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 ℓ² is considered, where the matrix of a linear operator describing the system is an infinite matrix with $\lambda \in ℝ$ 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. Further we show a dependence of the stability on the norm, i.e. the same system considered ${\ell }^{\infty }$ is not asymptotically stable if λ = - 1.

Keywords: Controllable system; Differential equations in Banach spaces; Eigenvalues; Gramian operators; Stability.