Observability of Singularly Perturbed Linear Time-Dependent Differential Systems with Small Delay

A singularly perturbed linear time-dependent differential system with multiple point and distributed time-delays in state variables is considered. Values of the delays are of order of a small positive parameter multiplying a part of derivatives in the system. Two much simpler systems (the reduced-order and the boundary-layer ones), associated with the original system, are introduced. These systems are in separate time scales, and they do not contain the small parameter any more. Connections between properties of observability of these systems and such a property of the original system itself, valid for all sufficiently small values of the parameter of singular perturbations, are established. These results are obtained for both, standard and nonstandard, types of the original system, and for two, Euclidean space and state-space, types of the observability.