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

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 nonlinearites, 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.