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

Abstract

In this work, the null controllability problem for a linear system in ℓ 2 is considered, where the matrix of a linear operator describing the system is an infinite matrix with λ ∈ R on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if λ ≤ −1, which shows the fine difference between the finite and the infinite-dimensional systems. When λ ≤ −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 ℓ ∞ is not asymptotically stable if λ = −1.