# Partial null controllability of parabolic linear systems

## TL;DR

This paper investigates the conditions under which certain components of solutions to parabolic linear systems can be driven to zero using localized controls, with results varying based on matrix constancy and dependency.

## Contribution

It establishes necessary and sufficient conditions for partial null controllability in systems with constant matrices and provides sufficient conditions for time-dependent matrices, including space-dependent cases.

## Key findings

- Necessary and sufficient conditions for constant matrices
- Sufficient conditions for time-dependent matrices
- Examples illustrating controllability and non-controllability

## Abstract

This paper is devoted to the partial null controllability issue of parabolic linear systems with n equations. Given a bounded domain in R N, we study the effect of m localized controls in a nonempty open subset only controlling p components of the solution (p, m < n). The first main result of this paper is a necessary and sufficient condition when the coupling and control matrices are constant. The second result provides, in a first step, a sufficient condition of partial null controllability when the matrices only depend on time. In a second step, through an example of partially controlled 2x2 parabolic system, we will provide positive and negative results on partial null controllability when the coefficients are space dependent.