论文标题
级联反应扩散系统具有奇数耦合术语的无效控制性
Null-controllability of cascade reaction-diffusion systems with odd coupling terms
论文作者
论文摘要
在本文中，我们考虑了两个抛物线方程的非线性系统，在第一个方程式中具有分布式控件，第二个方程式在第二个方程中具有奇数耦合项。我们证明了非线性系统是局部零件可控的小型系统。主要困难是线性化系统不能无效控制。为了克服这一障碍，我们在非线性设置中扩展了第一作者引入的策略，该策略包括为线性热方程构建奇数控件。证明依赖于三个主要步骤。首先，我们从经典的L^2抛物线卡尔曼估计中获得，并与最大规律性结果相结合，这是非均匀热方程的加权L^P可观察性不等式。其次，我们在反身Banach环境中执行二元论证，接近众所周知的Hilbert唯一性方法，以证明由于奇数控件而被源术语扰动的热方程式是无效控制的。最后，非线性用Schauder定点参数处理。
In this paper, we consider a nonlinear system of two parabolic equations, with a distributed control in the first equation and an odd coupling term in the second one. We prove that the nonlinear system is small-time locally null-controllable. The main difficulty is that the linearized system is not null-controllable. To overcome this obstacle, we extend in a nonlinear setting the strategy introduced by the first author that consists in constructing odd controls for the linear heat equation. The proof relies on three main steps. First, we obtain from the classical L^2 parabolic Carleman estimate, conjugated with maximal regularity results, a weighted L^p observability inequality for the nonhomogeneous heat equation. Secondly, we perform a duality argument, close to the well-known Hilbert Uniqueness Method in a reflexive Banach setting, to prove that the heat equation perturbed by a source term is null-controllable thanks to odd controls. Finally, the nonlinearity is handled with a Schauder fixed-point argument.
