Uniform attractors and asymptotic gain property for the PDE-ODE system
DOI:
https://doi.org/10.17721/1812-5409.2025/2.15Keywords:
asymptotic gain, uniform attractor, infinite-dimensional system, PDE-ODE systemAbstract
The paper investigates the qualitative behavior of weak solutions to a dissipative infinite-dimensional non-autonomous perturbed system, which consists of a parabolic reaction-diffusion system and a nonlinear system of ordinary differential equations. Perturbations are modeled by bounded functions included in the right-hand side of both systems. The considered system is known to possess a global attractor in the absence of perturbations, which determines the long-term dynamics of all trajectories. However, the robustness of such attractors under external disturbances remains a challenging issue, especially in the context of nonlinear and infinite-dimensional systems.
In this study, we adopt an approach based on the theory of uniform attractors for non-autonomous dynamical systems (semi-processes). A corresponding family of semi-processes associated with the perturbed PDE–ODE system is constructed. The existence of a uniform attractor is proven, and its convergence to the global attractor of the unperturbed system is established as the amplitude of disturbances tends to zero. This allows us to derive a nonlocal robust estimate of the asymptotic gain (AG) type, which complements the local ISS-based robustness result and provides an upper bound on the deviation of perturbed trajectories from the unperturbed attractor in terms of the perturbation magnitude. The results contribute to the theoretical foundation for robustness analysis in dissipative infinite-dimensional systems with complex long-term dynamics.
Pages of the article in the issue: 105 - 112
Language of the article: English
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Copyright (c) 2025 Oleksiy Kapustyan, Taras Yusypiv, Yuriy Perestyuk, Zoia Khaletska
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