论文标题
具有移动控件的一维非局部灰色模型的局部控制性
Local-controllability of the one-dimensional nonlocal Gray-Scott model with moving controls
论文作者
论文摘要
在本文中，我们证明了局部控制性，可用于两个耦合ODE方程的非线性系统的正恒定轨迹，该系统在具有非局部空间非线性的一维空间设置中构成，并且仅使用一个带有移动支持的局部化控制。当我们处理的模型源自众所周知的非线性反应 - 扩散灰色 - 灰色模型时，当第一种化学物种的扩散系数$ d_u $倾向于$ 0 $，而第二种化学物种$ {d_v} $的扩散系数倾向于$++ \ \ \ iffty $。证明的策略包括两个主要步骤。首先，我们建立了源自灰色 - 斯科特模型$ d_u = 0 $的局部控制性，并且相对于扩散参数$ {d_v} \ in（1， +\ infty）$均匀。为此，我们证明了线性化系统的（均匀）无效控制性，这要归功于通过对Carleman的ODE-PDE估计获得的可观察性估计值。要传递到非线性系统，我们使用精确的反映射参数，其次，我们应用了阴影限制$ {d_v} \ rightArrow + \ infty $，以简化为初始系统。
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.
