Null-controllability of non-autonomous Ornstein-Uhlenbeck equations

We study the null-controllability of parabolic equations associated to non-autonomous Ornstein-Uhlenbeck operators. When a Kalman type condition holds for some positive time $T>0$, these parabolic equations are shown to enjoy a Gevrey regularizing effect at time $T>0$. Thanks to this regularizing effect, we prove by adapting the Lebeau-Robbiano method that these parabolic equations are null-controllable in time greater than or equal to $T>0$ from control regions, for which null-controllability is classically known to hold in the case of the heat equation.