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

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Dynamic phenomena in social and biological sciences can often be modeled by reaction-diffusion equations. When addressing the control 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 (taking values in [0,1])). 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 monostable and bistable semilinear heat equations, with constraints in the state, and using boundary controls. We prove that controlling the system to a constant steady state may become impossible when the diffusivity is too small due to the existence of barrier functions. 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 to ensure controllability whenever it is possible. 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.