By A.A. Martynyuk

"Presents new techniques to qualitative research of continuing, discreteptime, and impulsive nonlinear platforms through Liapunov matrix-valued services that introduce more desirable assessments for fixing difficulties of estimating the domain names of asymptotic stability."

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L. 13. 7). (t, n(t,z) =¢(a(t, where ~eC(R axt,R+), +~. ~(0) =0, ~(s) >0 for s>0, and lim ¢(s) 16 1. 15) are scalar, the ordinary technique of the Liapunov functions method is used to check their property of having a fixed sign, decreasing and radially unboundedhess. 3. 16) then II(t,x) becomes an ordi- val(t,,) ... v~,~(t,x) , u(t,x) ".. ... \v,~(t,x) ... , m, and (cf. Djordjevid [2]) _a,r¢81(llxsll)¢r1(llxrll <~,r¢,2(llzsII)¢r2(llx~ll) )

2. 1. 1. FollowingGrujid, et al. [1], Hahn[2], Liapunov[1], Massera[1, 2], Yoshizawa [1] the next result follows (see Martynyuk[20]). 3. 7) be continuous on R x Af (on T~ x Af). 7 ) <_ -¢(x) for all (t,x,y) ~ R x ~ "~ (retail (t,x,y) 6 % x ~ x R’~). 2. 7) is uniformly asymptotically stable (on Tr). According to Barbashin and Krasovskii [1, 2] and Grujid, et al. [1], and the preceding proof in which we choose ~ ~ KR it is easy to prove (see Martynyuk [20]). 4. 2. 7) be continuous on R x R’~ (on T~ x R").

5) and functions v~(t,x), i ~ j, i, j 42 1. 5). In the presence of the matrix-valued function constructed in such a way, its further application in the framework of the direct Liapunov method is carried out in two ways, either by construction of a scalar function or by construction of a vector function (including the cone-valued one). Below, the main theorems of the method of matrix Liapunov functions are presented in the framework of scalar approach. 7 provides a version of theorems of the method of matrix-valued Liapunov functions available for application in stability investigation of large scale systems.