Admissibility of retarded diagonal systems with one-dimensional input space

TL;DR

This paper studies the conditions under which a control operator ensures infinite-time admissibility for a class of diagonal, delayed dynamical systems in Hilbert spaces, using complex analysis techniques.

Contribution

It provides new criteria for admissibility based on eigenvalues and control operator sequences in diagonal, delayed systems.

Findings

Derived criteria for infinite-time admissibility involving eigenvalues and control sequences.

Connected admissibility conditions to properties of the Laplace transform and Hardy space.

Extended existing theory to include systems with state delays and diagonal structure.

Abstract

We investigate infinite-time admissibility of a control operator $B$ in a Hilbert space state-delayed dynamical system setting of the form $\dot{z}(t)=Az(t)+A_1 z(t-\tau)+Bu(t)$, where $A$ generates a diagonal $C_0$-semigroup, $A_1\in\mathcal{L}(X)$ is also diagonal and $u\in L^2(0,\infty;\mathbb{C})$. Our approach is based on the Laplace embedding between $L^2$ and the Hardy space $H^2(\mathbb{C}_+)$. The results are expressed in terms of the eigenvalues of $A$ and $A_1$ and the sequence representing the control operator.