# Stability and Stabilization of Discontinuous Systems and Nonsmooth Lyapunov Functions

Andrea Bacciotti; Francesca Ceragioli

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

- Volume: 4, page 361-376
- ISSN: 1292-8119

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topBacciotti, Andrea, and Ceragioli, Francesca. " Stability and Stabilization of Discontinuous Systems and Nonsmooth Lyapunov Functions." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 361-376. <http://eudml.org/doc/116570>.

@article{Bacciotti2010,

abstract = {
We study stability and stabilizability properties of systems with discontinuous
righthand side (with solutions intended in Filippov's sense) by means of
locally Lipschitz continuous and regular Lyapunov functions. The stability
result is obtained in the more general context of
differential inclusions. Concerning stabilizability, we focus on systems
affine with respect to the input: we give some sufficient conditions for a
system to be stabilized by means of a feedback law of the Jurdjevic-Quinn type.
},

author = {Bacciotti, Andrea, Ceragioli, Francesca},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Stability; stabilization; Clarke gradient; Lyapunov functions;
LaSalle principle.; stability; stabilizability; differential inclusions; Jurdevic-Quinn type; LaSalle principle},

language = {eng},

month = {3},

pages = {361-376},

publisher = {EDP Sciences},

title = { Stability and Stabilization of Discontinuous Systems and Nonsmooth Lyapunov Functions},

url = {http://eudml.org/doc/116570},

volume = {4},

year = {2010},

}

TY - JOUR

AU - Bacciotti, Andrea

AU - Ceragioli, Francesca

TI - Stability and Stabilization of Discontinuous Systems and Nonsmooth Lyapunov Functions

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 4

SP - 361

EP - 376

AB -
We study stability and stabilizability properties of systems with discontinuous
righthand side (with solutions intended in Filippov's sense) by means of
locally Lipschitz continuous and regular Lyapunov functions. The stability
result is obtained in the more general context of
differential inclusions. Concerning stabilizability, we focus on systems
affine with respect to the input: we give some sufficient conditions for a
system to be stabilized by means of a feedback law of the Jurdjevic-Quinn type.

LA - eng

KW - Stability; stabilization; Clarke gradient; Lyapunov functions;
LaSalle principle.; stability; stabilizability; differential inclusions; Jurdevic-Quinn type; LaSalle principle

UR - http://eudml.org/doc/116570

ER -

## References

top- J.P. Aubin and A. Cellina, Differential Inclusions, Springer Verlag (1984).
- J. Auslander and P. Seibert, Prolongations and Stability in Dynamical Systems. Ann. Inst. Fourier (Grenoble)14 (1964) 237-268. Zbl0128.31303
- A. Bacciotti, Local Stabilizability Theory of Nonlinear System, World Scientific (1992). Zbl0757.93061
- A. Bacciotti and L. Rosier, Liapunov and Lagrange Stability: Inverse Theorems for Discontinuous Systems. Mathematics of Control, Signals and Systems 11 (1998) 101-128.
- N.P. Bhatia and G.P. Szëgo, Stability Theory of Dynamical Systems, Springer Verlag (1970).
- F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley and Sons (1983). Zbl0582.49001
- F.H. Clarke, Yu.S. Ledyaev, E.D. Sontag and A.I. Subbotin, Asymptotic Controllability Implies Feeedback Stabilization. IEEE Trans. Automat. Control42 (1997) 1394-1407. Zbl0892.93053
- F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Qualitative Properties of Control Systems: A Survey. J. Dynam. Control Systems1 (1995) 1-47. Zbl0951.49003
- J.M. Coron and L. Rosier, A Relation between Continuous Time-Varying and Discontinuous Feedback Stabilization. J. Math. Systems, Estimation and Control 4 (1994) 67-84. Zbl0925.93827
- K. Deimling, Multivalued Differential Equations, de Gruyter (1992).
- L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC (1992). Zbl0804.28001
- A.F. Filippov, Differential Equations with Discontinuous Righthandside, Kluwer Academic Publishers (1988).
- R.A. Freeman and P.V. Kokotovic, Backstepping Design with Nonsmooth Nonlinearities, IFAC NOLCOS, Tahoe City, California (1995) 483-488.
- V. Jurdjevic and J.P. Quinn, Controllability and Stability. J. Differential Equations28 (1978) 381-389.
- O. Hájek, Discontinuous Differential Equations. I, II. J. Differential Equations 32 (1979) 149-170, 171-185. Zbl0365.34017
- L. Mazzi and V. Tabasso, On Stabilization of Time-Dependent Affine Control Systems. Rend. Sem. Mat. Univ. Politec. Torino54 (1996) 53-66. Zbl0887.93055
- E.J. McShane, Integration, Princeton University Press (1947). Zbl0033.05302
- B. Paden and S. Sastry, A Calculus for Computing Filippov's Differential Inclusion with Application to the Variable Structure Control of Robot Manipulators. IEEE Trans. Circuits and Systems Cas-34 (1997) 73-81. Zbl0632.34005
- E.P. Ryan, An Integral Invariance Principle for Differential Inclusions with Applications in Adaptive Control. SIAM J. Control36 (1998) 960-980. Zbl0911.93046
- D. Shevitz and B. Paden, Lyapunov Stability Theory of Nonsmooth Systems. IEEE Trans. Automat. Control39 (1994) 1910-1914. Zbl0814.93049
- E.D. Sontag, A Lyapunov-like Characterization of Asymptotic Controllability. SIAM J. Control Optim.21 (1983) 462-471. Zbl0513.93047
- E.D. Sontag and H. Sussmann, Nonsmooth Control Lyapunov Functions, Proc. IEEE Conf. Decision and Control, New Orleans, IEEE Publications (1995) 2799-2805.

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