ON STABILITY-PRESERVING MAPPINGS OF DYNAMICAL-SYSTEMS .1. THEORY

In Part I of this two-part paper we utilize stability-preserving mappi ngs to identify general dynamical systems having equivalent qualitativ e properties. We establish sufficient conditions for a mapping to have stability-preserving properties and we employ stability-preserving ma ppings to formulate a comparison theorem for general dynamical systems . Qualitative aspects of dynamical systems which we address include st ability, asymptotic stability, and exponential stability of invariant sets (with an emphasis on equilibria) and boundedness of motions. The results developed herein include most of the corresponding comparison results reported in the literature as special cases. In addition, the present results are applicable to certain classes of general dynamical systems (such as discrete event systems) which cannot be addressed by corresponding previous results. For such contemporary systems, the mo tions which make up a dynamical system are not determined by equations , as is required by the existing results. We demonstrate the applicabi lity of the results developed herein by means of a class of discrete e vent systems in Part II of this paper.