Uniform attractors in sup-norm for semi linear parabolic problem and application to the robust stability theory

Oleksiy Kapustyan; Olena Kapustian; Oleksandr Stanzytskyi; Ihor Korol

Archivum Mathematicum (2023)

  • Issue: 2, page 191-200
  • ISSN: 0044-8753

Abstract

top
In this paper we establish the existence of the uniform attractor for a semi linear parabolic problem with bounded non autonomous disturbances in the phase space of continuous functions. We applied obtained results to prove the asymptotic gain property with respect to the global attractor of the undisturbed system.

How to cite

top

Kapustyan, Oleksiy, et al. "Uniform attractors in sup-norm for semi linear parabolic problem and application to the robust stability theory." Archivum Mathematicum (2023): 191-200. <http://eudml.org/doc/298987>.

@article{Kapustyan2023,
abstract = {In this paper we establish the existence of the uniform attractor for a semi linear parabolic problem with bounded non autonomous disturbances in the phase space of continuous functions. We applied obtained results to prove the asymptotic gain property with respect to the global attractor of the undisturbed system.},
author = {Kapustyan, Oleksiy, Kapustian, Olena, Stanzytskyi, Oleksandr, Korol, Ihor},
journal = {Archivum Mathematicum},
keywords = {parabolic equations; attractor; stability},
language = {eng},
number = {2},
pages = {191-200},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Uniform attractors in sup-norm for semi linear parabolic problem and application to the robust stability theory},
url = {http://eudml.org/doc/298987},
year = {2023},
}

TY - JOUR
AU - Kapustyan, Oleksiy
AU - Kapustian, Olena
AU - Stanzytskyi, Oleksandr
AU - Korol, Ihor
TI - Uniform attractors in sup-norm for semi linear parabolic problem and application to the robust stability theory
JO - Archivum Mathematicum
PY - 2023
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
IS - 2
SP - 191
EP - 200
AB - In this paper we establish the existence of the uniform attractor for a semi linear parabolic problem with bounded non autonomous disturbances in the phase space of continuous functions. We applied obtained results to prove the asymptotic gain property with respect to the global attractor of the undisturbed system.
LA - eng
KW - parabolic equations; attractor; stability
UR - http://eudml.org/doc/298987
ER -

