he main aim of this work is to present some natural applications of Gronwall
type inequalities in the Differential Games. We study the evasion problem for differential game of
the first order when geometrical constraint imposed on control function of pursuer and Gronwall
type constraint imposed on control function of evader. The Gronwall type constraint generalizes
geometrical constraint. To solve the evasion problem, we suggest a particular strategy for evader
and study its structure depending on the parameters. Here the new sufficient solvability conditions
for solution of evasion problem will be proposed.
he main aim of this work is to present some natural applications of Gronwall
type inequalities in the Differential Games. We study the evasion problem for differential game of
the first order when geometrical constraint imposed on control function of pursuer and Gronwall
type constraint imposed on control function of evader. The Gronwall type constraint generalizes
geometrical constraint. To solve the evasion problem, we suggest a particular strategy for evader
and study its structure depending on the parameters. Here the new sufficient solvability conditions
for solution of evasion problem will be proposed.
Ushbu maqolada differensial o‘yinlar nazariyasida Gronuoll
tipidagi tengsizliklarni qo‘llanishi ko‘rilgan. Bunda quvlovchining boshqaruv funksiyasiga
geometrik chegaralanish va qochuvchining boshqaruv funksiyasiga Gronuoll tipidagi chegaralanish
qo‘yilgan holda birinchi tartibli differensial o’yin uchun qochish masalasi o‘rganiladi. Gronuoll
tipidagi chegaralanish geometrik chegaralanishni umumlashtiradi. Bu yerda qochish masalasini
yechish uchun qochuvchiga alohida strategiya taklif etiladi va uni strukturasi parametrlarga bog‘liq
holda o‘rganiladi. Bunda qochish masalasini yechish uchun yangi yetarlilik shartlari taklif etiladi.
В данной работе рассматривается неравенства Грануолла в теории
дифференциальных игр. Здесь рассматриваются задача убегания для дифференциальных игр
первого порядка, которая для игроков задаются преследователя геометрическая
ограничения и убегавшего ограничения Грануолла. Ограничения Грануолла являются
обобщением геометрического ограничения. Для решения задачи убегания применяется
стратегия параллельного сближения игроков ( -стратегия) и изучается ее структура в
зависимости от заданных параметров. Здесь получены новые достаточные условия
разрешимости задачи убегания.
| № | Author name | position | Name of organisation |
|---|---|---|---|
| 1 | Samatov .T. | o'qituvchi | NamDU. |
| 2 | Juraev B.I. | o'qituvchi | NamDU. |
| № | 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.38-4 |
| 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 theory 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 | Chikrii A.A. Conflict-controlled processes, Boston-London-Dordrecht: Kluwer Academ. Publ., 1997, 424 p. |
| 7 | Fleming W. H. The convergence problem for differential games, J. Math. Anal. Appl. - 1961. - N 3. - P. 102-116. |