Robustness of global attractors: Abstract framework and application to dissipative wave equations

Sergey Dashkovskiy; Oleksiy Kapustyan

doi:10.3934/eect.2021054

Article Contents

2022, Volume 11, Issue 5: 1565-1577. Doi: 10.3934/eect.2021054

This issue Previous Article $ L^p $-exact controllability of partial differential equations with nonlocal terms Next Article Boundary controllability for a coupled system of degenerate/singular parabolic equations

Robustness of global attractors: Abstract framework and application to dissipative wave equations

Sergey Dashkovskiy^1, , and
Oleksiy Kapustyan^2,

1.
University of Würzburg, Institute of Mathematics, Emil-Fischer-Straße 40, 97074 Würzburg, Germany
2.
Taras Shevchenko National University of Kyiv, Department of Mathematics and Mechanics, Volodymyrska Street 64, 01601 Kyiv, Ukraine

^* Corresponding author: Sergey Dashkovskiy
^* Corresponding author: Sergey Dashkovskiy

Received: February 28, 2021

Revised: June 30, 2021

Early access: October 2021

Published: October 2022

This work is partially supported by the German Research Foundation (DFG) grant DA 767/12-1 and the National Research Foundation of Ukraine (NRFU) through the joint German-Ukrainian grant "Stability and robustness of attractors of nonlinear infinite-dimensional systems with respect to disturbances"

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Abstract

We establish local input-to-state stability and the asymptotic gain property for a class of infinite-dimensional systems with respect to the global attractor of the respective undisturbed system. We apply our results to a large class of dissipative wave equations with nontrivial global attractors.

Keywords:

Mathematics Subject Classification: Primary: 35B40, 35B41, 93B35; Secondary: 93D09, 35B35, 35L05.

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