Patchy vector fields and asymptotic stabilization

Fabio Ancona; Alberto Bressan

ESAIM: Control, Optimisation and Calculus of Variations (1999)

  • Volume: 4, page 445-471
  • ISSN: 1292-8119

How to cite


Ancona, Fabio, and Bressan, Alberto. "Patchy vector fields and asymptotic stabilization." ESAIM: Control, Optimisation and Calculus of Variations 4 (1999): 445-471. <>.

author = {Ancona, Fabio, Bressan, Alberto},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {asymptotic controllability; feedback asymptotic stabilization; discontinuous feedback; discontinuous vector fields},
language = {eng},
pages = {445-471},
publisher = {EDP Sciences},
title = {Patchy vector fields and asymptotic stabilization},
url = {},
volume = {4},
year = {1999},

AU - Ancona, Fabio
AU - Bressan, Alberto
TI - Patchy vector fields and asymptotic stabilization
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 1999
PB - EDP Sciences
VL - 4
SP - 445
EP - 471
LA - eng
KW - asymptotic controllability; feedback asymptotic stabilization; discontinuous feedback; discontinuous vector fields
UR -
ER -


  1. [1] Z. Artstein, Stabilization with relaxed controls. Nonlinear Anal. 7 ( 1983) 1163-1173. Zbl0525.93053MR721403
  2. [2] A. BacciottiLocal stabilizability of nonlinear control systems. Series on advances in mathematics for applied sciences 8, World Scientific, Singapore ( 1992). Zbl0757.93061MR1148363
  3. [3] R.W. Brockett, Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory, R.W. Brockett, R.S. Millman and H.J. Sussmann, Eds., Birkhauser, Boston ( 1983) 181-191. Zbl0528.93051MR708502
  4. [4] F.H. Clarke, Yu.S. Ledyaev, E.D. Sontag and A.I. Subbotin, Asymptotic controllability implies feedback stabilization . IEEE Trans. Automat. Control 42 ( 1997 ) 1394-1407. Zbl0892.93053MR1472857
  5. [5] F.H. Clarke, Yu.S. Ledyaev, L. Rifford and R.J. Stern, Feedback stabilization and Lyapunov functions , to appear. Zbl0961.93047MR1780907
  6. [6] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Qualitative properties of trajectories of control systems: A survey . J. Dynamic Control Systems 1 ( 1995) 1-47. Zbl0951.49003MR1319056
  7. [7] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth analysis and control theory 178, Springer-Verlag, New York ( 1998). Zbl1047.49500MR1488695
  8. [8] G. Colombo, On extremal solutions of differential inclusions. Bull. Polish. Acad. Sci. 40 ( 1992) 97-109. Zbl0771.34017MR1401862
  9. [9] J.-M. Coron, A necessary condition for feedback stabilization . Systems Control Lett. 14 ( 1990) 227-232. Zbl0699.93075MR1049357
  10. [10] 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. Zbl0925.93827MR1298548
  11. [11] J.-M. Coron, Global asymptotic stabilization for controllable systems without drift . Math. of Control, Signals, and Systems 5 ( 1992) 295-312. Zbl0760.93067MR1164379
  12. [12] J.-M. Coron, Stabilization in finite time of locally controllable systems by means of continuous time-varying feedback laws . SIAM J. Control Optim. 33 ( 1995) 804-833. Zbl0828.93054MR1327239
  13. [13] J.-M. Coron, L. Praly and A. Teel, Feedback stabilization of nonlinear systems: sufficient conditions and Lyapunov and input-output techniques, in Trends in Control: A European Perspective, A. Isidori, Eds., Springer, London ( 1995) 293-348. MR1448452
  14. [14] A.F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Kluwer Acad. Publ. ( 1988). Zbl0664.34001
  15. [15] O. Hájek, Discontinuos differential equations, I-II. J. Differential Equations 32 ( 1979) 149-185. Zbl0365.34017MR534546
  16. [16] H. Hermes, Discontinuous vector fields and feedback control, in Differential Equations and Dynamical Systems, J.K. Hale and J.P. La Salle, Eds., Academic Press, New York, ( 1967) 155-165. Zbl0183.15905MR222424
  17. [17] H. Hermes, On the synthesis of stabilizing feedback controls via Lie algebraic methods. SIAM J. Control Optim. 10 ( 1980) 352-361. Zbl0477.93046MR579546
  18. [18] N.N. Krasovskii and A.I. Subbotin, Positional differential games, Nauka, Moscow, ( 1974) [in Russian]. Revised English translation: Game-theoretical control problems, Springer-Verlag, New York ( 1988). Zbl0298.90067MR437107
  19. [19] Yu.S. Ledyaev and E.D. Sontag, A remark on robust stabilization of general asymptotically controllable systems, in Proc. Conf. on Information Sciences and Systems (CISS 97), Johns Hopkins, Baltimore, MD ( 1997) 246-251. 
  20. [20] Yu.S. Ledyaev and E.D. Sontag, A Lyapunov characterization of robust stabilization. J. Nonlinear Anal, to appear. Zbl0947.34054MR1695080
  21. [21] S. Nikitin, Piecewise-constant stabilization. SIAM J. Control Optim. to appear. Zbl0922.93043MR1680818
  22. [22] E.P. Ryan, On Brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback. SIAM J. Control Optim. 32 ( 1994) 1597-1604. Zbl0806.93049MR1297100
  23. [23] E.D. Sontag and H.J. Sussmann, Remarks on continuous feedback, in Proc. IEEE Conf. Decision and Control, Aulbuquerque, IEEE Publications, Piscataway ( 1980) 916-921. 
  24. [24] E.D. SontagNonlinear regulation: The piecewise linear approach . IEEE Trans. Automat. Control 26 ( 1981) 346-358. Zbl0474.93039MR613541
  25. [25] E.D. Sontag, Feedback stabilization of nonlinear systems, in Robust Control of Linear Systems and Nonlinear Control, M.A. Kaashoek, J.H. van Shuppen and A.C.M. Ran, Eds., Birkhäuser, Cambridge, MA ( 1990) 61-81. Zbl0735.93063MR1115377
  26. [26] E.D. Sontag, Mathematical control theory, deterministic finite dimensional systems, Springer-Verlag, New York ( 1990). Zbl0703.93001MR1070569
  27. [27] E.D. Sontag, Stability and stabilization: Discontinuities and the effect of disturbances, in Proc. NATO Advanced Study Institute - Nonlinear Analysis, Differential Equations, and Control (Montreal, Jul/Aug 1998), F.H. Clarke and R.J. Stern, Eds., Kluwer ( 1999) 551-598. Zbl0937.93034MR1695014
  28. [28] H.J. Sussmann, Subanalytic sets and feedback control . J. Differential Equations 31 ( 1979) 31-52. Zbl0407.93010MR524816

Citations in EuDML Documents

  1. Fabio Ancona, Alberto Bressan, Stability rates for patchy vector fields
  2. Fabio Ancona, Alberto Bressan, Stability rates for patchy vector fields
  3. Rafal Goebel, Andrew R. Teel, Direct design of robustly asymptotically stabilizing hybrid feedback
  4. Ludovic Rifford, Stratified semiconcave control-Lyapunov functions and the stabilization problem
  5. Francis H. Clarke, Ludovic Rifford, R. J. Stern, Feedback in state constrained optimal control
  6. Fabio Ancona, Alberto Bressan, Nearly time optimal stabilizing patchy feedbacks
  7. Francis H. Clarke, Ludovic Rifford, R. J. Stern, Feedback in state constrained optimal control

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