By Ljubomir T. Grujic, A.A. Martynyuk, M. Ribbens-Pavella
From the Preface: This e-book constitutes an up to the moment presentation and improvement of balance conception within the Liapunov experience with a variety of extensions and functions. detailed definitions of renowned and new balance houses are given via the authors who current common effects at the Liapunov balance houses of non-stationary structures that are out of the classical balance thought framework. The learn consists of using time various units and is broadened to time various Lur'e-Postnikov structures and singularly perturbed systems... in line with the volume and significance of definitions and balance standards provided I examine that this booklet, at the beginning released in Russian, represents the main whole one on balance concept proposed at this date. It pursuits every body excited about balance difficulties within the biggest experience and with defense, reliability and robustness. Professor Pierre Borne, Lille, France #1
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Extra info for Large Scale Systems Stability under Structural and Singular Perturbations
M i n o r of the m a t r i x H hili I hili 2 -'- hi2i 1 hi2i 2 ... ik] : . hlki 2 form V(x) : respectively; iff its q u a d r a t i c hiki I w h i c h has : semi-definite, definite, of these [see S. " the f u n d a m e n t a l For its p r e s e n t a t i o n A matrix is p o s i t i v e Let a (1970)]. m, that equation r e f e r r e d to as the L i a p u n o v m a t r i x nett and C. R. +PsnXn ) ~ s:l negative it , integers, then it is always possible homogenous function m of the form . "" . form V(x) : xTHx • he d e n o t e d by hili k hi • 2mk .
Massera (1967), ence of an open connected decrescent positive hold D+v(t,X) and sufficient on locally calculation and L a s a l l e definite of the system X=0 then ToXTrxR n ) positive respectively, Lipschitzian is also preceding [McShane T~ ) of (on ToXRXR n is necessary xR n) (on is l o c a l l y in the p r e c e d i n g on unbounded decrescent in the whole Remark 4. D+v (7) is continuous of a radially whole function Outline of the Liapunov stability theory in general . Let o =8+o* . Then last of I.
The conditions tively (i), invariant (iii) (on and (v) prove Yr ) with respect X(to) ~ S ( t o) ~ x ( t ; t o , X o ) 6 S ( t ) due to L e m m a i. B. The c o n d i t i o n (vi) means C. A and B prove (b). D. Let toeR Let ( t o e Y 7) , and ~(to,e) e ]0,+~[ Existence (ii)) , vt 6 [to,+~[ lim e>0 be a r b i t r a r i l y number ~(to,e ) of S(t) of the ( v t o e Y T) (7), , = 0 . chosen. obeying definiteness BE is posi- system F~(to,e)(t)c_ B e , v t e by both the p o s i t i v e and t i m e - i n v a r i a n c e set , VtoeR [S(t) :t-~+~] be such that of the p o s i t i v e tion is a s s u r e d tion that that the to motions [to;+~[ the last of (Proposition v .