t a k
z
is a linear function
(Fig-2).
(Figure-2)
1.3.
If
0
k
, then the function (6) is decreasing and it equals to zero at
*
1
t
k
(Fig-3).
(Figure-3)
2. Let be
.
2.1. If
0
k
, then the function (6) is equal to zero at
2
2
0
0
0
2
/ (
)
T
z k
z
k
z
(Fig-4).
(Figure-4)
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In this case, a maximal time of unapproach is equal to
0
0
/ (
)
t
z k
and
therefore,
a
maximal
distance
between
them
equals
to
2
2
0
0
0
( )
2
/ 2(
)
f t
z
z
k
.
2.2. If
0
k
, then the function (6) decreases monotonically, and this
function turns to zero at time
T
as in the case 2.1 (Fig-5).
(Figure-5)
2.3. If
0
k
, then the function (6) becomes in the form
2
, , , ,
2
t
f t a k
a
and the pursuit time equals to the following
(Fig-6):
0
2
z
T
(Figure-6)
In conclusion, the relation (3) is true at some time
*
t
according to the
inequality
2
0
(
1)
(
) / 2
z t
z
kt
t
and properties of (5), and it is
determined that a relation
*
t
T
is correct, i.e., the pursuit problem is solved.
Proved.
Summary
In the theory of differential games, issues of chase and escape occupy a special
place in Aloxi. One of them is the breadth of implementation of various methods,
as well as the specificity of the results obtained. This feature is especially
obvious in model questions. In accordance with the condition given in the
lemma, the theorem is conditioned and provides a proof. In the theory of
differential games, questions in which geometric, integral and their joint
constraints are imposed on controls have been sufficiently studied. This article
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includes new control classes in a control function called delimitation, using
Gromwell’s lemma. The chase-escape problem in a second-order differential
game was studied, and new adequacy conditions were proposed for the pursuer
and the evader.
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