Symmetry groupoids for pattern-selective feedback stabilization of the Chafee–Infante equation

Abstract

Reaction–diffusion equations are ubiquitous in various scientific domains and their patterns represent a fascinating area of investigation. However, many of these patterns are unstable and, therefore, challenging to observe. To overcome this limitation, we present new noninvasive feedback controls based on symmetry groupoids. As a concrete example, we employ these controls to selectively stabilize unstable equilibria of the Chafee–Infante equation under Dirichlet boundary conditions on the interval. Unlike conventional reflection-based control schemes, our approach incorporates additional symmetries that enable us to design new convolution controls for stabilization. By demonstrating the efficacy of our method, we provide a new tool for investigating and controlling systems with unstable patterns, with potential implications for a wide range of scientific disciplines.