H2-optimality conditions for reduced time-delay systems of dimension one

In this note, we first attempt to generalize the H2 optimal interpolation conditions for more general reduced order models. To this aim, we first expose the necessary optimality conditions in the case where the reduced system is of dimension one and have a single state delay structure. This can be viewed as a first step toward the H2 optimal model approximation where the reduced system correspond to an infinite dimensional system. Finally, we illustrate the results with an academic example.