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 l(2) is considered, where the matrix of a linear operator describing the system is an infinite matrix with lambda is an element of R on the main diagonal and is above it. We show that the system is asymptotically stable if and only if lambda <= -1, which shows the fine difference between the finite and the infinite-dimensional systems. When lambda <= -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 l(infinity) is not asymptotically stable if lambda = -1.