Phase portrait control for 1D monostable and bistable reaction–diffusion equations

Abstract

We consider the problem of controlling parabolic semilinear equations arising in population dynamics, either in finite time or infinite time. These are the monostable and bistable equations on for a density of individuals , with Dirichlet controls taking their values in . We prove that the system can never be steered to extinction (steady state ) or invasion (steady state ) in finite time, but is asymptotically controllable to independently of the size , and to if the length of the interval domain is less than some threshold value , which can be computed from transcendental integrals. In the bistable case, controlling to the other homogeneous steady state is much more intricate. We rely on a staircase control strategy to prove that can be reached in finite time if and only if . The phase plane analysis of those equations is instrumental in the whole process. It allows us to read obstacles to controllability, compute the threshold value for domain size as well as design the path of steady states for the control strategy.