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We consider differential inclusions where
a positive semidefinite function of the solutions satisfies a
in terms of time and a second positive semidefinite function of the
We show that a smooth converse Lyapunov function, i.e., one whose
derivative along solutions can be
used to establish the class-
estimate, exists if and
only if the class-
is robust, i.e., it holds for a larger, perturbed differential
It remains an open question whether all class-
estimates are robust.
One sufficient condition for robustness is that the original
differential inclusion is locally Lipschitz.
Another sufficient condition is that the two positive semidefinite
functions agree and
a backward completability condition holds. These special cases unify
and generalize many results
on converse Lyapunov theorems for differential equations and
differential inclusions that have appeared in the literature.
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