  In this work, the null controllability problem for a linear system in
$\ell^2$ is considered, where the matrix of a linear operator describing the
system is an infinite matrix with $\lambda\in \mathbb R$ on the main diagonal
and 1s above it. We show that the system is asymptotically stable if and only
if $\lambda\le-1$, which shows the fine difference between the finite and the
infinite-dimensional systems. When $\lambda\le-1$ we also show that the system
is null controllable in large. We also show a dependence of the stability on
the norm i.e. the same system considered in $\ell^\infty$ is not asymptotically
stable if $\lambda=-1$.
