Stability of solutions of Wazevski’s autonomous impulsive systems(in Ukrainian)

Author
O. I. Dvirny, V. I. Slyn’ko
The Arctic University of Norway, Tromso; Institute of Mechanics of Name of S. P. Timoshenko NAS of Ukraine
Abstract
In present paper we consider the problem of stability of autonomous systems of differential equations with impulsive action. Right sides of the system satisfy Wazewski’s condition. We propose new method for study of stability of trivial solutions for this class of the systems. Obtained results are illustrated by examples of the systems of differential equations in critical cases.
Keywords
impulsive differential equations; comparison principle; asymptotic stability
Reference
1. Samoilenko A.M., Perestyuk N.A., Impulsive differential equations. – World Scientific, Singapore, 1995.

2. Perestyuk N.A. On the stability of the equilibrium impulsive systems// Vuz : Appl. math. Sophia. – 1976. – V.11, ¹1. – P. 145–150. (in Russian)

3. Perestyuk N.A. Stability of solutions of linear systems with impulsive action// Vesti. Kiev. Univer. Mathematics and Mechanics. – 1977. – V.19. – P. 71–76. (in Russian)

4. Perestyuk N.A., Chernikova O.S. Some modern aspects of the theory of differential equations with impulse effect// Ukr. Math. J. – 2008. – V.60. – P. 81–94.

5. Ignat’ev A.O., Ignat’ev O.A., Soliman A.A. Asymptotic stability and instability of the solutions of systems with impulse action// Mathematical Notes. – 2006. – V.80, ¹4. – P. 491–499.

6. Ignat’ev A.O. On the stability of invariant sets of systems with impulse effect// Nonlinear Analisis. – V.69. – 2008. – P. 53–72.

7. Ignat’ev A.O., Ignat’ev O.A., Stability of solutions of systems with impulse effect, In:Progress in Nonlinear Analisys Research, Nova Science Publishers, Inc., 2009, P. 363–389.

8. Martynyuk A.A., Slyn’ko V.I. Stability of a nonlinear impulsive system// Int. Appl. Mech. – 2004. – V.40. – P. 231–239.

9. Dvirny A.I., Slyn’ko V.I. The conditions of global stability of solutions of nonstationary differential equations with impulse action in the pseudolinear form// Ukr. Math. Bull. – 2010. – V.8. – P. 182–202.

10. Dvirnyi A.I., Slyn’ko V.I. Global stability of solutions of nonstationary monotonic differential equations with impulsive action in the pseudo-linear form// Nonlinear Oscillations. – 2011. – V.14, ¹2. – P. 187–202.

11. Dvirnyi A.I., Slyn’ko V.I. Stabylity of solutions to impulsive differential equations in critical cases// Siberian Mathematical Journal. – 2011. – V.52, ¹1. – P. 54–62.

12. Haddad W.M., Impulsive and hybrid dynamical systems. Stability, dissipativity, and control, Princeton and Oxford: Princeton University Press, 2006, 504 p.

13. Dvirnyi A.I., Slyn’ko V.I. Stability criteria for quasilinear impulsive systems// Int. appl. mech. – 2004. – V.40, ¹5. – P. 592–599.

14. Matrosov V.M. The method of vector Lyapunov’s functions in systems with feedback// Automat. and Rem. cont. – 1972. – V.9. – P. 22–33. (in Russian)

15. Bailey F.N. The application of Lyapunov’s second method to interconnected systems// J. Soc. Indust. and Appl. Math. Ser. A. Control. – 1965. – V.3, ¹3. – P. 443–462.

16. Bellman R. Vector Lyapunov functions// J. Soc. Indust. and Appl. Math. Ser. A. Control. – 1962. – V.1, ¹1. – P. 32–34.

17. Martynyuk A.A., Lakshmikantham V., Leela S., Stability of Motion: a Method of Comparison, Nauk. Dumka, Kiev, 1991, 248 p. (in Russian)

18. Bainov D.D., Kulev G.K. Second method of Lyapunov and comparison principle for systems with impulse effect// J. Comput. and Appl. Math. – 1988. – V.22, ¹2. – P. 305–321.

19. Martynyk A.A., Obolensky A.Yu., Stability investigation of autonomous systems of comparison, Kiev, Institute of Mathematics USSR Academy of Sciences, 1978, 24 p. (in Russian)

20. Obolensky A.Yu. Stability of comparison systems// Dop. AS. Ukr. SSR. – 1979. – ¹8. – P. 607–611. (in Russian)

21. Obolensky A.Yu. Stability of linear comparison systems// Math. phys. and nonlinear mech. – 1984. – V.1. – P. 51–55. (in Russian)

22. Obolensky A.Yu. The stability of solutions of autonomous Wazewski systems with delay// Ukr. Math. J. – 1983. – V.35, ¹5. – P. 35–42. (in Russian)

23. Obolensky A.Yu., Criterias of motion stability of some nonlinear systems, Phoenix, Kiev, 2010, 228 p. (in Russian)

24. Dvirnyi A.I., Slyn’ko V.I. The stability of linear impulsive systems relative to the cone// Dop. NAS of Ukraine. – 2004. – V.4. – P. 37–43. (in Russian)

25. Dvirny A.I., Stability tests of impulsive systems via Lyapunov’s multicomponent functions, Thesis for a candidate’s degree, Kyiv, 2005, 167 p. (in Russian)

26. Slyn’ko V.I., Stability of motion of mechanical systems: hybrid models, Abstract of Thesis for a doctor’s degree, Kiev, 2009, 24 p. (in Russian)

27. Dvirnyi A.I., Slyn’ko V.I. The stability by two measures of abstract monotonic differential equations with impulsive action// Ukr. Math. J. – 2011. – V.63, ¹7. – P. 904–923.

28. Hartman Ph. Ordinary differential equations. M.: Mir, 1970, 720 p. (in Russian)

29. Krasnosel’skii M.A., Lifshitz E.A., Sobolev A.V., Positive linear systems, Nauka, Moskow, 1985, 256 p. (in Russian)
Pages
62-72
Volume
41
Issue
1
Year
2014
Journal
Matematychni Studii
Full text of paper
pdf
Table of content of issue