论文标题
Fredholm BackSteppping for关键操作员，并应用线性水波快速稳定
Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves
论文作者
论文摘要
Coron和Lü引入的Fredholm-type替补转换已成为过去十年中快速发展的快速稳定工具。它的强度在于其系统的方法，从而从近似可控性中推断出快速稳定。但是，对于$ | d_x |^α$的运营商而存在的限制，$ | d_x |^α$ for $α\在（1,3/2] $中。我们在这里提出了一种新的紧凑性/二元性方法，可以在Fredholm上替代Fredholm的替代方案，以克服$ a = 3/2 $的替代品，可以使用threshold。对于验证$α> 1 $的偏度型操作员，构建Fredholm BackStepping Transformation的关键步骤，通常的方法仅适用于$α> 3/2 $，显示了这种新方法的插图。
Fredholm-type backstepping transformation, introduced by Coron and Lü, has become a powerful tool for rapid stabilization with fast development over the last decade. Its strength lies in its systematic approach, allowing to deduce rapid stabilization from approximate controllability. But limitations with the current approach exist for operators of the form $|D_x|^α$ for $α\in (1,3/2]$. We present here a new compactness/duality method which hinges on Fredholm's alternative to overcome the $α=3/2$ threshold. More precisely, the compactness/duality method allows to prove the existence of a Riesz basis for the backstepping transformation for skew-adjoint operator verifying $α>1$, a key step in the construction of the Fredholm backstepping transformation, where the usual methods only work for $α>3/2$. The illustration of this new method is shown on the rapid stabilization of the linearized capillary-gravity water wave equation exhibiting an operator of critical order $α=3/2$.
