Issue |
ESAIM: COCV
Volume 5, 2000
|
|
---|---|---|
Page(s) | 313 - 367 | |
DOI | https://doi.org/10.1051/cocv:2000113 | |
Published online | 15 August 2002 |
A smooth Lyapunov function from a class-
estimate
involving two positive semidefinite functions
1
ECE Department, University of California,
Santa Barbara, CA 93106, U.S.A.; teel@ece.ucsb.edu.
2
Centre Automatique et Systèmes,
École des Mines de Paris,
35 rue Saint Honoré, 77305 Fontainebleau Cedex, France; praly@cas.ensmp.fr.
Received:
3
November
1999
Revised:
24
May
2000
We consider differential inclusions where
a positive semidefinite function of the solutions satisfies a
class- estimate
in terms of time and a second positive semidefinite function of the
initial condition.
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-
estimate
is robust, i.e., it holds for a larger, perturbed differential
inclusion.
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.
Mathematics Subject Classification: 34A60 / 34D20 / 34B25
Key words: Differential inclusions / Lyapunov functions / uniform asymptotic stability.
© EDP Sciences, SMAI, 2000
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