We consider a dissipative evolutionary system formed by a parabolic system of reaction-diffusion type and a system of ordinary differential equations perturbed by bounded external signals. We prove that the global attractor of the unperturbed system is stable in the input-to-state stability sense with respect to the value of perturbations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. D. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, Springer, New York (1998).
E. D. Sontag, “Smooth stabilization implies coprime factorization,” IEEE Trans. Automat. Control, 34, No. 4, 435–443 (1989).
S. Dashkovskiy and A. Mironchenko, “Input-to-state stability of infinite-dimensional control systems,” Math. Control Sign. Syst., 25, 1–35 (2013).
A. Mironchenko and F. Wirth, “Characterizations of input-to-state stability for infinite-dimensional systems,” IEEE Trans. Automat. Control, 63, No. 6, 1602–1617 (2018).
A. Mironchenko and Ch. Prieur, “Input-to-state stability of infinite-dimensional systems: recent results and open questions,” SIAM Rev., 62, 529–614 (2020).
J. Schmid, “Weak input-to-state stability: characterization and counterexamples,” Math. Control Sign. Syst., 31, No. 4, 433–454 (2019).
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York (1997).
J. C. Robinson, Infinite-Dimensional Dynamical Systems. An introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Univ. Press, Cambridge (2001).
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., 49, American Mathematical Society, Providence, RI (2002).
O. V. Kapustyan, P. O. Kasyanov, and J. Valero, “Structure of the global attractor for weak solutions of a reaction-diffusion equation,” Appl. Math. Inf. Sci., 9, No. 5, 2257–2264 (2015).
N. V. Gorban, A. V. Kapustyan, E. A. Kapustyan, and O. V. Khomenko, “Strong global attractors for the three-dimensional Navier–Stokes system of equations in unbounded domain of channel type,” J. Autom. Inform. Sci., 47(11), 48–59 (2015).
P. O. Kasianov, L. Toscano, and N. V. Zadoianchuk, “Long-time behavior of solutions for autonomous evolution hemivariational inequality with multidimensional ‘reaction–displacement’ law,” Abstr. Appl. Anal. (2012).
S. Dashkovskiy, O. Kapustyan, and J. Schmid, “A local input-to-state stability result w.r.t. attractors of nonlinear reaction-diffusion equations,” Math. Control Signals Systems, 32, No. 3, 309–326 (2020).
J. Schmid, V. Kapustyan, and S. Dashkovskiy, Asymptotic Gain Results for Attractors of Semilinear Systems, Preprint arXiv:1909.06302 [math.AP] (2020).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Neliniini Kolyvannya, Vol. 24, No. 3, pp. 336–341, July–September, 2021.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kapustyan, O.V., Yusypiv, T.V. Stability Under Perturbations for the Attractor of a Dissipative PDF-ODF-Type System. J Math Sci 272, 236–243 (2023). https://doi.org/10.1007/s10958-023-06413-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06413-1