Control of the Fisher-Stefan System

Abstract This paper addresses the exact controllability of trajectories in the one-dimensional Fisher-Stefan problem—a reaction-diffusion equation that models the spatial propagation of biological, chemical, or physical populations within a free-end domain, governed by Stefan’s law. We establish the local exact controllability of the trajectories by reformulating the problem as the local null controllability of a nonlinear system with distributed controls. Our approach leverages the Lyusternik-Graves theorem to achieve local inversion, leading to the desired controllability result. Finally, we illustrate our theoretical findings through several numerical experiments based on the physics-informed neural networks (PINNs) approach.