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Relative Stability in the Sup-Norm and Input-to-State Stability in the Spatial Sup-Norm for Parabolic PDEs

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In this article, 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 partial differential equations… Click to show full abstract

In this article, 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 partial differential equations (PDEs). 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 superlinear 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 article.

Keywords: stability; parabolic pdes; sup norm; spatial sup

Journal Title: IEEE Transactions on Automatic Control
Year Published: 2022

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