# Positive and negative results on the internal controllability of parabolic equations coupled by zero and first order terms

## TL;DR

This paper investigates the controllability of coupled parabolic equations with zero and first order terms, establishing new conditions for null controllability and providing examples where approximate controllability fails.

## Contribution

It introduces a new sufficient condition for null controllability and presents the first example of a system that is not approximately controllable under certain coupling conditions.

## Key findings

- Null controllability achieved under a new condition
- First example of non-approximate controllability with intersecting support
- Insights into the influence of coupling terms on controllability

## Abstract

This paper is devoted to studying the null and approximate controllability of two linear coupled parabolic equations posed on a smooth domain of R^N (N>1) with coupling terms of zero and first orders and one control localized in some arbitrary nonempty open subset of the domain. We prove the null controllability under a new sufficient condition and we also provide the first example of a not approximately controllable system in the case where the support of one of the nontrivial first order coupling terms intersects the control domain.