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 $\gamma \geq 1$ , we establish a resolvent estimate for the Baouendi–Grushin-type operator $\Delta _{\gamma }=\partial _x^2+\left \lvert x\right \rvert ^{2\gamma }\partial _y^2$ , which has step $\gamma +1$ . We then derive consequences for the observability of the Schrödinger-type equation $i\partial _tu-\left (-\Delta _{\gamma }\right )^{s}u=0$ , where $s\in \mathbb N$ . We identify three different cases: depending on the value of the ratio $(\gamma +1)/s$ , observability may hold in arbitrarily small time or only for sufficiently large times or may even fail for any time.

As a corollary of our resolvent estimate, we also obtain observability for heat-type equations $\partial _tu+\left (-\Delta _{\gamma }\right )^su=0$ and establish a decay rate for the damped wave equation associated with $\Delta _{\gamma }$ .

在这篇文章中，我们研究了一些子椭圆演化方程的可观测性（或等价地，可控性）取决于它们的步骤。这揭示了这些方程的传播速度，特别是在亚椭圆结构的“退化方向”。

首先，对于任何 $\gamma \geq 1$ ，我们为 Baouendi–Grushin 型算子建立分解估计 $\Delta _{\gamma }=\partial _x^2+\left \lvert x\right \rvert ^ {2\gamma }\partial _y^2$ ，其步长为 $\gamma +1$ 。然后，我们推导出薛定谔型方程 $i\partial _tu-\left (-\Delta _{\gamma }\right )^{s}u=0$ 的可 观察性结果，其中 $s\in \mathbb N $ 。我们确定了三种不同的情况：根据比率 $(\gamma +1)/s$ 的值，可观察性可能在任意短的时间内成立或仅在足够长的时间内成立，甚至可能在任何时候都失败。

作为我们分解估计的推论，我们还获得了热型方程 $\partial _tu+\left (-\Delta _{\gamma }\right )^su=0$ 的可 观测性，并为阻尼波动方程建立了衰减率与 $\Delta _{\gamma }$ 相关联。