GRONUOLL TIPIDAGI CHEGARALANISHLI CHIZIQLI DIFFERENSIAL O‘YIN  UCHUN QOCHISH MASALASI

Samatov Bahrom  Tadjiakhmatovich; Soyibboev Ulmasjon Boykuzi ugli; Akbarov Adakhambek Khasanbaevich

413

Ushbu maqolada differensial o‘yinlar nazariyasida Gronuoll
tipidagi tengsizliklarni qo‘llanishi ko‘rilgan. Bunda o‘yinchilarning boshqaruv funksiyalari,
geometrik chegaralanishlarni umumlashtiruvchi Gronuoll tipidagi chegaralanishlar uchun chiziqli
differensial o‘yinlarda qochish masalasi o‘rganiladi. Bu yerda qochish masalasini yechish uchun
qochuvchiga alohida strategiya taklif etiladi va o‘yinchilar orasidagi masofani aniqlovchi
funksiyaning xossalari o‘rganiladi. Maqolada Ayzeks, Petrosyan, Pshenichniy va boshqa
tadqiqotchilar, shuningdek mualliflarning avvalgi ishlari rivojlantiriladi va kengaytiriladi. Bunda
qochish masalasini yechish uchun yangi yetarlilik shartlari taklif etiladi.

Name of journalNamangan davlat universiteti ilmiy axborotnomasi
Number of edition2019-йил 10-сон
View count 413

Web Address

DOI

Date of creation in the UzSCI system 25-01-2021

Read count 405

Date of publication 25-01-2021

Main LanguageO'zbek

Pages26-33

Tags

geometrik chegaralanish

Differensial o‘yin

Gronuoll tengsizliklari

o‘yinchilar

qochish

Ўзбек

Ushbu maqolada differensial o‘yinlar nazariyasida Gronuoll
tipidagi tengsizliklarni qo‘llanishi ko‘rilgan. Bunda o‘yinchilarning boshqaruv funksiyalari,
geometrik chegaralanishlarni umumlashtiruvchi Gronuoll tipidagi chegaralanishlar uchun chiziqli
differensial o‘yinlarda qochish masalasi o‘rganiladi. Bu yerda qochish masalasini yechish uchun
qochuvchiga alohida strategiya taklif etiladi va o‘yinchilar orasidagi masofani aniqlovchi
funksiyaning xossalari o‘rganiladi. Maqolada Ayzeks, Petrosyan, Pshenichniy va boshqa
tadqiqotchilar, shuningdek mualliflarning avvalgi ishlari rivojlantiriladi va kengaytiriladi. Bunda
qochish masalasini yechish uchun yangi yetarlilik shartlari taklif etiladi.

Tags

geometrik chegaralanish

Differensial o‘yin

Gronuoll tengsizliklari

o‘yinchilar

qochish

Русский

Основная цель настоящей работы является применение
неравенства Грануолла в теории дифференциальных игр. Здесь рассматривается задача
убегания для линейных дифференциальных игр с ограничением типа Грануолла, которое в
некотором смысле обобщает геометрическое ограничение на управления игроков. Для
решения задачи предлагается специальная стратегия для убегающего игрока и изучается
функция определяющая расстояния между игроками. В настоящей статье развиваются
идеи предложенные в работах Айзекса, Петросяна, Пшеничного и других, а так же авторов.
Здесь получены новые достаточные условия разрешимости задачи убегания.

Tags

геометрическое ограничение

Дифференциальные игры

неравенство Гронуолла

игроки

убегания

English

The main aim of this work is to present some natural applications of Gronwall
type inequalities in the Differential Games. In the present, the evasion problem is studied in linear
differential games when Gronwall type constraints imposed on control functions of players. The
Gronwall type constraint generalizes geometrical constraint. To solve the evasion problem, we
propose a particular strategy for evader and study its structure depending on the parameters. This
work develops and extends the ideas of works of Isaacs, Petrosyan, Pshenichnii and other
researchers, including the author. Here the new suﬃcient solvability conditions for evader will be
proposed.

Tags

Differential game

Gronwall’s inequalities

players

geometrical constraint

evasion

№ Author name position Name of organisation

1 Samatov B.T.

2 Soyibboev U.B.

3 Akbarov A.K.

№ Name of reference

1 Gronwall T.H. Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math., 1919, 20(2): 293-296.

2 Azamov A.A. About the quality problem for the games of simple pursuit with the restriction, Serdika. Bulgarian math. spisanie, 12, 1986, - P.38-43.

