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

TL;DR

This paper introduces the concept of relative equi-stability (RKES) for nonlinear parabolic PDEs, establishing input-to-state stability in the spatial sup-norm and analyzing stability in coupled PDE systems using De Giorgi iteration.

Contribution

It proposes the novel RKES notion to analyze stability and input-to-state stability in nonlinear parabolic PDEs with boundary disturbances.

Findings

Established ISS in the spatial sup-norm for certain nonlinear PDEs.

Illustrated stability results with two example PDEs.

Applied RKES to analyze stability in coupled PDE systems.

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.