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.

在本文中，我们证明了两个耦合 ODE 方程的非线性系统的正恒定轨迹的局部可控性，在一维空间设置中提出，具有非局部空间非线性，并且仅使用一个带有移动支撑的局部控制。我们处理的模型源自著名的非线性反应-扩散格雷-斯科特模型，当第一种化学物质的扩散系数\(d_u\)趋于 0，而第二种化学物质的扩散系数\({d_v }\)倾向于\(+ \infty \)。证明的策略包括两个主要步骤。首先，我们建立了从 Gray-Scott 模型推导出的反应-扩散 ODE-PDE 的局部可控性，取\(d_u=0\)，并且均匀地关于扩散参数\({d_v} \in (1, +\infty )\)。为了做到这一点，我们证明了线性化系统的（均匀）零可控性，这要归功于通过 ODE-PDE 的自适应卡尔曼估计获得的可观察性估计。为了传递到非线性系统，我们使用精确的逆映射参数，其次，我们应用阴影限制\({d_v} \rightarrow + \infty \)来减少到初始系统。