3 Azamov A.A., Samatov B.T. П-Strategy. An Elementary introduction to the Theory of Differential Games. - T.: National Univ. of Uzb., 2000. - 32 p.

4 Azamov A.A., Samatov B.T. The П-Strategy: Analogies and Appli-cations, The Fourth International Conference Game Theory and Management, June 28-30, 2010, St. Petersburg, Russia, Collected papers. - P.33-47.

5 Azamov A., Kuchkarov A.Sh. Generalized 'Lion Man' Game of R. Rado, Contributions to game theori and management. Second International Conference "Game Theory and Management" - St.Petersburg, Graduate School of Manage-ment SPbU. - St.Petersburg, 2009. - Vol.11. - P. 8-20.

6 Azamov A.A., Kuchkarov A.Sh., Samatov B.T. The Relation between Problems of Pursuit, Controllability and Stability in the Large in Linear Systems with Different Types of Constraints, J.Appl.Maths and Mechs. - Elsevier. - Netherlands, 2007. - Vol. 71. - N 2. - P. 229-233.

7 Barton J.C, Elieser C.J. On pursuit curves, J. Austral. Mat. Soc. B. - London, 2000. - Vol. 41.- N 3. - P. 358-371.

8 Borovko P., Rzymowsk W., Stachura A. Evasion from many pursuers in the simple case, J. Math. Anal. And Appl. - 1988. - Vol.135. - N 1. - P. 75-80.

9 Chikrii A.A. Conflict-controlled processes, Boston-London-Dordrecht: Kluwer Academ. Publ., 1997, 424 p.

10 Fleming W. H. The convergence problem for differential games, J. Math. Anal. Appl. - 1961. - N 3. - P. 102-116.

11 A. Friedman. Differential Games, New York: Wiley, 1971, - 350 p.

12 Hajek O. Pursuit Games: An Introduction to the Theory and Appli-cations of Differential Games of Pursuit and Evasion. - NY.:Dove. Pub. 2008. - 288 p.

13 Isaacs R. Differential Games, J. Wiley, New York-London-Sydney, 1965, 384 p.

14 Ibragimov G.I. Collective pursuit with integral constrains on the controls of players, Siberian Advances in Mathematics, 2004, v.14, No.2, - P.13-26.

15 Ibragimov G.I., Azamov A.A., Khakestari M. Solution of a linear pursuit-evasion game with integral constraints, ANZIAM Journal. Electronic Supplement. - 2010. - Vol.52. - P. E59-E75.

16 Krasovskii A.N., Choi Y.S. Stochastic Control with the Leaders-Stabilizers. - Ekaterinburg: IMM Ural Branch of RAS, 2001. - 51 p.

17 Krasovskii A.N., Krasovskii N.N. Control under Lack of Information. - Berlin etc.: Birkhauser, 1995. – 322, p.

18 Kuchkarov A.Sh. Solution of Simple Pursuit-Evasion Problem When Evader Moves on a Given Curve, International Game Theory Review. - World Scientific Publishing Company, 2010. - Vol.12. - N 3, - P. 223-238.

19 Miller B., Rubinovich E.Y. Impulsive Control in Continuous and Discrete-Continuous Systems. - N.Y.: Kluwer Academic/Plenum Publishers, 2003. - 447 p.

20 Nahin P.J. Chases and Escapes: The Mathematics of Pursuit and Evasion. Princeton University Press, Princeton, 2012, - 260.

21 Petrosyan L.A. About some of the family differential games at a survival in the space n R , Dokl. Akad. Nauk SSSR, 1965, 161, No1, -P.52-54.

22 Petrosyan L.A. The Differential Games of pursuit, Leningrad, LSU, 1977, - 224 p.

23 Petrosyan L.A., Rixsiev B.B. Presledovanie na ploskosti [Pursuit on the plane], Nauka, Moscow, 1991, - 96 p.

24 Pontryagin L.S. Lineyniy differentsialnie igri presledovaniya ["Linear Differential Pursuit Games"], Math. Sb. [Math. USSR-Sb], 112, No.3, -P.307-330.

