Hostname: page-component-74656f9884-2dmjx Total loading time: 0 Render date: 2026-11-10T06:34:51.626Z Has data issue: false hasContentIssue false
  • English
  • Français

Patchy Vector Fields and Asymptotic Stabilization

Published online by Cambridge University Press:  15 August 2002

Fabio Ancona  and
Alberto Bressan
Fabio Ancona
Affiliation:
Dipartimento di Matematica and CIRAM, Università di Bologna, piazza Porta S. Donato 5, Bologna 40127, Italy; ancona@ciram3.ing.unibo.it.
Alberto Bressan
Affiliation:
S.I.S.S.A., via Beirut 4, Trieste 34014, Italy; bressan@sissa.it.
Get access

Abstract

This paper is concerned with the structure ofasymptoticallystabilizing feedbacks for a nonlinear control system on $\mathbb{R}^n$ . We first introduce a family of discontinuous, piecewise smooth vector fieldsand derive a number of properties enjoyed bysolutions of the corresponding O.D.E's.We then define a class of “patchy feedbacks”which are obtained by patching together a locally finitefamily of smooth controls.Our main result shows that,if a system is asymptotically controllable at the origin, thenit can be stabilized by a piecewise constantpatchy feedback control.

Keywords

Asymptotic controllabilityfeedback asymptotic stabilizationdiscontinuous feedbackdiscontinuous vector fields.

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 445 - 471
Copyright
© EDP Sciences, SMAI, 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Artstein, Z., Stabilization with relaxed controls. Nonlinear Anal. 7 (1983) 1163-1173. CrossRef
A. Bacciotti, Local stabilizability of nonlinear control systems. Series on advances in mathematics for applied sciences 8, World Scientific, Singapore (1992).
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.
Clarke, F.H., Ledyaev, Yu.S., Sontag, E.D. and Subbotin, A.I., Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Control 42 (1997) 1394-1407. CrossRef
F.H. Clarke, Yu.S. Ledyaev, L. Rifford and R.J. Stern, Feedback stabilization and Lyapunov functions, to appear.
Clarke, F.H., Ledyaev, Yu.S., Stern, R.J. and Wolenski, P.R., Qualitative properties of trajectories of control systems: A survey. J. Dynamic Control Systems 1 (1995) 1-47. CrossRef
F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth analysis and control theory 178, Springer-Verlag, New York (1998).
Colombo, G., On extremal solutions of differential inclusions. Bull. Polish. Acad. Sci. 40 (1992) 97-109.
Coron, J.-M., A necessary condition for feedback stabilization. Systems Control Lett. 14 (1990) 227-232. CrossRef
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.
J.-M. Coron, Global asymptotic stabilization for controllable systems without drift. Math. of Control, Signals, and Systems 5 (1992) 295-312.
Coron, J.-M., Stabilization in finite time of locally controllable systems by means of continuous time-varying feedback laws. SIAM J. Control Optim. 33 (1995) 804-833. CrossRef
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.
A.F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Kluwer Acad. Publ. (1988).
Hájek, O., Discontinuos differential equations, I-II. J. Differential Equations 32 (1979) 149-185. CrossRef
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.
Hermes, H., On the synthesis of stabilizing feedback controls via Lie algebraic methods. SIAM J. Control Optim. 10 (1980) 352-361. CrossRef
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).
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.
Yu.S. Ledyaev and E.D. Sontag, A Lyapunov characterization of robust stabilization. J. Nonlinear Anal. to appear.
S. Nikitin, Piecewise-constant stabilization. SIAM J. Control Optim. to appear.
Ryan, E.P., Brockett's, On condition for smooth stabilizability and its necessity in a context of nonsmooth feedback. SIAM J. Control Optim. 32 (1994) 1597-1604. CrossRef
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.
Sontag, E.D., Nonlinear regulation: The piecewise linear approach. IEEE Trans. Automat. Control 26 (1981) 346-358. CrossRef
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.
E.D. Sontag, Mathematical control theory, deterministic finite dimensional systems, Springer-Verlag, New York (1990).
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.
Sussmann, H.J., Subanalytic sets and feedback control. J. Differential Equations 31 (1979) 31-52. CrossRef