PROLONGATIONS AND GENERALIZED LIAPUNOV FUNCTIONS
Abstract
The purpose of this note is to study various types of stability in autonomous systems by means of generalized Liapunov functions in the sense of S. Lefschetz (Bol. soc. Mat. Mexicana (2) 3:25-39, 1958) and prolongations in the sense of T. Ura (Ann. sci. Ecole Norm. Sup. (3) 70:287360, 1953 and Funkcialaj Ekvacioj 2:143-200, 1959). The concept of prolongation leads to a whole hierarchy of new types of stability, all of which occupy an intermediate place between stability in the sense of Liapunov and asymptotic stability. The most significant among them, called absolute stability, is characterized by the existence of a continuous generalized Liapunov function with a non-positive generalized total derivative. If the latter is strictly negative, we have asymptotic stability. The second part of the report is devoted to stability under permanent perturbations or total stability with small bounded perturbation terms. We characterize total stability as invariance under a certain map which has essential properties in common with Ura's concept of prolongation. In order to study properties of different kinds of stability simultaneously, we define prolongations abstractly and consider their invariant sets, thus obtaining stability theorems as corollaries. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1961
- Accession Number
- AD0262350
Entities
People
- Peter Seibert
Organizations
- Glenn L. Martin Company