# Patchy Vector Fields and Asymptotic Stabilization

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

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

## Access Full Article

top## Abstract

top## How to cite

topAncona, Fabio, and Bressan, Alberto. "Patchy Vector Fields and Asymptotic Stabilization." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 445-471. <http://eudml.org/doc/116552>.

@article{Ancona2010,

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

author = {Ancona, Fabio, Bressan, Alberto},

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

keywords = {
Asymptotic controllability;
feedback asymptotic stabilization;
discontinuous feedback; discontinuous vector fields.
; asymptotic controllability; feedback asymptotic stabilization; discontinuous feedback; discontinuous vector fields},

language = {eng},

month = {3},

pages = {445-471},

publisher = {EDP Sciences},

title = {Patchy Vector Fields and Asymptotic Stabilization},

url = {http://eudml.org/doc/116552},

volume = {4},

year = {2010},

}

TY - JOUR

AU - Ancona, Fabio

AU - Bressan, Alberto

TI - Patchy Vector Fields and Asymptotic Stabilization

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 4

SP - 445

EP - 471

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

LA - eng

KW -
Asymptotic controllability;
feedback asymptotic stabilization;
discontinuous feedback; discontinuous vector fields.
; asymptotic controllability; feedback asymptotic stabilization; discontinuous feedback; discontinuous vector fields

UR - http://eudml.org/doc/116552

ER -

## References

top- Z. Artstein, Stabilization with relaxed controls. Nonlinear Anal.7 (1983) 1163-1173. Zbl0525.93053
- 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. Zbl0528.93051
- F.H. Clarke, Yu.S. Ledyaev, E.D. Sontag and A.I. Subbotin, Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Control42 (1997) 1394-1407. Zbl0892.93053
- F.H. Clarke, Yu.S. Ledyaev, L. Rifford and R.J. Stern, Feedback stabilization and Lyapunov functions, to appear. Zbl0961.93047
- 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 Systems1 (1995) 1-47. Zbl0951.49003
- F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth analysis and control theory 178, Springer-Verlag, New York (1998).
- G. Colombo, On extremal solutions of differential inclusions. Bull. Polish. Acad. Sci.40 (1992) 97-109. Zbl0771.34017
- J.-M. Coron, A necessary condition for feedback stabilization. Systems Control Lett.14 (1990) 227-232. Zbl0699.93075
- J.-M. Coron and L. Rosier, A relation between continuous time-varying and discontinuous feedback stabilization. J. Math. Systems, Estimation, and Control4 (1994) 67-84. Zbl0925.93827
- J.-M. Coron, Global asymptotic stabilization for controllable systems without drift. Math. of Control, Signals, and Systems5 (1992) 295-312. Zbl0760.93067
- 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. Zbl0828.93054
- 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). Zbl0664.34001
- O. Hájek, Discontinuos differential equations, I-II. J. Differential Equations32 (1979) 149-185.
- 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. Zbl0183.15905
- H. Hermes, On the synthesis of stabilizing feedback controls via Lie algebraic methods. SIAM J. Control Optim.10 (1980) 352-361. Zbl0477.93046
- 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). Zbl0298.90067
- 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. Zbl0947.34054
- S. Nikitin, Piecewise-constant stabilization. SIAM J. Control Optim. to appear. Zbl0922.93043
- 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. Zbl0806.93049
- 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.
- E.D. Sontag, Nonlinear regulation: The piecewise linear approach. IEEE Trans. Automat. Control26 (1981) 346-358. Zbl0474.93039
- 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. Zbl0735.93063
- E.D. Sontag, Mathematical control theory, deterministic finite dimensional systems, Springer-Verlag, New York (1990). Zbl0703.93001
- 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. Zbl0937.93034
- H.J. Sussmann, Subanalytic sets and feedback control. J. Differential Equations31 (1979) 31-52. Zbl0407.93010

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.