# Quadratic differential equations : partial Gelfand-Shilov smoothing effect and null-controllability

## TL;DR

This paper investigates the regularizing and decay properties of solutions to certain quadratic evolution equations and demonstrates that these properties imply null-controllability from thick control sets, extending known geometric control conditions.

## Contribution

It establishes that thick control subsets suffice for null-controllability of parabolic equations linked to non-globally hypoelliptic quadratic operators, broadening control theory applications.

## Key findings

- Solutions exhibit partial Gelfand-Shilov regularization and exponential decay.
- Null-controllability holds in any positive time from thick control sets.
- Thickness of control sets is sufficient for null-controllability of these operators.

## Abstract

We study the partial Gelfand-Shilov regularizing effect and the exponential decay for the solutions to evolution equations associated to a class of accretive non-selfadjoint quadratic operators, which fail to be globally hypoelliptic on the whole phase space. By taking advantage of the associated Gevrey regularizing effects, we study the null-controllability of parabolic equations posed on the whole Euclidean space associated to this class of possibly non-globally hypoelliptic quadratic operators. We prove that these parabolic equations are null-controllable in any positive time from thick control subsets. This thickness property is known to be a necessary and sufficient condition for the null-controllability of the heat equation posed on the whole Euclidean space. Our result shows that this geometric condition turns out to be a sufficient one for the null-controllability of a large class of quadratic differential operators.