25 Pshenichnii B.N. The simple pursuit with some objects, Cybernetics, 1976, No.3, -P.145- 146.

26 Rikhsiev B.B. The differential games with simple motions, Tashkent: Fan, 1989, - 232 p.

Waiting

№	Name of reference
1	Gronwall T.H. Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math., 1919, 20(2): 293-296.
2	Azamov A.A. About the quality problem for the games of simple pursuit with the restriction, Serdika. Bulgarian math. spisanie, 12, 1986, - P.38-43.
3	Azamov A.A., Samatov B.T. П-Strategy. An Elementary introduction to the Theory of Differential Games. - T.: National Univ. of Uzb., 2000. - 32 p.
4	Azamov A.A., Samatov B.T. The П-Strategy: Analogies and Appli-cations, The Fourth International Conference Game Theory and Management, June 28-30, 2010, St. Petersburg, Russia, Collected papers. - P.33-47.
5	Azamov A., Kuchkarov A.Sh. Generalized 'Lion Man' Game of R. Rado, Contributions to game theori and management. Second International Conference "Game Theory and Management" - St.Petersburg, Graduate School of Manage-ment SPbU. - St.Petersburg, 2009. - Vol.11. - P. 8-20.
6	Azamov A.A., Kuchkarov A.Sh., Samatov B.T. The Relation between Problems of Pursuit, Controllability and Stability in the Large in Linear Systems with Different Types of Constraints, J.Appl.Maths and Mechs. - Elsevier. - Netherlands, 2007. - Vol. 71. - N 2. - P. 229-233.
7	Barton J.C, Elieser C.J. On pursuit curves, J. Austral. Mat. Soc. B. - London, 2000. - Vol. 41.- N 3. - P. 358-371.
8	Borovko P., Rzymowsk W., Stachura A. Evasion from many pursuers in the simple case, J. Math. Anal. And Appl. - 1988. - Vol.135. - N 1. - P. 75-80.
9	Chikrii A.A. Conflict-controlled processes, Boston-London-Dordrecht: Kluwer Academ. Publ., 1997, 424 p.
10	Fleming W. H. The convergence problem for differential games, J. Math. Anal. Appl. - 1961. - N 3. - P. 102-116.
11	A. Friedman. Differential Games, New York: Wiley, 1971, - 350 p.
12	Hajek O. Pursuit Games: An Introduction to the Theory and Appli-cations of Differential Games of Pursuit and Evasion. - NY.:Dove. Pub. 2008. - 288 p.
13	Isaacs R. Differential Games, J. Wiley, New York-London-Sydney, 1965, 384 p.
14	Ibragimov G.I. Collective pursuit with integral constrains on the controls of players, Siberian Advances in Mathematics, 2004, v.14, No.2, - P.13-26.
15	Ibragimov G.I., Azamov A.A., Khakestari M. Solution of a linear pursuit-evasion game with integral constraints, ANZIAM Journal. Electronic Supplement. - 2010. - Vol.52. - P. E59-E75.
16	Krasovskii A.N., Choi Y.S. Stochastic Control with the Leaders-Stabilizers. - Ekaterinburg: IMM Ural Branch of RAS, 2001. - 51 p.
17	Krasovskii A.N., Krasovskii N.N. Control under Lack of Information. - Berlin etc.: Birkhauser, 1995. – 322, p.
18	Kuchkarov A.Sh. Solution of Simple Pursuit-Evasion Problem When Evader Moves on a Given Curve, International Game Theory Review. - World Scientific Publishing Company, 2010. - Vol.12. - N 3, - P. 223-238.
19	Miller B., Rubinovich E.Y. Impulsive Control in Continuous and Discrete-Continuous Systems. - N.Y.: Kluwer Academic/Plenum Publishers, 2003. - 447 p.
20	Nahin P.J. Chases and Escapes: The Mathematics of Pursuit and Evasion. Princeton University Press, Princeton, 2012, - 260.
21	Petrosyan L.A. About some of the family differential games at a survival in the space n R , Dokl. Akad. Nauk SSSR, 1965, 161, No1, -P.52-54.
22	Petrosyan L.A. The Differential Games of pursuit, Leningrad, LSU, 1977, - 224 p.
23	Petrosyan L.A., Rixsiev B.B. Presledovanie na ploskosti [Pursuit on the plane], Nauka, Moscow, 1991, - 96 p.
24	Pontryagin L.S. Lineyniy differentsialnie igri presledovaniya ["Linear Differential Pursuit Games"], Math. Sb. [Math. USSR-Sb], 112, No.3, -P.307-330.
25	Pshenichnii B.N. The simple pursuit with some objects, Cybernetics, 1976, No.3, -P.145- 146.
26	Rikhsiev B.B. The differential games with simple motions, Tashkent: Fan, 1989, - 232 p.