Associate Professor of the Department of Mathematics, Andijan State University, Republic of Uzbekistan, Andijan
THE PROBLEM OF CAPTURING ONE FUGITIVE IN A GROUP DIFFERENTIAL GAME
ABSTRACT
In this paper, using the example of L. S. Pontryagin, we study the issue of the pursuit of one fugitive by several pursuers. All players have the same dynamic options. Such issues have been studied in the literature. In this paper, the interception problem is solved under the assumption that the solution of the Cauchy problem for a system-compatible homogeneous system is recurrent.
АННОТАЦИЯ
В данной работе на примере Л. С. Понтрягина исследуется вопрос о преследовании одного беглеца несколькими преследователями. Все игроки имеют одинаковые динамические возможности. Такие вопросы изучались в литературе. В данной работе задача перехвата решается в предположении рекуррентности решения задачи Коши для системно-совместимой однородной системы.
Keywords: differential game, group pursuit, capture problem, Pontryagin's example.
Ключевые слова: дифференциальная игра, групповое преследование, задача поимки, пример Понтрягина.
we consider a differential game with
players in a space where
are the pursuers and E is the fugitive. The law of motion of each hunter for Pi
The law of motion of every hunter Pi
has a look. E - evasive law of motion
has a look. Here and then [t_0,∞) are continuous in the interval
- is a compact set with a rigid convex smooth boundary, a part of the space
.
.
given initial condition [1], that's all
.
Denote the game under consideration by .
(1) – (4) instead of these systems
System
look at the initial state. let it be.
is the history of the control
of the evader at time
.
reported to the collection.
1- definition. If back to original state,
va
voluntarily fugitive
control history values
receiver from the collection
size function fit
if a reflection is detected
say that the quasi-strategy is given.
2-definition. If so
So
if quasi-strategies are found, optional
for the dimensional function
equality holds
number and
if there is a moment, then it is
we say that it can be caught in the game.
through
the equation of
let's determine the solution in the initial conditions. The following function
and we will enter the set.
3-definition. If optional so for
if found, optional
is for
inequality would be appropriate if available,
the function is called recurrent (abbreviated recurrent) in the sense of Zubov.
If for all
can be chosen independently of all t, then
is said to be almost periodic [2].
4-definition. If all at points
satisfying equation
if there is a recursive function
function
in the interval it is called recurrent (recurrent for short) in the sense of Zubov.
1-lemma. Let's assume all so for s
vectors exist
let the relationship be fulfilled and
let the functions be recurrent [3]. That's how it is
va
the following confirmations are appropriate:
1. and all
in
relationship is fulfilled here
;
2. Each so for
moments are found such that
the inequality holds.
Proof. set ends
consists of a convex polyhedron with points. From the condition of the lemma
originates.
– open set. So it is
number is optional
for
attitude is appropriate.
of the lemma from the relation 1-it seems that the confirmation is correct.
functions are recursive, then optional
so for
there is every one
so for
moments are found such that
the inequality holds. The lemma is proved [4]. In the next places
va
2.1- we consider that it is chosen according to the conditions of the lemma.
We define the following functions:
This
we enter the designations.
2-lemma. The following conditions must be met:
1. functions
recurrent in;
2. ;
3. All so for is
vectors exist
attitude will be appropriate.
That's how it is the moment is found that for arbitrary control
and arbitrary
there exists such a number
for which the inequality
is fulfilled.
Isbot. From the condition of the lemma for arbitrary
the fulfillment of the inequality follows.
[3] according to lemma 1.3.13 in the article, the function λ is continuous in every set , from which
In turn, D is in the set functions are continuous.
taking into account the compactness of the set
we generate the inequality. From this
So
The inequality holds for the moment
and some number
defined by the condition. The lemma is proved.
let it be According to Lemma 2, the inequality holds.
1- Theorem. The following conditions must be met:
1. da
functions recursive;
2. There are such vectors that the relation
is fulfilled;
3. There are moments for which
relationship is fulfilled.
U xolda o’yinda tutib olish mumkin.
Proof. In arbitrary admissible controls, the solution of problem (5), (6) is based on the Cauchy formula
has a view. Let us assume that - moments satisfy the conditions of the theorem, and
- be an arbitrary admissible control of the escaper
, where
.
This
Let's look at the function. Let us denote the smallest root of this function by It should be noted that according to the 3 condition of the theorem, there is a moment
and the inequality
holds for at least one i.
Moreover, there exists a number shunday such that for
equality holds. We control the chaser in all
as follows:
here larda
we think that.
In that case, it is based on the Cauchy formula
according to the determination of,
the expression in parentheses above the number becomes zero, so
. The theorem is proved.
1-Result. The following conditions must be met:
1. da
functions recursive;
2. relation is fulfilled;
In that case, can be caught in the game.
An example. (4) let the system be as follows
here
it is not difficult to show that the function is recurrent.
let it be
, here
So the function is recurrent in
and, in turn, the function
is recurrent.
Confirmation. If it is possible to catch in this game.
References:
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- Понтрягин, Л.С. Избранные научные труды : в 3-х т. Т. 2. Дифференциальные уравнения. Теория операторов. Оптимальное управление. Дифференциальные игры/Л.С. Понтрягин; отв. ред. Р.В.Гамкрелидзе. — М.: Наука, 1988. — 575 с
- Чикрий А.А. Конфликтно управляемые процессы. Киев: Наукова думка, 1992. 380с.
- Umrzaqov, Nodirbek (2021) "Sufficient condition for the possibility of completing the pursuit," Scientific Bulletin. Physical and Mathematical Research: Vol. 3 : Iss. 1 , Article 14.