Relative Stability in the Sup-norm and Input-to-state Stability in the Spatial Sup-norm for Parabolic PDEs
Abstract
In this paper, we introduce the notion of relative -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. Two examples, concerned respectively with a super-linear parabolic PDE with Robin boundary condition and a -D parabolic PDE with a destabilizing term, are 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.
Keywords
Cite
@article{arxiv.2008.01932,
title = {Relative Stability in the Sup-norm and Input-to-state Stability in the Spatial Sup-norm for Parabolic PDEs},
author = {Jun Zheng and Guchuan Zhu and Sergey Dashkovskiy},
journal= {arXiv preprint arXiv:2008.01932},
year = {2021}
}
Comments
This is a reduced version of our manuscript written in the summer of 2019, which presents the application of the De Giorgi iteration in the analysis of ISS in $L^\infty$-norm for nonlinear parabolic PDEs with Robin or mixed boundary disturbances