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The set 
)
,
,
,
(
1
0
0
B
A
t
t
K
is a convex compact of 
n
R
. Taking into account the equality (6), the 
formula (5) takes the form: 
)
(
)
,
(
)
,
,
,
(
)
,
,
,
,
(
1
1
0
1
0
0
1
0
t
M
t
t
Ф
B
A
t
t
K
B
A
M
t
t
K
A


. 
(7) 
If 
)
(
1
t
M
is a convex compact, then equality (7) can be written as the geometric difference 
)
(
)
,
(
)
,
,
,
(
)
,
,
,
,
(
1
1
0
1
0
0
*
1
0
t
M
t
t
Ф
B
A
t
t
K
B
A
M
t
t
K
A


. 
Let 

)
,
,
,
(
1
0
0
B
A
t
t
X
T
)
,
,
0
,
,
(
1
0
B
A
t
t
X
T
be the reachability set of system (3) at 
0
0

x
. 
Corollary 2. The formula is valid
)]
(
)
,
,
,
(
)[
,
(
)
,
,
,
,
(
1
1
0
0
1
0
1
0
t
M
B
A
t
t
X
t
t
Ф
B
A
M
t
t
K
T
A



. 
Theorem 2. Let 
1
0
t
t
t


. Then: 
)
,
,
,
,
(
)
,
,
,
,
(
0
1
0
B
A
M
t
t
K
B
A
M
t
t
K

, 
where 
)
,
,
,
,
(
)
(
1
B
A
M
t
t
K
t
M

. 
In fact, using the formula (5), we have: 








t
t
A
t
t
A
A
dt
t
B
t
t
Ф
t
M
t
t
Ф
dt
t
B
t
t
Ф
B
A
M
t
t
K
0
1
0
)
(
)
,
(
)
(
)
,
(
)
(
)
,
(
)
,
,
,
,
(
0
1
1
0
0
1
0
).
,
,
,
,
(
)
,
(
)
(
)
,
(
]
)
,
(
)
(
)
,
(
)[
,
(
1
0
0
0
1
0
0
1
B
A
M
t
t
K
t
t
Ф
dt
t
B
t
t
Ф
M
t
t
Ф
dt
t
B
t
t
Ф
t
t
Ф
t
t
A
A
t
t
A
A
A








Corollary 3. Let 
1
0
t
t
t


. Then the relation 
)
,
,
,
,
(
)
,
,
,
,
(
0
1
0
B
A
M
t
t
K
B
A
M
t
t
K

holds if and only if 
)
(
)
,
,
,
,
(
1
t
M
B
A
M
t
t
K

. 
Theorem 3. Let 
A
t
A

)
(
,
B
t
B

)
(
at 
]
,
[
1
0
t
t
T
t


. Then: 

)
(
1
0
1
0
1
)
,
,
,
,
(
)
,
,
,
,
(
t
M
B
A
t
t
X
B
A
M
t
t
K






. (8) 
In fact, using the Cauchy formula (4), we can write the following representation 










1
0
1
)
,
(
)
(
)
,
(
)
,
,
,
,
(
1
)
(
1
0
1
1
0
t
t
A
t
M
A
Bdt
t
t
t
M
t
t
B
A
t
t
X



. 
It 
is 
not 
difficult 
to 
see 
that 
),
,
(
)
,
(
1
0
0
1
t
t
t
t
A
A




)
,
(
)
,
(
1
1
t
t
t
t
A
A




, 
)
,
(
)
,
(
0
1
0
1
s
t
t
s
t
t
A
A





at 
]
,
[
1
0
t
t
s

. Now, taking into account these relations and making 
the substitution of variables 
t
t
t
s



0
1
in the integral



1
0
)
,
(
1
t
t
A
Bdt
t
t
, 
we get: 








1
0
1
)
,
(
)
(
)
,
(
)
,
,
,
,
(
0
)
(
1
1
0
1
0
t
t
A
t
M
A
Bds
s
t
t
M
t
t
B
A
t
t
X



. 
By virtue of theorem 1, the right-hand side of the last equality is the set 
).
,
,
,
,
(
1
0
B
A
M
t
t
K
4. Discussion of the results and conclusion.
The paper focuses on studying the properties of the auxiliary set 
)
,
,
,
(
1
0
F
M
t
t
K
, which 
can 
be 
used 
to 
study 
the 
properties 
of 
the 
M-controllability 
set 
)
(
\
))
,
,
,
(
(
)
,
(
0
1
0
0
1
t
M
F
M
t
t
K
F
M
W
t
t



. From the results obtained, we should note theorem 1, 

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5th Global Congress on Contemporary Sciences & Advancements 
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42
which gives the formula (5) for the representation of the set 
)
,
,
,
(
1
0
F
M
t
t
K
for the differential 
inclusion (3). This result allows us to find out some properties of the set of M-controllability.
In particular, the conditions of convexity and compactness of the set 
)
,
,
,
,
(
1
0
B
A
M
t
t
K
are 
specified. Theorem 2 and its corollary give an idea of the dynamics of the set 
)
,
,
,
,
(
1
0
B
A
M
t
t
K
. 
In Theorem 3, the formula (8) is given, which indicates a close connection of the sets of 
M-
controllability with the set of reachability of the differential inclusion (3) at 
A
t
A

