Backstepping Control Laws for Higher-Dimensional PDEs: Spatial Invariance and Domain Extension Methods

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.