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Relative stability in the sup-norm and input-to-state stability in the spatial sup-norm for parabolic PDEs

Jun Zheng, Guchuan Zhu and Sergey Dashkovskiy

Article (2022)

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Abstract

In this paper, we introduce the notion of relative K-equi-stability (RKES) to characterize the uniformly continuous dependence of (weak) solutions on external disturbances for nonlinear parabolic PDE systems. Based on the RKES, we prove the input-to-state stability (ISS) in the spatial sup-norm for a class of nonlinear parabolic PDEs with either Dirichlet or Robin boundary disturbances. An example concerned with a super-linear parabolic PDE with Robin boundary condition is provided to illustrate the obtained ISS results. Besides, as an application of the notion of RKES, we conduct stability analysis for a class of parabolic PDEs in cascade coupled over the domain or on the boundary of the domain, in the spatial and time sup-norm, and in the spatial sup-norm, respectively. The technique of De Giorgi iteration is extensively used in the proof of the results presented in this paper.

Uncontrolled Keywords

Nonlinear PDEs, relative stability, input-tostate stability, De Giorgi iteration, cascade of PDE systems.

Subjects: 2500 Electrical and electronic engineering > 2500 Electrical and electronic engineering
Department: Department of Electrical Engineering
Funders: NSFC, CRSNG/NSERC, German Research Foundation (DFG)
Grant number: 11901482, RGPIN-2018-04571, DA 767/12-1
PolyPublie URL: https://publications.polymtl.ca/10400/
Journal Title: IEEE Transactions on Automatic Control (vol. 67, no. 10)
Publisher: IEEE
DOI: 10.1109/tac.2022.3192325
Official URL: https://doi.org/10.1109/tac.2022.3192325
Date Deposited: 27 Jul 2022 13:53
Last Modified: 19 Jul 2023 12:12
Cite in APA 7: Zheng, J., Zhu, G., & Dashkovskiy, S. (2022). Relative stability in the sup-norm and input-to-state stability in the spatial sup-norm for parabolic PDEs. IEEE Transactions on Automatic Control, 67(10), 5361-5375. https://doi.org/10.1109/tac.2022.3192325

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