On the stability and null-controllability of an infinite system of linear differential equations

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 $\lambda=-1$.