Controllability of the Fisher-Stefan system

TL;DR

The paper addresses local exact controllability to trajectories for the 1D Fisher-Stefan free-boundary system, governed by $\varrho_t - \sigma\varrho_{xx} = r\varrho\left(1 - \frac{\varrho}{k}\right)$ on $(0,L(t))$ with $L'(t) = -\mu\varrho_x(L(t),t)$. By reformulating to a fixed cylinder via $\psi(x,t)=\varrho(xL(t),t)$ and $h(t)=L(t)^2$, the authors reduce the problem to a nonlinear parabolic system with distributed control, then prove local null controllability using Lyusternik-Graves inverse mapping theory. The linearized system is shown to be null-controllable through Carleman-based observability and adjoint analysis, and these results lift to the nonlinear setting via a local inverse argument. Numerically, Physics-Informed Neural Networks (PINNs) are employed to approximate the nonlinear and linearized problems, with experiments illustrating that the nonlinear logistic term affects controllability patterns, while preserving nonnegativity and trajectory matching. Altogether, the work provides a rigorous framework for steering both density and free boundary in Fisher-Stefan dynamics and demonstrates a modern PINN-based computational avenue for free-boundary controllability problems.

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 to 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.

Figures (6)

• Figure 1: The extended domain $(-1,1)$ of $(0,1)$ and the control region $\omega$.
• Figure 2: PINN-based algorithm for approximating the exact states and control. The neural networks $\hat{z}\left(x,t,\boldsymbol{\theta}\right)$, $\hat{k}\left(t,\boldsymbol{\theta}\right)$ and $\hat{v}\left(x,t,\boldsymbol{\theta}\right)$ are required to satisfy, in the least squares sense, the ODE-PDE system, boundary condition, initial conditions, and exact controllability conditions. Then, the residual on training points $\mathcal{L}\left(\boldsymbol{\theta};\mathcal{T}\right)$ is minimized to get the optimal set of parameters $\boldsymbol{\theta}^\ast$ of the neural networks. This leads to the PINN exact states $\hat{z}\left(x,t;\boldsymbol{\theta}^\ast\right)$, $\hat{k}\left(t;\boldsymbol{\theta}^\ast\right)$ as well as the PINN control $\hat{v}\left(x,t;\boldsymbol{\theta}^\ast\right)$.
• Figure 3: Experiment 1. Convergence history (left), and control (right).
• Figure 4: Experiment 1. (Left) Plot of the densities associated with the linearized system at different times, and state variable $L(t)$(right).
• Figure 5: Experiment 2. $T=1$. (Left) Convergence history. (Right) Control $u(t)$.

Theorems & Definitions (18)

• Definition 1.1
• Theorem 1.2
• Remark 1
• Theorem 1.3
• Definition 2.1
• Remark 2
• Proposition 2.2
• Proposition 2.3
• proof
• Proposition 2.4