References

top
  1. Asrorov, F., Sobchuk, V., Kurylko, O., 10.15587/1729-4061.2019.178635, East-Europ. J. Enterprise Technol. 6 (4(102)) (2019), 14–20. (2019) DOI10.15587/1729-4061.2019.178635
  2. Barabash, O., Dakhno, N., Shevchenko, H., Sobchuk, V., Unmanned aerial vehicles flight trajectory optimisation on the basis of variational enequality algorithm and projection method, Proceeding 2019 IEEE 5th International Conference “Actual Problems of Unmanned Aerial Vehicles Developments” (APUAVD), National Aviation University, Kyiv, Ukraine, 2019, pp. 136–139. (2019) 
  3. Chepyzkov, V.V., Vishik, M.I., Attractors for equations of mathematical physics, vol. 49, AMS Colloquium Publications, 2002. (2002) MR1868930
  4. Dashkovskiy, S., Feketa, P., Kapustyan, O., Romaniuk, I., 10.1016/j.jmaa.2017.09.001, J. Math. Anal. Appl. 458 (1) (2018), 193–218. (2018) MR3711900DOI10.1016/j.jmaa.2017.09.001
  5. Dashkovskiy, S., Kapustyan, O., Romaniuk, I., Global attractors of impulsive parabolic inclusions, Discrete Contin. Dyn. Syst. Ser. B 22 (5) (2017), 1875–1886. (2017) MR3627133
  6. Dashkovskiy, S., Kapustyan, O., Schmid, J., 10.1007/s00498-020-00256-w, Math. Control Signals Systems 32 (3) (2020), 309–326. (2020) MR4149749DOI10.1007/s00498-020-00256-w
  7. Dashkovskiy, S., Mironchenko, A., 10.1007/s00498-012-0090-2, Math. Control Signal Systems 25 (2013), 1–35. (2013) MR3022292DOI10.1007/s00498-012-0090-2
  8. Haraux, A., Kirane, M., 10.5802/afst.598, Ann. Fac. Sci. Toulouse Math. 5 (1983), 265–280. (1983) DOI10.5802/afst.598
  9. Kapustyan, O.V., Kapustian, O.A., Gorban, N.V., Khomenko, O.V., 10.1615/JAutomatInfScien.v47.i11.40, J. Automat. Inform. Sci. 47 (11) (2015), 48–59. (2015) DOI10.1615/JAutomatInfScien.v47.i11.40
  10. Kapustyan, O.V., Kasyanov, P.O., Valero, J., Structure of the global attractor for weak solutions of a reaction-diffusion equation, Appl. Math. Inform. Sci. 9 (5) (2015), 2257–2264. (2015) MR3358694
  11. Kichmarenko, O., Stanzhytskyi, O., Sufficient conditions for the existence of optimal controls for some classes of functional-differential equations, Nonlinear Dyn. Syst. Theory 18 (2) (2018), 196–211. (2018) MR3820833
  12. Manthey, R., Zausinger, T., Stochastic equations in L ρ 2 , Stochastic Rep. 66 (1977), 370–373. (1977) 
  13. Mironchenko, A., Prieur, Ch., 10.1137/19M1291248, SIAM Rev. 62 (2020), 529–614. (2020) MR4131339DOI10.1137/19M1291248
  14. Mironchenko, A., Wirtz, F., 10.1109/TAC.2017.2756341, IEEE Trans. Automat. Control 63 (6) (2018), 1602–1617. (2018) MR3805142DOI10.1109/TAC.2017.2756341
  15. Nakonechnyi, A.G., Mashchenko, S.O., Chikrii, V.K, 10.1615/JAutomatInfScien.v50.i1.40, J. Automat. Inform. Sci. 50 (1) (2018), 54–75. (2018) MR3821216DOI10.1615/JAutomatInfScien.v50.i1.40
  16. Nakonechnyi, O.G., Kapustian, O.A., Chikrii, A.O., 10.1007/s10559-019-00189-6, Cybernet. Systems Anal. 55 (5) (2019), 785–795. (2019) MR4017248DOI10.1007/s10559-019-00189-6
  17. Pazy, A., Semigroups of linear operators and applications to PDE, Springer-Verlag New York, 1983. (1983) 
  18. Pichkur, V.V., Sobchuk, V.V., Mathematical models and control design of a functionally stable technological process, J. Optim. Differ. Equ. Appl. (JODEA) 21 (1) (2021), 1–11. (2021) 
  19. Robinson, J., Infinite-dimensional dynamical systems. An introduction to dissipative parabolic PDEs and the theory of global attractors, Cambridge University Press, 2001. (2001) MR1881888
  20. Samoilenko, A.M., Stanzhitskii, A.N., 10.1134/S0012266106040070, Differ. Equ. 42 (4) (2006), 505–511. (2006) MR2296521DOI10.1134/S0012266106040070
  21. Schmid, J., Kapustyan, O., Dashkovskiy, S., 10.3934/mcrf.2021044, Math. Control Relat. Fields 12 (3) (2022), 763–788. (2022) MR4459660DOI10.3934/mcrf.2021044
  22. Sell, G., You, Y., Dynamics of evolutionary equations, Springer New York, NY, 2000. (2000) 
  23. Sontag, E.D., 10.1109/9.28018, IEEE Trans. Automat. Control 34 (4) (1989), 435–443. (1989) DOI10.1109/9.28018
  24. Sontag, E.D., Mathematical control theory. Deterministic finite-dimensional systems, Springer, N.Y., 1998. (1998) 
  25. Stanzhitskii, A.M., 10.1023/A:1010437625118, Ukrainian Math. J. 53 (2) (2001), 323–327. (2001) MR1833535DOI10.1023/A:1010437625118
  26. Stanzhyts’kyi, O., 10.1023/A:1015259031308, Ukrainian Math. J. 53 (11) (2001), 1882–1894. (2001) DOI10.1023/A:1015259031308

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.