By Bildhauer M.
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28 1. 2. Let the vector function f nuous on R x Rn (on T~ x Rn). 2. 1. 1. FollowingGrujid, et al. , Hahn, Liapunov, Massera[1, 2], Yoshizawa  the next result follows (see Martynyuk). 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. , and the preceding proof in which we choose ~ ~ KR it is easy to prove (see Martynyuk ).
3 1. , m. 12). 5) is established in terms of one of the following functions (see Lakshmikantham,Matrosov, et al. 9) v(t,x) = Q(Y(t,x)), Q(0) Q ¯ C(R~, R+), the function Q(u) is nondecreasing in u. 12. 2. 5). ). 11) II(t, x) = ... ".. ~Vtl(t,X) ... ,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.
2. (1) (a) qi(uj) (2) Assume tha~: > qii(u~), uj e R+, j = 1,2,... ,~)ll m ~ ~ e~g~(v(~,~))~(~(t,~)); (4) I~(t,o) = a~(t,o,... ,m. 4) is asymptotically stable. 2 is a typical result in the stability analysis of large scale system via the method of vector Liapunov functions. Comment1. 5)). The assumption A(u) contained in (1. 7) imp lies tha the comparison matrix function A(u) must be Metzler with quasi-dominant main diagonal property. This means that if the comparison matrix is neither Metzler nor possesses the restrictive quasi-dominant diagonal property, then the methodfails to yield the required stability results.
Convex variational problems by Bildhauer M.