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We study the null-controllability of some hypoelliptic quadratic parabolicequations posed on the whole Euclidean space with moving control supports, andprovide necessary or sufficient geometric conditions on the moving controlsupports to ensure null-controllability. The first class of equations is theone associated to non-autonomous Ornstein-Uhlenbeck operators satisfying ageneralized Kalman rank condition. In particular, when the moving controlsupports comply with the flow associated to the transport part of theOrnstein-Uhlenbeck operators, a necessary and sufficient condition fornull-controllability on the moving control supports is established. The secondclass of equations is the class of accretive non-selfadjoint quadraticoperators with zero singular spaces for which some sufficient geometricconditions on the moving control supports are also given to ensurenull-controllability.

Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports