A smooth Lyapunov function from a class- 𝒦ℒ estimate involving two positive semidefinite functions

Andrew R. Teel; Laurent Praly

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

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

Abstract

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

How to cite

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Teel, 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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