This paper introduces a novel method, namely the approximations of Lyapunov functionals, for input-to-state stability (ISS) analysis of a class of higher dimensional nonlinear parabolic partial differential equations (PDEs) with variable coefficients. Specifically, for any $q\in [1,+\infty ]$ and the considered nonlinear parabolic PDEs with different types of boundary disturbances in $L_{loc}^q({\mathbb{R}}_{+};L^1\left(\partial \Omega \right))$ and initial data in $L^1\left(\Omega \right)$, we show that ISS-like estimates in $L^1$-norm (or weighted $L^1$-norm) can be established by constructing approximations of (coercive and non-coercive) Lyapunov functionals. Some examples are provided to illustrate the application of the proposed method.

本文介绍了一种新颖的方法，即Lyapunov泛函的逼近，用于一类具有变系数的高维非线性抛物型偏微分方程（PDE）的输入状态稳定性（ISS）分析。具体来说，对于任何$q\in [\mathrm{1个}，+\infty ]$ 并考虑了不同边界干扰类型的非线性抛物线偏微分方程 ${\mathrm{大号}}_{升ØC}^q（{\mathbb{[R}}_{+};{\mathrm{大号}}^{\mathrm{1个}}（\partial \Omega ））$ 和初始数据 ${\mathrm{大号}}^{\mathrm{1个}}（\Omega ）$，我们证明在 ${\mathrm{大号}}^{\mathrm{1个}}$-范数（或加权 ${\mathrm{大号}}^{\mathrm{1个}}$-范数）可以通过构造（强制和非强制）Lyapunov函数的近似来建立。提供了一些示例来说明所提出方法的应用。