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

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

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

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topRifford, 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 -

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