ABSTRACT THEORY OF ABSOLUTE STABILITY AND ITS APPLICATIONS.

The paper gives an outline of the abstract theory of absolute stability previously developed by the author. The theory allows considering the stability of 'abstract systems' defined by sets in the functional space of a 'process'. If such a system is stable, it means that all of its processes have a finite L//2 norm. Special systems are considered; their stability criterion is defined as the condition of positive certainty of a quadratic form at the intersection of subspaces defined by the linear part of the system and various stable processes. The application of this criterion is illustrated by the derivation of two special conditions of absolute stability for integral equations. The first of these has been known previously as regards sufficiency. The second criterion is new. It defines the necessary and sufficient conditions of absolute stability in the class of nonlinear blocks defined by an integral quadratic relation.