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

TL;DR

This paper analyzes the controllability of 1D reaction-diffusion equations in population dynamics, showing conditions under which states can be reached or avoided, with a focus on domain size effects and phase plane analysis.

Contribution

It introduces a phase portrait control method for monostable and bistable equations, identifying domain size thresholds for controllability and designing control strategies based on phase plane analysis.

Findings

System cannot be steered to extinction or invasion in finite time.

Asymptotic controllability to the invasion state is always possible.

Controllability to intermediate states depends on domain size, with a computable threshold.

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 $(0,L)$ for a density of individuals $0 \leq y(t,x) \leq 1$, with Dirichlet controls taking their values in $[0,1]$. We prove that the system can never be steered to extinction (steady state $0$) or invasion (steady state $1$) in finite time, but is asymptotically controllable to $1$ independently of the size $L$, and to $0$ if the length $L$ of the interval domain is less than some threshold value $L^\star$, which can be computed from transcendental integrals. In the bistable case, controlling to the other homogeneous steady state $0 <\theta< 1$ is much more intricate. We rely on a staircase control strategy to prove that $\theta$ can be reached in finite time if and only if $L< L^\star$. 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.