# A smooth Lyapunov function from a class-$\mathrm{\mathcal{K}\mathcal{L}}$ estimate involving two positive semidefinite functions

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 5, page 313-367
- ISSN: 1292-8119

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topTeel, Andrew R., and Praly, Laurent. "A smooth Lyapunov function from a class-${\mathcal{KL}}$ estimate involving two positive semidefinite functions." ESAIM: Control, Optimisation and Calculus of Variations 5 (2010): 313-367. <http://eudml.org/doc/197354>.

@article{Teel2010,

abstract = {
We consider differential inclusions where
a positive semidefinite function of the solutions satisfies a
class-$\{\mathcal\{KL\}\}$ 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-$\{\mathcal\{KL\}\}$ estimate, exists if and
only if the class-$\{\mathcal\{KL\}\}$ estimate
is robust, i.e., it holds for a larger, perturbed differential
inclusion.
It remains an open question whether all class-$\{\mathcal\{KL\}\}$
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.
},

author = {Teel, Andrew R., Praly, Laurent},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Differential inclusions; Lyapunov functions; uniform asymptotic stability.; differential inclusions; uniform asymptotic stability; smooth converse Lyapunov function},

language = {eng},

month = {3},

pages = {313-367},

publisher = {EDP Sciences},

title = {A smooth Lyapunov function from a class-$\{\mathcal\{KL\}\}$ estimate involving two positive semidefinite functions},

url = {http://eudml.org/doc/197354},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Teel, Andrew R.

AU - Praly, Laurent

TI - A smooth Lyapunov function from a class-${\mathcal{KL}}$ estimate involving two positive semidefinite functions

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 5

SP - 313

EP - 367

AB -
We consider differential inclusions where
a positive semidefinite function of the solutions satisfies a
class-${\mathcal{KL}}$ 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-${\mathcal{KL}}$ estimate, exists if and
only if the class-${\mathcal{KL}}$ estimate
is robust, i.e., it holds for a larger, perturbed differential
inclusion.
It remains an open question whether all class-${\mathcal{KL}}$
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.

LA - eng

KW - Differential inclusions; Lyapunov functions; uniform asymptotic stability.; differential inclusions; uniform asymptotic stability; smooth converse Lyapunov function

UR - http://eudml.org/doc/197354

ER -

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