By A.A. Martynyuk
"Presents new ways to qualitative research of continuing, discreteptime, and impulsive nonlinear structures through Liapunov matrix-valued services that introduce more desirable exams for fixing difficulties of estimating the domain names of asymptotic stability."
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Additional info for Qualitative Methods in Nonlinear Dynamics (Pure and Applied Mathematics)
In the monographby Zubov  the following assertions are proved. 1) obey the bilateral inequality , <_~2(t)~ ~(t)~~(t) < ~(t, ~(t ~, )~) where~oi(t) > 0 for all 6 Toandp(t) = (xT(t to, Xo)x(t; to, Xo)) (ii) function g(t, x, v) satisfy the estimates --¢1 (t)p2(t) where ¢i(t) _> 0 for all t 6 To and functions ¢i(t)/qoi(t), are integrable. 2. 2. (t)p where ¢i(t), i = 1, 2 are positive piece-wise continuous functions for all t ~ A, kl ~ k2. 2. 7) the estimates 1 1 to are valid for all t ~ A, v0 = v(to,xo,a), Ai = ~, ,~i > 1 for i = 1,2.
X S iff (i) holds for A/" = (iii) positive definite in the wholeon 7-r iff (i) holds for Af = Rn; (iv) negative definite (in the whole) on 7-r (on Tr xAf) iff (-v) is positive definite (in the whole) on Tr (on Tr x iV’), respectively. The expression "on 7"r " is omitted iff all corresponding requirements hold for every r ~ R. 8. 3. 10. 11. A function v: R x Rn --+ R is: (i) decreasing on 7"~, r ~ R, iff there is a time-invariant neighborhood Af of x = 0 and a positive definite function w on Af, w: Rn -+ R, such that v(t, x) <_w(x) for all (t, x) e T~ x Af; (ii) decreasing on T~ x S iff (i) holds for Af = (iii) decreasing in the whole on 7-r iff (i) holds for Af Rn.
3. 12 is based on the results by Grujid , and Grujid, et al. . For the details see also Barbashin and Krasovskii [1, 2], Coppel , Demidovich , Hahn , Hirsch and Smale , Krasovskii , Liapunov , Massera , Nemytskii and Stepanov , Yoshizawa, Zubov, etc. 4. Classical results of qualitative analysis of systems in terms of the direct Liapunov methodbased on the scalar auxiliary function can be found in the well-known monographs and manuals by Barbashin , Bhatia and Szeg5 , Chetaev , Hahn , Kalmanand Bertman , Krasovskii , Lakshmikantham,Leela, et al.