Admissibility of Diagonal State-Delayed Systems with a One-Dimensional Input Space

In this paper we investigate admissibility of the control operator B in a Hilbert space state-delayed dynamical system setting of the form \({\dot{z}}(t)=Az(t-\tau )+Bu(t)\), where A generates a diagonal semigroup and u is a scalar input function. Our approach is based on the Laplace embedding between \(L^2\) and the Hardy space. The sufficient conditions for infinite-time admissibility are stated in terms of eigenvalues of the generator and in terms of the control operator itself.