By Anthony N. Michel
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The equilibrium x = 0 of composite system (9') with decomposition (Xi) is exponentially stable in the large if the following conditions are satisfied. possesses Property B ; (i) Each isolated subsystem (q) (ii) all comparison functions t,hG, i = 1, . , I, and t E J ; and (iv) given oi in hypothesis (i), there exists an I-vector such that the matrix S = [sG] specified by (ai
In the former case Du(,, is specified by Eq. 12) while in the latter case Do(,) is given by Eq. 13). We now characterize several properties of u-functions in terms of special types of comparison functions. 14. Definition. , cp E K , if cp(0) = 0 and if cp is strictly increasing on [O,r,] (or on [0, m)). If cp: R + -+ R', if cp E K , and if lim,+m cp(r) = co,then cp is said t o belong t o class KR. 15. Definition. 2 LYAPUNOV STABILITY AND RELATED RESULTS 17 k , and k , such that k , rpl(r) 2 rp2(r) I k,rp,(r) for all r E [O,r,] (or for all r E co, a>>.
Z,') E R", g i :R" x J + R r i , f , : Rntx Rrcx J + R"', and hi:R" x J + R"'. Letting hT(x,t ) = [ h , ( x , t ) T ,... , h l ( x ,r)T], Eq. 20) can equivalently be represented by (9") i= h ( x , r ) . 3 LARGE SCALE SYSTEMS 25 We assume that Eq. 21) possesses only one equilibrium, x = 0, and that for every to E R+ and every xo E R”, Eq. 21) has a unique solution x ( t ; x o ,1 , ) for all t 2 to. In the present case we speak of composite system (9”) with decomposition (C;) and leave the notion of isolated subsystem undefined.