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arXiv · 2008.01932

Relative Stability in the Sup-norm and Input-to-state Stability in the Spatial Sup-norm for Parabolic PDEs

Jun Zheng·Guchuan Zhu·Sergey Dashkovskiy
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Abstract

In this paper, we introduce the notion of relative $\mathcal{K}$-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. Two examples, concerned respectively with a super-linear parabolic PDE with Robin boundary condition and a $1$-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.

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Jun Zheng, Guchuan Zhu, Sergey Dashkovskiy. 2020-08-05. Relative Stability in the Sup-norm and Input-to-state Stability in the Spatial Sup-norm for Parabolic PDEs. https://arxiv.org/abs/2008.01932

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