# 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 -

## References

top- Z. Artstein, Stabilization with relaxed controls. Nonlinear Anal.7 (1983) 1163-1173. Zbl0525.93053
- J.-P. Aubin, Viability theory. Birkhäuser Boston Inc., Boston, MA (1991). Zbl0755.93003
- J.P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag (1984).
- J.P. Aubin and H. Frankowska, Set-valued analysis. Birkhäuser (1990).
- C.I. Byrnes and A. Isidori, New results and examples in nonlinear feedback stabilization. Systems Control Lett.12 (1989) 437-442. Zbl0684.93059
- F.H. Clarke, Yu.S. Ledyaev, L. Rifford and R.J. Stern, Feedback stabilization and Lyapunov functions. SIAM J. Control Optim.39 (2000) 25-48. Zbl0961.93047
- F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983). Republished as Classics Appl. Math. 5 (1990).
- F.H. Clarke, Yu.S. Ledyaev and R.J. Stern, Asymptotic stability and smooth Lyapunov functions. J. Differential Equations149 (1998) 69-114. Zbl0907.34013
- F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory. Springer-Verlag, New York, Grad. Texts in Math. 178 (1998). Zbl1047.49500
- J.-M. Coron, On the stabilization of some nonlinear control systems: Results, tools, and applications, in Nonlinear analysis, differential equations and control (Montreal, QC, 1998). Kluwer Acad. Publ., Dordrecht (1999) 307-367.
- J.-M. Coron, Some open problems in control theory, in Differential geometry and control (Boulder, CO, 1997). Providence, RI, Amer. Math. Soc. (1999) 149-162.
- J.-M. Coron and L. Rosier, A relation between continuous time-varying and discontinuous feedback stabilization. J. Math. Systems Estim. Control4 (1994) 67-84. Zbl0925.93827
- K. Deimling, Multivalued Differential Equations. de Gruyter, Berlin (1992). Zbl0760.34002
- A.F. Filippov, Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers (1988).
- R. Freeman and P.V. Kokotovic, Robust Nonlinear Control Design. State-Space and Lyapunov Techniques. Birkhäuser (1996). Zbl0857.93001
- R.A. Freeman and P.V. Kokotovic, Backstepping design with nonsmooth nonlinearities, in Proc. of the IFAC Nonlinear Control Systems design symposium. Tahoe City, California (1995).
- O. Hájek, Discontinuous differential equations. I, II. J. Differential Equations32 (1979) 149-170, 171-185. Zbl0365.34017
- J.-B. Hiriart-Urruty and C. Imbert, Les fonctions d'appui de la jacobienne généralisée de Clarke et de son enveloppe plénière. C. R. Acad. Sci. Paris Sér. I Math.326 (1998) 1275-1278.
- N.N. Krasovskiĭ, Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay. Stanford University Press, Stanford, California (1963). Translated by J.L. Brenner. Zbl0109.06001
- J. Kurzweil, On the inversion of Lyapunov's second theorem on stability of motion. Amer. Math. Soc. Transl. Ser. 224 (1956) 19-77.
- Yu.S. Ledyaev and E.D. Sontag, A Lyapunov characterization of robust stabilization. Nonlinear Anal.37 (1999) 813-840. Zbl0947.34054
- Y. Lin, E.D. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability. SIAM J. Control Optim.34 (1996) 124-160. Zbl0856.93070
- J.L. Massera, Contributions to stability theory. Ann. of Math. (2)64 (1956) 182-206. Zbl0070.31003
- E. Michael, Continuous selections. I. Ann. of Math. (2)63 (1956) 361-382. Zbl0071.15902
- L. Praly and A.R. Teel, On assigning the derivative of a disturbance attenuation clf, in Proc. of the 37th IEEE conference on decision and control. Tampa, Florida (1998). Zbl0965.93048
- L. Rifford, Existence of Lipschitz and semiconcave control-Lyapunov functions. SIAM J. Control Optim.39 (2000) 1043-1064. Zbl0982.93068
- L. Rosier, Étude de quelques problèmes de stabilisation, Ph.D. Thesis. ENS de Cachan (1993).
- E.D. Sontag, A ``universal'' construction of Artstein's theorem on nonlinear stabilization. Systems Control Lett.13 (1989) 117-123. Zbl0684.93063
- E.D. Sontag, Mathematical Control Theory. Springer-Verlag, New York, Texts Appl. Math. 6 (1990) (Second Edition, 1998). Zbl0703.93001
- E.D. Sontag, Stability and stabilization: Discontinuities and the effect of disturbances, in Nonlinear analysis, differential equations and control (Montreal, QC, 1998). Kluwer Acad. Publ., Dordrecht (1999) 551-598. Zbl0937.93034
- A.R. Teel and L. Praly, A smooth Lyapunov function from a class- $\mathcal{K}\mathcal{L}$ estimate involving two positive semidefinite functions. ESAIM: COCV5 (2000) 313-367. Zbl0953.34042
- J. Tsinias, A Lyapunov description of stability in control systems. Nonlinear Anal.13 (1989) 3-74. Zbl0695.93083
- J. Tsinias, Sufficient Lyapunov-like conditions for stabilization. Math. Control Signals Systems2 (1989) 343-357. Zbl0688.93048
- J. Tsinias, A local stabilization theorem for interconnected systems. Systems Control Lett.18 (1992) 429-434. Zbl0763.93076
- J. Tsinias, An extension of Artstein's theorem on stabilization by using ordinary feedback integrators. Systems Control Lett.20 (1993) 141-148. Zbl0782.93080

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