# Controllability of a 2x2 parabolic system by one force with a space-dependent coupling term of order one

## TL;DR

This paper investigates the controllability of a coupled 2x2 parabolic system with space-dependent coupling, establishing conditions for null and approximate controllability, and determining minimal control times using the moment method.

## Contribution

It provides new controllability results for coupled parabolic systems with space-dependent coupling terms, including necessary and sufficient conditions and minimal control times.

## Key findings

- Null controllability when control and coupling domains intersect.
- Minimal time for null controllability when domains do not intersect.
- Necessary and sufficient conditions for approximate controllability.

## Abstract

This paper is devoted to the controllability of linear systems of two coupled parabolic equations when the coupling involves a space dependent first order term. This system is set on an bounded interval, and the first equation is controlled by a force supported in a subinterval of I or on the boundary. In the case where the intersection of the coupling and control domains is nonempty, we prove null controllability at any time. Otherwise, we provide a minimal time for null controllability. Finally we give a necessary and sufficient condition for the approximate controllability. The main technical tool for obtaining these results is the moment method