Geodesic
On the existence of nonsmooth control-Lyapunov functions in the sense of generalized gradients
ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 593-611

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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.

Classification : 93D05, 93D15, 93D20, 93D30, 93D09, 93B05
Keywords: asymptotic stabilizability, converse Lyapunov theorem, nonsmooth analysis, differential inclusion, Filippov and krasovskii solutions, feedback 
@article{COCV_2001__6__593_0,
     author = {Rifford, Ludovic},
     title = {On the existence of nonsmooth {control-Lyapunov} functions in the sense of generalized gradients},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {593--611},
     publisher = {EDP-Sciences},
     volume = {6},
     year = {2001},
     mrnumber = {1872388},
     zbl = {1002.93058},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/COCV_2001__6__593_0/}
}
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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, Tome 6 (2001), pp. 593-611. http://geodesic.mathdoc.fr/item/COCV_2001__6__593_0/