•  
  •  
 

Scientific Bulletin of Namangan State University

Abstract

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.

First Page

25

Last Page

33

References

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 , 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. 27. Satimov N.Yu. Methods of solving of pursuit problem in differential games, Tashkent: NUUz, 2003, - 245 p. 28. Samatov B.T. On a Pursuit-Evasion Problem under a Linear Change of the Pursuer Resource, Siberian Advances in Mathematics. – Allerton Press, Inc. Springer. - New York, 2013. V. 23. - No 4. - P. 294-302. 29. Samatov B.T. The Resolving Functions Method for the Pursuit Problem with Integral Constraints on Controls, Journal of Automation and Information Sciences. - Begell House, Inc. (USA). 2013. - V. 45, No 8. - P.41-58. 30. Samatov B.T. The П-strategy in a differential game with linear control constraints, J.Appl.Maths and Mechs. - Elsevier. - Netherlands, 2014. - V. 78. - No 3. - P. 258-263.

Erratum

???????

Share

COinS
 
 

To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.