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Let us recall that in the topological setting, useful informations about the dynamical behavior of a system can be deduced from the inspection of certain sets. These sets depcnd in general on the initial state. Thus, they can be reviewed as set valued maps. 13), namely r+(x0) = (y e ~ ~ y = x(t; ~0) for s o m e t ~ O} . Nonlinear systems: stability 39 Another example is the positive limit set. We adopt the following agreements about notation. Let Q(x) be a set valued map from ~'~ to ll~'~. For f2 C ~'~, we denote Q(f2) = o , e n Q ( x ).

In other words, V(~:(7")) <_c < V(z(tl)). e.. 27) applies and V(x(t)) turns out to be decreasing. This yields V(x(r)) > V(x(tl)), which is a contradiction. > We can now complete the argument. ) are given. )ll~) for each t >_ 0. If z0 ~ K, the existence of a function fl E s that the inequality such ib(t)ii < Z(t, ii~0ii) holds as long as z(t) remains outside K, can be proved by virtue of the comparison principle for solutions of differential equations (see for instance [63], p. 26) and Lemma 1 of [117].

5 (with Br replaced by Br), will be called a smooth weak Liapunov function in the large. 7 A smooth strict Liapunov function in the small is a weak Lia- punov function such that V(x) is negative definite; in other words, it satisfies, instead of (iv), (v) v v ( x ) , f(~) < o for each ~ 9 B, (~ # 0). A function V(x) defined for all z E ~'~, which is radially unbounded and fulfills the properties (i), (ii}, (iii), (v) with B~ replaced by I~n, will be called a smooth global strict Liapunov function.

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A unified theory of elastic degradation and damage based on a loading surface by Carol I.


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