A weak maximum principle-based approach for input-to-state stability analysis of nonlinear parabolic PDEs with boundary disturbances

TL;DR

This paper presents a novel approach based on the weak maximum principle to analyze input-to-state stability of nonlinear parabolic PDEs with boundary disturbances, offering a less computationally intensive method.

Contribution

The paper extends the weak maximum principle to provide a new ISS analysis framework for nonlinear parabolic PDEs with boundary disturbances.

Findings

Established ISS estimates for linear reaction-diffusion PDEs

Extended maximum estimate results to nonlinear PDEs

Proposed a computationally efficient ISS analysis scheme

Abstract

In this paper, we introduce a weak maximum principle-based approach to input-to-state stability (ISS) analysis for certain nonlinear partial differential equations (PDEs) with boundary disturbances. Based on the weak maximum principle, a classical result on the maximum estimate of solutions to linear parabolic PDEs has been extended, which enables the ISS analysis for certain {}{nonlinear} parabolic PDEs with boundary disturbances. To illustrate the application of this method, we establish ISS estimates for a linear reaction-diffusion PDE and a generalized Ginzburg-Landau equation with {}{mixed} boundary disturbances. Compared to some existing methods, the scheme proposed in this paper involves less intensive computations and can be applied to the ISS analysis for a {wide} class of nonlinear PDEs with boundary disturbances.