zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Global asymptotic stabilization for controllable systems without drift. (English) Zbl 0760.93067
Summary: This paper proves that the accessibility rank condition on $\bbfR\sp n\backslash\{0\}$ is sufficient to guarantee the existence of a global smooth time-varying (but periodic) feedback stabilizer, for systems without drift. This implies a general result on the smooth stabilization of nonholonomic mechanical systems, which are generically not smoothly stabilizable using time-invariant feedback.

93D20Asymptotic stability of control systems
Full Text: DOI
[1] R. W. Brockett, Asymptotic stability and feedback stabilization, inDifferential Geometric Control Theory (R. W. Brockett, R. S. Millman, and H. J. Sussmann, eds.), Birkhäuser, Basel, 1983. · Zbl 0528.93051
[2] G. Campion, B. d’Andréa-Novel, and G. Bastin, Modelling and state feedback control of nonholonomic mechanical systems, preprint, LAAS Louvain la Neuve, ENSMP Fontainebleau, 1990.
[3] J.-M. Coron, Linearized control systems and applications to smooth stabilization, preprint, Université Paris-Sud, 1992.
[4] J.-M. Coron and B. d’Andréa-Novel, Smooth stabilizing time-varying control laws for a class of nonlinear systems. Application to mobile robots, preprint, Université Paris-Sud and ENSMP Fontainebleau, 1991.
[5] J.-M. Coron and J.-B. Pomet, A remark on the design of time-varying stabilizing feedback laws for controllable systems without drift, preprint, Université Paris-Sud and ECNantes, 1991.
[6] J. Dieudonné,Foundations of Modern Analysis, Academic Press, New York, 1960.
[7] M. Gromov,Partial Differential Relations, Ergebnisse Mathematik 3, Folge 9, Springer-Verlag, Berlin, 1986. · Zbl 0651.53001
[8] A. Ilchmann, I. Nürnberger, and W. Schmale, Time-varying polynomial matrix systems,Internat. J. Control,40 (1984), 329-362. · Zbl 0545.93045 · doi:10.1080/00207178408933278
[9] J. Kurzweil, On the inversion of Lyapunov’s second theorem on stability of motion,Ann. Math. Soc. Transl., Ser. 2,24 (1956), 19-77.
[10] J.-B. Pomet, Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift,Systems Control Lett., to appear. · Zbl 0744.93084
[11] C. Samson, Velocity and torque feedback control of a wheeled mobile robot: Stability analysis, preprint, INRIA, Sophia-Antipolis, 1990.
[12] R. Sepulchre, private communication, 1991.
[13] L. M. Silverman and H. E. Meadows, Controllability and observability in time variable linear systems,SIAM J. Control,5 (1967), 64-73. · Zbl 0163.11001 · doi:10.1137/0305005
[14] E. D. Sontag, Finite-dimensional open-loop control generators for non-linear systems,Internat. J. Control,47 (1988), 537-556. · Zbl 0641.93035 · doi:10.1080/00207178808906030
[15] E. D. Sontag, Feedback stabilization of nonlinear systems, inRobust Control of Linear Systems and Nonlinear Control (M. A. Kaashoek, J. H. van Schuppen, and A. C. M. Ran, eds.), Birkhäuser, Cambridge, MA, 1990, pp. 61-81. · Zbl 0735.93063
[16] E. D. Sontag,Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer-Verlag, New York, 1990. · Zbl 0703.93001
[17] E. D. Sontag and H. J. Sussmann, Remarks on continuous feedback, inProceedings of the 19th IEEE Conference on Decision and Control, Albuquerque, NM, 1980, pp. 916-921.
[18] J. Tsinias, Sufficient Lyapunov-like conditions for stabilization,Math. Control Signals Systems,2 (1989), 343-357. · Zbl 0688.93048 · doi:10.1007/BF02551276