Title: Backstepping Control Laws for Higher-Dimensional PDEs: Spatial Invariance and Domain Extension Methods Show Chinese title

Title: 高维偏微分方程的反步控制律：空间不变性和区域扩展方法

Abstract: This paper extends backstepping to higher-dimensional PDEs by leveraging domain symmetries and structural properties. We systematically address three increasingly complex scenarios. First, for rectangular domains, we characterize boundary stabilization with finite-dimensional actuation by combining backstepping with Fourier analysis, deriving explicit necessary conditions. Second, for reaction-diffusion equations on sector domains, we use angular eigenfunction expansions to obtain kernel solutions in terms of modified Bessel functions. Finally, we outline a domain extension method for irregular domains, transforming the boundary control problem into an equivalent one on a target domain. This framework unifies and extends previous backstepping results, offering new tools for higher-dimensional domains where classical separation of variables is inapplicable. Show Chinese abstract

Abstract: 本文通过利用域的对称性和结构特性，将后推法扩展到高维偏微分方程。我们系统地处理了三个越来越复杂的情况。首先，对于矩形域，我们结合后推法与傅里叶分析来表征具有有限维激励的边界稳定化，并推导出明确的必要条件。其次，对于扇形域上的反应扩散方程，我们使用角度特征函数展开以修正的贝塞尔函数的形式获得核解。最后，我们概述了一种用于不规则域的域扩展方法，将边界控制问题转化为目标域上的等效问题。此框架统一并扩展了以前的后推结果，为经典变量分离不适用的高维域提供了新的工具。