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

TL;DR

This paper establishes geometric conditions on moving control supports that guarantee null-controllability of certain hypoelliptic quadratic parabolic equations, including non-autonomous Ornstein-Uhlenbeck and accretive quadratic operators.

Contribution

It provides necessary and sufficient geometric conditions for null-controllability of hypoelliptic quadratic parabolic equations with moving supports, extending control theory to complex operator classes.

Findings

Necessary and sufficient conditions for Ornstein-Uhlenbeck equations.

Sufficient geometric conditions for accretive quadratic operators.

Control support flow alignment ensures null-controllability.

Abstract

We study the null-controllability of some hypoelliptic quadratic parabolic equations posed on the whole Euclidean space with moving control supports, and provide necessary or sufficient geometric conditions on the moving control supports to ensure null-controllability. The first class of equations is the one associated to non-autonomous Ornstein-Uhlenbeck operators satisfying a generalized Kalman rank condition. In particular, when the moving control supports comply with the flow associated to the transport part of the Ornstein-Uhlenbeck operators, a necessary and sufficient condition for null-controllability on the moving control supports is established. The second class of equations is the class of accretive non-selfadjoint quadratic operators with zero singular spaces for which some sufficient geometric conditions on the moving control supports are also given to ensure null-controllability.