On the existence of nonsmooth control-Lyapunov functions in the sense of generalized gradients

Ludovic Rifford

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

  • Volume: 6, page 593-611
  • ISSN: 1292-8119

Abstract

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Let x ˙ = f ( x , u ) be a general control system; the existence of a smooth control-Lyapunov function does not imply the existence of a continuous stabilizing feedback. However, we show that it allows us to design a stabilizing feedback in the Krasovskii (or Filippov) sense. Moreover, we recall a definition of a control-Lyapunov function in the case of a nonsmooth function; it is based on Clarke's generalized gradient. Finally, with an inedite proof we prove that the existence of this type of control-Lyapunov function is equivalent to the existence of a classical control-Lyapunov function. This property leads to a generalization of a result on the systems with integrator.

How to cite

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Rifford, Ludovic. "On the existence of nonsmooth control-Lyapunov functions in the sense of generalized gradients." ESAIM: Control, Optimisation and Calculus of Variations 6 (2010): 593-611. <http://eudml.org/doc/197327>.

@article{Rifford2010,
abstract = { Let $\dot\{x\}=f(x,u)$ be a general control system; the existence of a smooth control-Lyapunov function does not imply the existence of a continuous stabilizing feedback. However, we show that it allows us to design a stabilizing feedback in the Krasovskii (or Filippov) sense. Moreover, we recall a definition of a control-Lyapunov function in the case of a nonsmooth function; it is based on Clarke's generalized gradient. Finally, with an inedite proof we prove that the existence of this type of control-Lyapunov function is equivalent to the existence of a classical control-Lyapunov function. This property leads to a generalization of a result on the systems with integrator. },
author = {Rifford, Ludovic},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Asymptotic stabilizability; converse Lyapunov theorem; nonsmooth analysis; differential inclusion; and Krasovskii solutions; feedback.; asymptotic stabilizability; nonsmooth analysis; Filippov and Krasovskij solutions; feedback; epigraph; viability property; control-Lyapunov function},
language = {eng},
month = {3},
pages = {593-611},
publisher = {EDP Sciences},
title = {On the existence of nonsmooth control-Lyapunov functions in the sense of generalized gradients},
url = {http://eudml.org/doc/197327},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Rifford, Ludovic
TI - On the existence of nonsmooth control-Lyapunov functions in the sense of generalized gradients
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 6
SP - 593
EP - 611
AB - Let $\dot{x}=f(x,u)$ be a general control system; the existence of a smooth control-Lyapunov function does not imply the existence of a continuous stabilizing feedback. However, we show that it allows us to design a stabilizing feedback in the Krasovskii (or Filippov) sense. Moreover, we recall a definition of a control-Lyapunov function in the case of a nonsmooth function; it is based on Clarke's generalized gradient. Finally, with an inedite proof we prove that the existence of this type of control-Lyapunov function is equivalent to the existence of a classical control-Lyapunov function. This property leads to a generalization of a result on the systems with integrator.
LA - eng
KW - Asymptotic stabilizability; converse Lyapunov theorem; nonsmooth analysis; differential inclusion; and Krasovskii solutions; feedback.; asymptotic stabilizability; nonsmooth analysis; Filippov and Krasovskij solutions; feedback; epigraph; viability property; control-Lyapunov function
UR - http://eudml.org/doc/197327
ER -

References

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