Null controllability in large time of a parabolic equation involving the Grushin operator with an inverse-square potential

We prove the null controllability in large time of the following linear parabolic equation involving the Grushin operator with an inverse-square potential

$$u_t-\Delta_{x} u-|x|^{2}\Delta_{y}u-\frac{\mu}{|x|^2}u=v1_\omega$$

in a bounded domain \({\Omega=\Omega_1\times \Omega_2\subset \mathbb{R}^{N_1}\times \mathbb{R}^{N_2} (N_1\geq 3, N_2\geq 1}\)) intersecting the surface {x = 0} under an additive control supported in an open subset \({\omega=\omega_1\times \Omega_2}\) of \({\Omega}\).