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)
Geometric homogeneity with applications to finite-time stability. (English) Zbl 1110.34033
The authors study properties of homogeneous systems in a geometric, coordinate-free setting. A key contribution of this paper is a result relating regularity properties of a homogeneous function to its degree of homogeneity and the local behavior of the dilation near the origin. This result makes it possible to extend previous results on homogeneous systems to the geometric framework. As an application of their results, the authors consider finite-time stability of homogeneous systems. The main result that links homogeneity and finite-time stability is that a homogeneous system is finite-time stable if and only if it is asymptotically stable and has a negative degree of homogeneity.

34D20Stability of ODE
34A26Geometric methods in differential equations
Full Text: DOI
[1] Bacciotti A, Rosier L (2001) Liapunov functions and stability in control theory. Springer-Verlag, London · Zbl 0968.93004
[2] Bhat SP, Bernstein DS (1995) Lyapunov analysis of finite-time differential equations. In: Proceedings American control conference, Seattle, pp 1831--1832
[3] Bhat SP, Bernstein DS (1997) Finite-time stability of homogeneous systems. In: Proceedings of American control conference, Albuquerque, pp 2513--2514
[4] Bhat SP, Bernstein DS (1998) Continuous, finite-time stabilization of the translational and rotational double integrators. IEEE Trans Automatic Control 43:678--682 · Zbl 0925.93821
[5] Bhat SP, Bernstein DS (2000) Finite-time stability of continuous autonomous systems. SIAM J Control Optim 38:751--766 · Zbl 0945.34039
[6] Bhatia NP, Hajek O (1969) Local semi-dynamical systems. Lecture Notes in Mathematics, vol 90. Springer, Berlin Heidelberg New York · Zbl 0176.39102
[7] Bhatia NP, Szegö GP (1970) Stability theory of dynamical systems. Springer, Berlin Heidelberg New York
[8] Coleman C (1960) Asymptotic stability in 3-space. In: Cesari L, LaSalle J, Lefschetz S (eds) Contributions to the theory of nonlinear oscillations, vol V. Princeton University Press, pp 257--268 · Zbl 0095.28905
[9] Dayawansa WP, Martin CF (1989) Some sufficient conditions for the asymptotic stabilizability of three dimensional homogeneous polynomial systems. In: Proceedings of conference on decision and control. Tampa, pp 1366--1369
[10] Filippov AF (1988) Differential equations with discontinuous righthand sides. Kluwer, Dordrecht
[11] Flett TM (1980) Differential analysis. Cambridge University Press, Cambridge
[12] Gavrilyako VM, Korobov VI, Skylar GM (1986) Designing a bounded control of dynamic systems in entire space with the aid of controllability function. Automation Remote Control 47:1484--1490 · Zbl 0616.93026
[13] Hahn W (1967) Stability of motion. Springer, Berlin Heidelberg New York · Zbl 0189.38503
[14] Haimo VT (1986) Finite time controllers. SIAM J Control Optim. 24:760--770 · Zbl 0603.93005
[15] Hartman P (1982) Ordinary differential equations. 2nd edn. Birkhäuser, Boston · Zbl 0476.34002
[16] Hermes H (1991) Homogeneous coordinates and continuous asymptotically stabilizing feedback controls. In: Elaydi S (ed) Differential equations, stability and control. Lecture Notes in Pure and Applied Mathematics, vol 127. Marcel Dekker, New York, pp 249--260
[17] Hermes H (1991) Nilpotent and high-order approximations of vector field systems. SIAM Rev 33:238--264 · Zbl 0733.93062
[18] Hermes H (1992) Vector field approximations; flow homogeneity. In: Wiener J, Hale JK (eds) Ordinary and delay differential equations. Longman Sci Tech, pp 80--89 · Zbl 0793.34033
[19] Hermes H (1995) Homogeneous feedback controls for homogeneous systems. Syst Control Lett 24:7--11 · Zbl 0877.93088
[20] Hong Y (2002) Finite-time stabilization and stabilizability of a class of controllable systems. Systems Control Lett 46:231--236 · Zbl 0994.93049
