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