Advertisement

Ukrainian Mathematical Journal

, Volume 70, Issue 1, pp 30–41 | Cite as

Stability of Global Attractors of Impulsive Infinite-Dimensional Systems

  • O. V. Kapustyan
  • M. O. Perestyuk
  • I. V. Romanyuk
Article

We prove the stability of global attractor for an impulsive infinite-dimensional dynamical system. The obtained abstract results are applied to a weakly nonlinear parabolic equation whose solutions are subjected to impulsive perturbations at the times of crossing a certain surface of the phase space.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. M. Samoilenko and A. D. Myshkis, “Systems with pushes at given times,” Mat. Sb., 74, Issue 2, 202–208 (1967).MathSciNetGoogle Scholar
  2. 2.
    A. M. Samoilenko, “Averaging method in systems with pushes,” Mat. Fiz., Issue 9, 101–117 (1971).Google Scholar
  3. 3.
    A. M. Samoilenko and N. A. Perestyuk, “Stability of solutions of differential equations with impulsive actions,” Differents. Uravn., 13, 1981–1992 (1977).MathSciNetGoogle Scholar
  4. 4.
    A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulsive Action [in Russian], Kiev State University, Kiev (1980).Google Scholar
  5. 5.
    N. A. Perestyuk, “Invariant sets of a class of discontinuous dynamical systems,” Ukr. Mat. Zh., 36, No. 1, 63–68 (1984); English translation: Ukr. Math. J., 36, No. 1, 58–62 (1984).Google Scholar
  6. 6.
    A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulsive Action [in Russian], Vyshcha Shkola, Kiev (1987).Google Scholar
  7. 7.
    A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore (1995).CrossRefzbMATHGoogle Scholar
  8. 8.
    V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore (1989).CrossRefGoogle Scholar
  9. 9.
    V. Rozko, “Stability in terms of Lyapunov of discontinuous dynamic systems,” Differents. Uravn., 11, No. 6, 1005–1012 (1975).MathSciNetGoogle Scholar
  10. 10.
    S. K. Kaul, “Stability and asymptotic stability in impulsive semidynamical systems,” J. Appl. Math. Stochast. Anal., 7, No. 4, 509–523 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    T. Pavlidis, “Stability of a class of discontinuous dynamical systems,” Inform. Contr., 9, 298–322 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    K. Ciesielski, “On stability in impulsive dynamical systems,” Bull. Pol. Acad. Sci. Math., 52, 81–91 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M. Akhmet, Principles of Discontinuous Dynamical Systems, Springer, New York (2010).CrossRefzbMATHGoogle Scholar
  14. 14.
    E. M. Bonotto, “Flows of characteristic 0+ in impulsive semidynamical systems,” J. Math. Anal. Appl., 332, 81–96 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yu. M. Perestyuk, “Discontinuous oscillations in one impulsive system,” Nelin. Kolyv., 15, No. 4, 494–503 (2012); English translation: J. Math. Sci., 194, No. 4, 404–413 (2013).Google Scholar
  16. 16.
    K. Li, C. Ding, F. Wang, and J. Hu, “Limit set maps in impulsive semidynamical systems,” J. Dynam. Control Syst., 20, No. 1, 47–58 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    P. Feketa and Yu. Perestyuk, “Perturbation theorems for a multifrequency system with pulses,” Nelin. Kolyv., 18, No. 2, 280–289 (2015); English translation: J. Math. Sci., 217, No. 4, 515–524 (2016).Google Scholar
  18. 18.
    J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI (1988).zbMATHGoogle Scholar
  19. 19.
    R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York (1988).CrossRefzbMATHGoogle Scholar
  20. 20.
    I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems [in Russian], ASTA, Kharkiv (1999).Google Scholar
  21. 21.
    V. S. Melnik, Multivalued Dynamics of Nonlinear Infinite-Dimensional Systems, Preprint No. 94-17, Institute of Cybernetics, Ukrainian National Academy of Sciences, Kyiv (1994).Google Scholar
