Approximate null-controllability with uniform cost for the hypoelliptic Ornstein-Uhlenbeck equations

TL;DR

This paper characterizes the approximate null-controllability with uniform cost for hypoelliptic Ornstein-Uhlenbeck equations on ^n using an integral thickness geometric condition, extending results from parabolic equations and fractional heat equations.

Contribution

It introduces a geometric condition for controllability of hypoelliptic equations and generalizes known results from parabolic and fractional heat equations to a broader class.

Findings

Controllability characterized by integral thickness condition.

Quantitative weak observability estimates provided.

Extension to fractional heat equations with ^n control supports.

Abstract

We prove that the approximate null-controllability with uniform cost of the hypoelliptic Ornstein-Uhlenbeck equations posed on $\mathbb R^n$ is characterized by an integral thickness geometric condition on the control supports. We also provide associated quantitative weak observability estimates. This result for the hypoelliptic Ornstein-Uhlenbeck equations is deduced from the same study for a large class of non-autonomous elliptic equations from moving control supports. We generalize in particular results known for parabolic equations posed on $\mathbb R^n$, for which the approximate null-controllability with uniform cost is ensured by the notion of thickness, which is stronger that the integral thickness condition considered in the present work. Examples of those parabolic equations are the fractional heat equations associated with the operator $(-\Delta)^s$, in the regime $s\geq1/2$. Our strategy also allows to characterize the approximate null-controllability with uniform cost from moving control supports for this class of fractional heat equations.