2. Definitions and preliminary results

This section aims to introduce key concepts and mathematical objects crucial for defining, modeling, and analyzing time-delay systems. The delay operator is fundamental in this context (second only to the derivative operator), as it enables the description of delayed signals. This operator is linear and time-invariant when the delay itself is constant. Consequently, we can define a transfer function for this operator, which in turn allows us to establish a characteristic equation for the system – analogous to the characteristic polynomial of a standard linear system.

We will also explore the significance of the characteristic roots – the zeros of the characteristic equation – and their relationship to the system stability. Notably, we will see that many fundamental properties and results from non-delayed systems extend to delayed systems, often with minor modifications....