# Indirect controllability of some linear parabolic systems of m equations with m-1 controls involving coupling terms of zero or first order

## TL;DR

This paper investigates the controllability of coupled linear parabolic systems with fewer controls than equations, providing necessary and sufficient conditions for null and approximate controllability in small time.

## Contribution

It introduces new controllability criteria for coupled parabolic systems with minimal controls, using algebraic and Carleman estimate techniques.

## Key findings

- Necessary and sufficient conditions for controllability with constant coefficients.
- Generic sufficient conditions for systems with variable coefficients in one dimension.
- Application of the fictitious control method combined with algebraic and Carleman techniques.

## Abstract

This paper is devoted to the study of the null and approximate controllability for some classes of linear coupled parabolic systems with less controls than equations. More precisely, for a given bounded domain in R^N, we consider a system of m linear parabolic equations (m > 2) with coupling terms of first and zero order, and m-1 controls localized in some arbitrary nonempty open subset. In the case of constant coupling coefficients, we provide a necessary and sufficient condition to obtain the null or approximate controllability in arbitrary small time. In the case m = 2 and N = 1, we also give a generic sufficient condition to obtain the null or approximate controllability in arbitrary small time for general coefficients depending on the space and times variables, provided that the supports of the coupling terms intersect the control domain. The results are obtained thanks to the fictitious control method together with an algebraic method and some appropriate Carleman estimates.