Local controllability of the one-dimensional nonlocal Gray–Scott model with moving controls

Abstract

In this paper, we prove the local controllability to positive constant trajectories of a nonlinear system of two coupled ODE equations, posed in the one-dimensional spatial setting, with nonlocal spatial nonlinearities, and using only one localized control with a moving support. The model we deal with is derived from the well-known nonlinear reaction–diffusion Gray–Scott model when the diffusion coefficient of the first chemical species \(d_u\) tends to 0 and the diffusion coefficient of the second chemical species \({d_v}\) tends to \(+ \infty \). The strategy of the proof consists in two main steps. First, we establish the local controllability of the reaction–diffusion ODE–PDE derived from the Gray–Scott model taking \(d_u=0\), and uniformly with respect to the diffusion parameter \({d_v} \in (1, +\infty )\). In order to do this, we prove the (uniform) null-controllability of the linearized system thanks to an observability estimate obtained through adapted Carleman estimates for ODE–PDE. To pass to the nonlinear system, we use a precise inverse mapping argument and, secondly, we apply the shadow limit \({d_v} \rightarrow + \infty \) to reduce to the initial system.