论文标题
通过解决估计值的Baouendi-Grushin型方程的可观察性
Observability of Baouendi-Grushin-Type Equations Through Resolvent Estimates
论文作者
论文摘要
在本文中，我们根据其步骤研究了某些亚细胞演化方程的可观察性（或等效地，是可控性）。这阐明了这些方程式的传播速度，特别是在层次结构的“退化方向”中。首先，对于任何$γ\ geq 1 $，我们为Baouendi-Grushin-type操作员建立了一个分解估计，$Δ_γ= \ partial_x^2+| x | x | x |^{2γ} \ partial_y^2 $，它具有步骤$γ+1 $。然后，我们为Schr \''odinger类型方程的可观察性提供后果，$ i \ partial_tu-（ - Δ_γ）^{s} u = 0 $ = 0 $其中$ s \ in n $。我们确定三种不同的情况：取决于比率$（γ+1）/s $的值，可观察性可能在任意的时间内或仅在足够大的时间内，甚至在任何时间失败。 作为我们的分解估计值的推论，我们还获得了热型方程的可观察性$ \ partial_tu+（ - Δ_γ）^su = 0 $，并为与$Δ_γ$相关的阻尼波方程建立了衰减率。
In this article, we study the observability (or, equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the "degenerated directions" of the subelliptic structure. First, for any $γ\geq 1$, we establish a resolvent estimate for the Baouendi-Grushin-type operator $Δ_γ=\partial_x^2+|x|^{2γ}\partial_y^2$, which has step $γ+1$. We then derive consequences for the observability of the Schr\''odinger type equation $i\partial_tu-(-Δ_γ)^{s}u=0$ where $s\in N$. We identify three different cases: depending on the value of the ratio $(γ+1)/s$, observability may hold in arbitrarily small time, or only for sufficiently large times, or even fail for any time. As a corollary of our resolvent estimate, we also obtain observability for heat-type equations $\partial_tu+(-Δ_γ)^su=0$ and establish a decay rate for the damped wave equation associated with $Δ_γ$.
