Vol.4, No 16, 2004 pp. 33-42
UDC 531:01+531.36:517.93(04)
Invited Paper
A NEW UNIFIED CONCEPT OF STABILITY
S. Leela
Professor Emerita, SUNY@Geneseo
Geneseo, NY 14454, USA
Abstract. In the useful and rich field of stability theory for nonlinear
systems, there have been many refinements, extensions and generalizations
[3, 4]. Basically, stability concerns with comparing phase-space positions
of solutions of perturbed and unperturbed equations, with classical Lyapunov
stability being too stringent a requirement and orbital stability being
too loose a demand. We define a new concept of stability that can unify
theses two extreme cases (and possibly many other appropriate notions between
these two) in terms of suitable topologies, following the idea of J. L.
Massera [6]. Also, a further unification is achieved by using two measures
[5]. In this unified frame work, we give sufficient conditions for these
concepts to hold, via Lyapunov functions.
Key words: clock stability, Lyapunov stability, orbital
stability.
NOVI UNIFICIRANI KONCEPT STABILNOSTI
U korisnoj i bogatoj oblasti teorije stabilnosti nonlinearnih sistema,
dogodila su se mnoga prečišćavanja, proširenja i generalizacije
[3, 4]. U osnovi stabilnost se odnosi na uporedjivanje položaja faznog
prostora rešenja poremećenih i neporemećenih jednačina sa klasičnom Ljapinovom
stabilnošću koja postavlja isuviše strog zahtev i orbitalne stabilnosti,
koja postavlja isuviše slab zahtev. Novi koncept stabilnosti se definiše
tako da može da objedinjuje ova dva ekstremna slučaja (i verovatno mnoge
druge odgovarajuće pomove i pojave izmedju ovih dvaju) na osnovu odgovarajuće
topologije prateći ideju J. L. Masera [6]. Dalja unifikacija se takodje
ostvaruje koristeći dve mere [5]. U ovom unificiranom okviru, ukazujemo
na potrebne uslove za održivost ovih koncepata preko Ljapinovih funkcija.
Ključne reči: stabilnost časovnika, Ljapinova stabilnost, orbitalna
stabilnost.
