Admissibility of retarded diagonal systems with one-dimensional input space

We investigate infinite-time admissibility of a control operator B in a Hilbert space state-delayed dynamical system setting of the form z ˙ ( t ) = A z ( t ) + A 1 z ( t - τ ) + B u ( t ) , where A generates a diagonal C 0 -semigroup, A 1 ∈ L ( X ) is also diagonal and u ∈ L 2 ( 0 , ∞ ; C ) . Our approach is based on the Laplace embedding between L 2 and the Hardy space H 2 ( C + ) . The results are expressed in terms of the eigenvalues of A and A 1 and the sequence representing the control operator.