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In this article, we prove null-controllability results for the heat equationassociated tofractional Baouendi-Grushin operators $$\partial_tu+\bigl(-\Delta_x-V(x)\Delta_y\bigr)^s u = \mathbb{1}_\Omega h$$ where $V$ is apotential that satisfies some power growth conditions and the set $\Omega$ isthick in some sense. This extends previously known results for potentials$V(x)=|x|^{2k}$. To do so, we study Zhu-Zhuge's spectral inequality forSchr\"odinger operators with power growth potentials, and give a precisedquantitative form of it.

Null-controllability of the Generalized Baouendi-Grushin heat like equations