[21] Hong Y, Huang J, Xu Y (2001) On an output feedback finite-time stabilization problem. IEEE Trans Automatic Control 46:305--309 · Zbl 0992.93075
[22] Horn RA, Johnson CR (1985) Matrix analysis. Cambridge University Press, Cambridge
[23] Iggidr A, Outbib R (1995) Feedback stabilization of homogeneous polynomial systems. In: Preprint, IFAC Symposium on nonlinear control systems design, Lake Tahoe · Zbl 0938.93053
[24] Kartsatos AG (1980) Advanced ordinary differential equations. Mariner Tampa · Zbl 0495.34001
[25] Kawski M (1988) Controllability, approximations and stabilization. In: Bowers K, Lund J (eds) Computation and control, vol 1. Birkhäuser, Boston, pp 155--167 · Zbl 0696.93009
[26] Kawski M (1988) Stabilizability and nilpotent approximations. In: Proceedings of conference on decision and control, Austin, pp 1244--1248
[27] Kawski M (1989) Homogeneous feedback laws in dimension three. In: Proceedings of conference on decision and control, Tampa, pp 1370--1375
[28] Kawski M (1989) Stabilization of nonlinear systems in the plane. Syst Control Lett 12:169--175 · Zbl 0666.93103
[29] Kawski M (1990) Homogeneous stabilizing feedback laws. Control-Theory Adv Technol 6:497--516
[30] Kawski M (1991) Families of dilations and asymptotic stability. In: Bonnard B, Bride B, Gauthier JP, Kupka I (eds) Analysis of controlled dynamical systems, progress in systems and control theory, vol 8. Birkhäuser, Boston, pp 285--294 · Zbl 0798.93050
[31] Kawski M (1999) Geometric homogeneity and stabilization. In: Krener A, Mayne D (eds) IFAC postprint volumes series. Elsevier. Amsterdam · Zbl 1092.93519
[32] Khalil HK (1996) Nonlinear systems. 2nd edn. Prentice-Hall, Upper Saddle River
[33] Komarov VA (1984) Design of constrained controls for linear systems. Automation Remote Control 45:1280--1286 · Zbl 0575.93023
[34] M’Closkey RT, Murray RM (1997) Exponential stabilization of driftless nonlinear control systems using homogeneous feedback. IEEE Trans Automatic Control 42:614--628
[35] Munkres JR (1975) Topology a first course. Prentice-Hall, Englewood Cliffs · Zbl 0306.54001
[36] Praly L (1997) Generalized weighted homogeneity and state dependent time scale for controllable systems. In: Proceedings of conference on decision and control, San Diego pp 4342--4347
[37] Rosier L (1992) Homogeneous Lyapunov function for homogeneous continuous vector field. Syst Control Lett 19:467--473 · Zbl 0762.34032
[38] Rosier L (1993) Etude de quelques problèmes de stabilisation. PhD Thesis, Ecole Normale Supérieure de Cachan, France
[39] Rouche N, Habets P, Laloy M (1977) Stability theory by Liapunov’s direct method. Springer, Berlin Heidelberg New York · Zbl 0364.34022
[40] Ryan EP (1982) Optimal relay and saturating control system synthesis. IEE control engineering series, vol 14. Peter Peregrinus Ltd · Zbl 0505.93001
[41] Ryan EP (1991) Finite-time stabilization of uncertain nonlinear planar systems. Dyn Control 1:83--94 · Zbl 0742.93068
[42] Ryan EP (1995) Universal stabilization of a class of nonlinear systems with homogeneous vector fields. Syst Control Lett 26:177--184
[43] Sepulchre R, Aeyels D (1995) Adding an integrator to a non-stabilizable homogeneous planar system. In: Preprint, IFAC symposium on nonlinear control systems design, Lake Tahoe
[44] Sepulchre R, Aeyels D (1996) Homogeneous Lyapunov functions and necessary conditions for stabilization. Math Control Signals Syst 9:34--58 · Zbl 0881.93063
[45] Sepulchre R, Aeyels D (1996) Stabilizability does not imply homogeneous stabilizability for controllable homogeneous systems. SIAM J Control Optimization 34:1798--1813 · Zbl 0879.93038
[46] Sontag ED (1990) Mathematical control theory, vol 6 of texts in applied mathematics. Springer, Berlin Heidelberg New York · Zbl 0703.93001
[47] Yoshizawa T (1966) Stability theory by Liapunov’s second method. The Mathematical Society of Japan · Zbl 0144.10802