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 finite-time stabilization of a class of uncertain nonlinear systems. (English) Zbl 1098.93032
Summary: This paper studies the problem of finite-time stabilization for nonlinear systems. We prove that global finite-time stabilizability of uncertain nonlinear systems that are dominated by a lower-triangular system can be achieved by Hölder continuous state feedback. The proof is based on the finite-time Lyapunov stability theorem and the nonsmooth feedback design method developed recently for the control of inherently nonlinear systems that cannot be dealt with by any smooth feedback. A recursive design algorithm is developed for the construction of a Hölder continuous, global finite-time stabilizer as well as a $C^1$ positive definite and proper Lyapunov function that guarantees finite-time stability.

93D15Stabilization of systems by feedback
93C41Control problems with incomplete information
Full Text: DOI
[1] Athans, M.; Falb, P. L.: Optimal control: an introduction to theory and its applications. (1966) · Zbl 0196.46303
[2] Bacciotti, A.: Local stabilizability of nonlinear control systems. (1992) · Zbl 0757.93061
[3] Bhat, S. P., & Bernstein, D. S. (1997). Finite-time stability of homogeneous systems. In Proceedings of the American control conference (pp. 2513-2514).
[4] Bhat, S. P.; Bernstein, D. S.: Continuous finite-time stabilization of the translational and rotational double integrators. IEEE transactions on automatic control 43, 678-682 (1998) · Zbl 0925.93821
[5] Bhat, S. P.; Bernstein, D. S.: Finite-time stability of continuous autonomous systems. SIAM journal on control and optimization 38, 751-766 (2000) · Zbl 0945.34039
[6] Coron, J. M.; Praly, L.: Adding an integrator for the stabilization problem. System and control letters 17, 89-104 (1991) · Zbl 0747.93072
[7] Dayawansa, W. (1992). Recent advances in the stabilization problem for low dimensional systems. In Proceedings of the second IFAC symposium on nonlinear control systems and design, Bordeaux (pp. 1-8). · Zbl 0976.93509
[8] Dayawansa, W.; Martin, C.; Knowles, G.: Asymptotic stabilization of a class of smooth two dimensional systems. SIAM journal on control and optimization 28, 1321-1349 (1990) · Zbl 0731.93076
[9] Freeman, R.; Kokotovic, P.: Robust nonlinear control design: state-space and Lyapunov techniques. (1996) · Zbl 0863.93075
[10] Haimo, V. T.: Finite-time controllers. SIAM journal on control and optimization 24, 760-770 (1986) · Zbl 0603.93005
[11] Hermes, H.: Homogeneous coordinates and continuous asymptotically stabilizing feedback controls. Differential equations stability and control, 249-260 (1991)
[12] Hong, Y.: Finite-time stabilization and stabilizability of a class of controllable systems. System & control letters 46, 231-236 (2002) · Zbl 0994.93049
[13] Hong, Y.; Huang, J.; Xu, Y.: On an output feedback finite-time stabilization problem. IEEE transactions on automatic control 46, 305-309 (2001) · Zbl 0992.93075
[14] Kawski, M.: Stabilization of nonlinear systems in the plane. System & control letters 12, 169-175 (1989) · Zbl 0666.93103
[15] Kawski, M.: Homogeneous stabilizing feedback laws. Control theory and advanced technology 6, 497-516 (1990)
[16] Praly, L.; Andrea-Novel, B.; Coron, J.: Lyapunov design of stabilizing controllers for cascaded systems. IEEE transactions on automatic control 36, 1177-1181 (1991)
[17] Qian, C.; Lin, W.: A continuous feedback approach to global strong stabilization of nonlinear systems. IEEE transactions on automatic control 46, 1061-1079 (2001) · Zbl 1012.93053
[18] Qian, C.; Lin, W.: Non-Lipschitz continuous stabilizer for nonlinear systems with uncontrollable unstable linearization. System & control letters 42, 185-200 (2001)
[19] Rosier, L.: Homogeneous Lyapunov function for homogeneous continuous vector field. System & control letters 19, 467-473 (1992) · Zbl 0762.34032
[20] Ryan, E. P.: Singular optimal controls for second-order saturating system. International journal of control 36, 549-564 (1979) · Zbl 0422.49006