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 ${\mathrm{\ell }}^2$ 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 ${\mathrm{\ell }}^{\mathrm{\infty }}$ is not asymptotically stable if $\lambda =-1$.