Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations

TL;DR

This paper investigates the controllability of multi-dimensional reaction-diffusion equations with state constraints, focusing on boundary control strategies for monostable and bistable systems, and introduces a novel approach using fictitious domain techniques.

Contribution

It extends controllability analysis to multi-dimensional systems with constraints, employing a new fictitious domain method to overcome the limitations of phase plane analysis.

Findings

Controllability may be impossible with small diffusivity or large domains due to barrier functions.

Constructed control strategies leverage system dissipativity and steady-state connectivity.

Fictitious domain techniques enable path construction in higher dimensions where phase plane methods fail.

Abstract

Dynamic phenomena in social and biological sciences can often be modeled by employing reaction-diffusion equations. When addressing the control of these modes, from a mathematical viewpoint one of the main challenges is that, because of the intrinsic nature of the models under consideration, the solution, typically a proportion or a density function, needs to preserve given lower and upper bounds. Controlling the system to the desired final configuration then becomes complex, and sometimes even impossible. In the present work, we analyze the controllability to constant steady-states of spatially homogeneous semilinear heat equations, with constraints in the state, and using boundary controls, which is indeed a natural way of acting on the system in the present context. The nonlinearities considered are among the most frequent: monostable and bistable ones. We prove that controlling the system to a constant steady-state may become impossible when the diffusivity is too small (or when the domain is large), due to the existence of barrier functions. When such an obstruction does not arise, we build sophisticated control strategies combining the dissipativity of the system, the existence of traveling waves and some connectivity of the set of steady-states. This connectivity allows building paths that the controlled trajectories can follow, in a long time, with small oscillations, preserving the natural constraints of the system. This kind of strategy was successfully implemented in one-space dimension, where phase plane analysis techniques allowed to decode the nature of the set of steady-states. These techniques fail in the present multi-dimensional setting. We employ a fictitious domain technique, extending the system to a larger ball, and building paths of radially symmetric solution that can then be restricted to the original domain.