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 P_i

The law of motion of every hunter P_i

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:

1. Григоренко, Н.Л. Математические методы управления несколькими динамическими процессами/Н.Л. Григоренко.—М.: МГУ, 1990.—197 с.
2. 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
3. Чикрий А.А. Конфликтно управляемые процессы. Киев: Наукова думка, 1992. 380с.
4. Umrzaqov, Nodirbek (2021) "Sufficient condition for the possibility of completing the pursuit," Scientific Bulletin. Physical and Mathematical Research: Vol. 3 : Iss. 1 , Article 14.