)
(
, 
B
t
B

)
(
. 
In this paper, the problem of controllability of the trajectories of differential inclusions 
for the case of mobility of the terminal set 
M. The studied properties of the set 
)
,
,
,
(
1
0
F
M
t
t
K
allow us to clarify the structure of the set 
M-controllability of the considered class of 
differential inclusions. The obtained results are developed by the research work [9]. 
Referenses 
1.
Borisovich Yu. G., Gelman B. D., Myshkis A.D., Obukhovsky V. V. Introduction to the 
theory of multi-valued maps and differential inclusions. Moscow: KomKniga (2005).
2.
Clark F. Optimization and nonsmooth analysis. John Wiley & Sons, New York (1983).
3.
Polovinkin E. S. Multi-valued analysis and differential inclusions. Moscow: Fizmatlit 
(2015).
4.
Otakulov S. Problems of controlling an ensemble of trajectories of differential 
inclusions. LAP Lambert Academic Publishing (2019). 
5.
Otakulov S., Kholiyarova F.Kh. Time optimal control problem of ensemble trajectories 
of differential inclusion with delays. Journal of Advanced Research in dynamic and 
Control Systems, vol. 12, issue 6 (2020). pp. 1043-1050. 
6.
Otakulov S., Rahimov B. Sh. About the controllability property of an ensemble of 
trajectories of differential inclusion. International Engineering Journal of Research and 
Development. vol. 5, issue 4 (2020). pp. 366-374. 
7.
Blagodatskikh V. I., Filippov A. F. Differential inclusions and optimal control. 
Proceedings of the Mathematical Institute of the USSR Academy of Sciences.. vol.169 
(1985). pp. 194-252. 
8.
Warga J. Optimal control of differential and functional equations. Academic Press New 
York and London (1974). 
9.
Otakulov S., Rahimov B. Sh. On the structural properties of the reachability set of a 
differential inclusion. Proceedings of the International Conference "Research 
Innovations in Interdisciplinary Sciences", March 2021. Received From New York, USA 
(2021). pp. 150–153. 

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43
PROSPECTS OF TEACHING THE TOPIC "USE OF MULTIMEDIA 
IN EDUCATION" WITH THE METHOD OF "AQUARIUM" 
Abdusamatova Shaxodat Khojiakbar qizi,
Akmurodova Anora Kurbanov qizi 
Muhammad al-Khwarizmi University of Information Technology students majoring in ICT 
Tel: [phone], 
[e-mail]
 
Annotation:This article discusses the use of multimedia as one of the most popular tools in the 
educational process, teaching students through one of the modern teaching methods, and the 
fact that the method is based on discussion, which is interesting for students. 
Keywords: Multimedia, method, educational technology, multimedia technology, multimedia 
applications 
Improving the quality and efficiency of the education system in Uzbekistan in recent 
years has led to the formation of modern knowledge and skills among students, close 
cooperation and integration between the education system and science, Systematic work is 
being done to ensure the continuity and continuity of the lim. Prospective plans for the 
implementation of consistent measures to create the necessary conditions for their effective 
operation have been identified, for example, the President of the Republic of Uzbekistan "On 
the development of education and science in Uzbekistan in the new era of development. 
development measures ”. It sets out the following future plans
1

Creation of a system of automation and comprehensive analysis of education 
management using modern information and communication technologies, further 
development of electronic resources and distance learning, popularization of IT professions 
among students; 

Lantirish To make science the main driving force of the economy, to expand the scope of 
scientific research, to encourage the innovative activities of talented young scientists, to 
further strengthen and develop the potential of existing scientific organizations, etc. 
It is also important to introduce and use modern media in the education system of our 
country. Informing students about the media, the organization of practical classes on their 
use will help students to develop skills in this area. To this end, in this article we will 
consider the prospects of teaching the topic of the use of multimedia in education using the 
method of "Aquarium". 
The aquarium method is a method of studying the subject on the basis of forcing 
students to study the subject in depth on the basis of the desire to gain more knowledge in 
competition with each other. 
The process of using the "aquarium" method 
Step 1. Formation of initial concepts about the topic studied by the teacher. 
Step 2. In order to reinforce the topic, the method of "Aquarium" is used. Students are 
introduced to the concepts of this method, for example, it is necessary to mention the 
following: First, students are divided into two groups, and the members of one group are fish, 
the members of the second group are hunters. Once these groups are attached to their 
communities, the process begins. The fish community discusses the given topic and explains 
1
Decree of the President of the Republic of Uzbekistan "On measures to develop education and science in 
the new period of development of Uzbekistan" dated 07.11.2020, No. 06/20/6108/1483)

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5th Global Congress on Contemporary Sciences & Advancements 
Hosted from Singapore 
10th May 2021