SIAM Journal on Control and Optimization, Volume 58, Issue 4, Page 2567-2587, January 2020.
The aim of this paper is to introduce a weak maximum principle--based approach for studying the input-to-state stability (ISS) with respect to boundary disturbances and states in certain classes for a class of one-dimensional nonlinear parabolic partial differential equations (PDEs) with nonlinear boundary conditions. To tackle the difficulties in ISS analysis due to, in particular, the nonlinear terms on the boundary, we establish first several maximum estimates for the solutions of linear parabolic PDEs with different nonlinear boundary conditions by means of the weak maximum principle. Then, using the technique of splitting and combining maximum estimates for the solutions of linear parabolic PDEs and the Lyapunov method, we establish ISS estimates for nonlinear parabolic PDEs with nonlinear boundary conditions. Two examples of specific parabolic equations with nonlinear boundary conditions are provided to illustrate the developed approach.

中文翻译：

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一类具有非线性边界条件的一维非线性抛物线PDE的输入状态稳定性

SIAM控制与优化杂志，第58卷，第4期，第2567-2587页，2020年1月。
本文的目的是介绍一种基于弱最大原理的方法，以研究一类一维非线性抛物型偏微分方程在某些类别中的边界状态和状态的输入到状态稳定性（ISS） （PDE）具有非线性边界条件。为了解决特别是边界上的非线性项所引起的ISS分析中的困难，我们借助弱最大原理建立了具有不同非线性边界条件的线性抛物型PDE的解的第一个最大估计。然后，使用将线性抛物型偏微分方程解的最大估计值分解和组合的技术和Lyapunov方法，建立了具有非线性边界条件的非线性抛物型偏微分方程的ISS估计。