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

arXiv: https://arxiv.org/abs/1805.10786 · HTML: https://mlsanity.com/paper/1805.10786/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 May 2018.

## Built on by (most cited first, 1 of 1)

- [Turnpike in optimal control of PDEs, ResNets, and beyond](https://mlsanity.com/paper/2202.04097/) (2022)

## Abstract (arXiv)

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 ≤ y(t,x) ≤ 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 <θ< 1 is much more intricate. We rely on a staircase control strategy to prove that θ 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.