Robust stability of the global attractor of the reaction-diffusion system
DOI:
https://doi.org/10.17721/1812-5409.2021/3.6Keywords:
input-to-state stability, asymptotic gain, global attractor, infinite-dimensional system, reaction-diffusion systemAbstract
In this paper we consider the problem of robust stability for a nonlinear system of equations in partial derivatives of the reaction-diffusion type. An undisturbed system is considered to have a global attractor. The main task is to estimate the deviation of the trajectory of the perturbed system from the global attractor of the perturbed system depending on the magnitude of the perturbations. Such an estimate can be obtained in the framework of the theory of input-to-state stability (ISS). The paper does not impose any conditions on the derivative of the nonlinear interaction function, so the unity of the solution of the initial problem is not ensured. The paper proposes a new approach to obtaining estimates of robust stability of the attractor in the case of a multivalued evolutionary decoupling operator. In particular, it is proved that the multivalued decoupling operator generated by weak solutions of a nonlinear reaction-diffusion system has the property of asymptotic gain (AG) with respect to the attractor of the undisturbed system.
Pages of the article in the issue: 46 - 50
Language of the article: Ukrainian
References
V.V. CHEPYZHOV, M.I. VISHIK (2002) Attractors for equations of mathematical physics. American Mathematical Society.
S. DASHKOVSKIY, AND A. MIRONCHENKO (2013) Input-to-state stability of infinite-dimensional control systems. Math. Contr. Sygn. Syst., 25 p. 1-35.
S. DASHKOVSKIY, O.V. KAPUSTYAN, I. ROMANIUK (2017) Global attractors of impulsive parabolic inclusions, Discr. Contin. Dyn. Syst. - Series B, 22 p. 1875-1886.
S. DASHKOVSKIY, O.V. KAPUSTYAN, J. SCHMID (2020) A local input-to-statestability result w.r.t. attractors of nonlinear reaction-diffusion equations. Mathematics of Control, Signals, and Systems, 32, p. 309-326.
B. JACOB, F. SCHWENNINGER (2018) Input-to-state stability of unbounded bilinear control systems. arXiv:1811.08470.
O.V. KAPUSTYAN, P.O. KASYANO, V.J. VALERO, M.Z. ZGUROVSKY (2014) Structure of uniform global attractor for general non-autonomous reaction-diffusion system. Continuous and distributed systems. vol. 211, p. 221-237.
KARAFYLLIS, I. AND KRSTIC, M. (2019) Input-to-state stability for PDEs. Communi-cations and Control Engineering Series. Springer, Cham.
A. MIRONCHENKO, CH. PRIEUR (2020) Input-to-state Stability of Innite-Dimensional Systems: Recent Results and Open Questions. SIAM Review., 62, p. 529-614.
A. MIRONCHENKO, F. WIRTH (2018) Characterizations of input-to-state stability for infinite-dimensional systems. IEEE Trans. Autom. Contr., 63, p. 1692-1707.
Downloads
Published
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