  22. 22.
    V. S. Melnik and J. Valero, “On attractors of multivalued semiflows and differential inclusions,” Set-Valued Anal., No. 6, 83–111 (1998).Google Scholar
  23. 23.
    V. S. Melnik and O. V. Kapustyan, “On global attractors of multivalued semidynamic systems and their approximations,” Dokl. Akad. Nauk, 366, No. 2, 445–448 (1998).Google Scholar
  24. 24.
    O. V. Kapustyan and D. V. Shkundin, “Global attractor of one nonlinear parabolic equation,” Ukr. Mat. Zh., 55, No. 4, 446–455 (2003); English translation: Ukr. Math. J., 55, No. 4, 535–547 (2003).Google Scholar
  25. 25.
    O. V. Kapustyan, P. O. Kasyanov, and J. Valero, “Regular solutions and global attractors for reaction-diffusion systems without uniqueness,” Comm. Pure Appl. Anal., 13, No. 5, 1891–1906 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI (2002).zbMATHGoogle Scholar
  27. 27.
    O. V. Kapustyan and M. O. Perestyuk, “Global attractor for an evolution inclusion with pulse influence at fixed moments of time,” Ukr. Mat. Zh., 55, No. 8, 1058–1068 (2003); English translation: Ukr. Math. J., 55, No. 8, 1283–1294 (2003).Google Scholar
  28. 28.
    B. Schmalfuss, “Attractors for nonautonomous and random dynamical systems perturbed by impulses,” Discrete Contin. Dynam. Syst., 9, 727–744 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    G. Iovane and O. V. Kapustyan, “Global attractor for impulsive reaction-diffusion equation,” Nelin. Kolyv., 8, No. 3, 319–328 (2005); English translation: Nonlin. Oscillat., 8, No. 3, 318–328 (2005).Google Scholar
  30. 30.
    O. V. Kapustyan, J. Valero, and G. Iovane, “Asymptotic behavior of reaction-diffusion equations with nondamped impulsive effects,” Nonlin. Anal., 68, 2516–2530 (2008).CrossRefzbMATHGoogle Scholar
  31. 31.
    X. Yan, Y. Wub, and C. Zhong, “Uniform attractors for impulsive reaction-diffusion equations,” Appl. Math. Comput., 216, 2534–2543 (2010).MathSciNetzbMATHGoogle Scholar
  32. 32.
    M. O. Perestyuk and O. V. Kapustyan, “Long-time behavior of evolution inclusion with nondamped impulsive effects,” Mem. Different. Equat. Math. Phys., 56, 89–113 (2012).zbMATHGoogle Scholar
  33. 33.
    E. M. Bonotto and D. P. Demuner, “Attractors of impulsive dissipative semidynamical systems,” Bull. Sci. Math., 137, 617–642 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    E. M. Bonotto, M. C. Bortolan, A. N. Carvalho, and R. Czaja, “Global attractors for impulsive dynamical systems — a precompact approach,” J. Different. Equat., 259, 2602–2625 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    E. M. Bonotto, M. C. Bortolan, R. Collegary, and R. Czaja, “Semicontinuity of attractors for impulsive dynamical systems,” J. Different. Equat., 261, 4358–4367 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    M. O. Perestyuk and O. V. Kapustyan, “Existence of global attractors for impulsive dynamical systems,” Dokl. Nats. Akad. Nauk. Ukr., Mathematics, 12, 13–18 (2015).Google Scholar
  37. 37.
    M. O. Perestyuk and O. V. Kapustyan, “Global attractors of impulsive infinite-dimensional systems,” Ukr. Mat. Zh., 68, No. 4, 517–528 (2016); English translation: Ukr. Math. J., 68, No. 4, 583–597 (2016).Google Scholar
  38. 38.
    S. Dashkovskiy, O. V. Kapustyan, and I. V. Romaniuk, “Global attractors of impulsive parabolic inclusions,” Discrete Contin. Dynam. Syst., Ser. B, 22, No. 5, 1875–1886 (2017).Google Scholar
  39. 39.
    N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems, Springer, New York (2002).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • O. V. Kapustyan
    • 1
  • M. O. Perestyuk
    • 1
  • I. V. Romanyuk
    • 1
  1. 1.Shevchenko Kyiv National UniversityKyivUkraine

Personalised recommendations