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Scientific Bulletin. Physical and Mathematical Research

Abstract

A scientific school on differential games was founded by N.Y.Satimov in Uzbekistan, and is being continued by A.Azamov now. The representatives of this school have achieved significant success in this area, have created remarkable ways in the theory of pursuit with a geometric constraint on the parameters of control. It is noteworthy that the theorem of A.Azamov [3] on this problem practically solved the famous problem created by Pontryagin. Significant results were also obtained with an integral constraint on the player control parameter. Influential world journals published the results obtained in this direction by N.Y.Satimov, B.B.Rikhsiev, G.I.Ibragimov, A.A.Khamdamov, and others [4-6]. Thanks to in-depth scientific research, A.Z.Fozilov and M.S.Nikolskiy achieved the transfer of the second Pontryagin method to a position with an integral constraint of control parameters [7]. We study a pursuit differential game of m pursuers and one evader Ewhich is described by the following differential equations (1) where — control parameter of the pursuer , and E of evader— . Assume controls of the pursuers and evader are defined as the measurable function , , subjected to contraints: (2) (3) (4) (5) where – positive numbers. Defenition 1.A function , Is referred to as the strategy of the pursuer iffor an arbitrary admissible controlof the evader , the initial value problem has a unique solution and along this solution the following coordinate-wise integral contraints are satisfied. Assume.There are sets , satisfying condition , for set and , let it satisfy . Theorem 1.If assume is performed, then ingame(1), (2)-(5) d multiple pursuit comes from any starting position . Theorem 2.If the following inequalities satisfy longd multiple pursuit occur. Theorem 3. If one-time pursuit occur.

First Page

86

Last Page

96

References

1. Azamov, A. (1982) O vtorom metode Pontryagina v linejnyh differencial'nyh igrah presledovaniya [On Pontryagin’s second method in linear differential games of pursuit]. Matematicheskij sbornik. Vol. 118(160).Issue 3(7). pp. 422-430. 2. Pontryagin, L.S. (1980) Linejnye differencial'nye igry presledovaniya [Linear differential games of pursuit]. Matematicheskij sbornik. Vol. 112(154).Issue 3(7). pp. 307-330. 3. Azamov, A. (1988) Semistability and Duality in the Theory of Pontryagin Alternating Integral. Soviet Mathematics Doklady. Vol. 37.Issue 2. pp. 355-359. 4. Kuchkarov, A., Ibragimov, G.I. and Khakestari, M. (2013) On a linear differential game of optimal approach of many pursuers with one evader. Journal of Dynamical and Control Systems.Vol. 19(1). pp. 1-15. 5. Alias, I.A., Ibragimov, G.I., Kuchkarov, A., Sotvoldiyev, A. (2016)Differential Game with Many Pursuers When Controls are Subjected to Coordinate-wise Integral Constraints.Malaysian of Mathematical Sciences.Vol. Issue 10(2). pp. 195-207. 6. Chikriy, A.A., Belousov, A.A. (2009) O linejnyh differencial'nyh igrah s integral'nymi ogranicheniyami [On linear differential games with integral constraints]. Trudy Instituta matematiki i mekhaniki UrO RAN.Vol. 15. №4. pp. 290-301. 7. Nikolskiy, M.S. (1969) The direct method in linear differential games with integral constraints. Upravlyaemaya sistema. IM, IK, SO AN SSSR. Issue2. pp. 49-59. 8. Satimov, N.Y., Rikhsiev, B.B., Khamdamov, A.A. [1982] O zadache presledovaniya dlya linejnyh differencial'nyh i diskretnyh igr mnogih lic s integral'nymi ogranicheniyami [On pursuit and evasion problems for multi-person linear differential and discrete games with integral constraints]. Matematicheskij sbornik. Vol. 118(160).Issue4(8). pp. 456–469. 9. Samatov, B.T. (2013)The Resolving Functions Method for the Pursuit Problem with Integral Constraints on Controls.Journal of Automation and Information Sciences. Vol 45.Issue8. pp. 41-58. 10. Satimov, N.Y., Fazilov, A.Z., Khamdamov. (1984) O zadachah presledovaniya i ukloneniya v differencial'nyh i diskretnyh igrah mnogih lic s integral'nymi ogranicheniyami [On a pursuit problem for a person linear differential and discrete games with integral constraints]. Differencialnye uravneniya.Vol. 20. Issue8. pp. 1388-1396. 11. Tukhtasinov, M., Kuchkarova, S.A. (2017) Integral chegarali differensial o’yinda yetarli shartlar [Sufficient conditions for differential equations with integral constraint]. Actual problems of differential equations and their applications. Tashkent. pp. 140-142. Mualliflar haqida ma’lumot To‘xtasinov Mo‘minjon¬ – fizika-matematika fanlari doktori, O‘zbekiston Milliy universiteti differensial tenglamalar va matematik fizika kafedrasi professori. E-mail: mumin51@mail.ru. Quchqarova Sarvinoz Atamuratovna – O‘zbekiston Milliy universiteti differensial tenglamalar va matematik fizika kafedrasi magistranti. E-mail: kuchkarov1@yandex.ru.

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