Dichotomy and stability by frequency-domain methods

In the last two decades, the frequency-domain criteria of stability have continuously progressed, overcoming previous limitations. In quest of a more complete meaning of these criteria, this paper shows that (if one renounces the auxiliary assumptions which were required in the past) they imply a property of conditional stability (or dichotomy) for a special type of relations. When applied to conventional systems, the obtained results give a general and flexible approach to the problem of stability (the auxiliary assumptions which were previously required, are eliminated). The paper is self-contained and does not assume any previous training in the theory of stability. Therefore, it can be also used as an introduction, which starts almost from scratch and leads to a branch of contemporary research in this field.