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)
Finite-time stability and instability of stochastic nonlinear systems. (English) Zbl 1235.93254
Summary: This paper presents a new definition of finite-time stability for stochastic nonlinear systems. This definition involves stability in probability and finite-time attractiveness in probability. An important Lyapunov theorem on finite-time stability for stochastic nonlinear systems is established. A theorem extending the stochastic Lyapunov theorem is also proved. Moreover, an example and a lemma are presented to illustrate the scope of extension. A useful inequality, extended from Bihari’s inequality, is derived, which plays an important role in showing the Lyapunov theorem. Finally, a Lyapunov theorem on finite-time instability is proved, which states that almost surely globally asymptotical stability is not equivalent to finite-time stability for some stochastic systems. Two simulation examples are given to illustrate the theoretical analysis.

93E15Stochastic stability
93C10Nonlinear control systems
Full Text: DOI
[1] Bhat, S. P.; Bernstein, D. S.: Finite-time stability of continuous autonomous systems, SIAM journal on control and optimization 38, No. 3, 751-766 (2000) · Zbl 0945.34039 · doi:10.1137/S0363012997321358
[2] Deng, H.; Krstić, M.: Stochastic nonlinear stabilization-I: a backstepping design, Systems control letters 32, 143-150 (1997) · Zbl 0902.93049 · doi:10.1016/S0167-6911(97)00068-6
[3] Deng, H.; Krstić, M.; Williams, R. J.: Stabilization of stochastic nonlinear systems driven by noise of unknown covariance, IEEE transactions on automatic control 46, 1237-1253 (2001) · Zbl 1008.93068 · doi:10.1109/9.940927
[4] Florchinger, P.: Lyapunov-like techniques for stochastic stability, SIAM journal on control and optimization 33, 1151-1169 (1995) · Zbl 0845.93085 · doi:10.1137/S0363012993252309
[5] Has’minskii, R. Z.: Stochastic stability of differential equations, (1980)
[6] Hong, Y.; Huang, J.; Yu, Y.: On an output feedback finite-time stabilization problem, IEEE transactions on automatic control 46, 305-309 (2001) · Zbl 0992.93075 · doi:10.1109/9.905699
[7] Hong, Y.; Jiang, Z. P.; Feng, G.: Finite-time input-to-state stability and applications to finite-time control design, SIAM journal on control and optimization 48, 4395-4418 (2010) · Zbl 1210.93066 · doi:10.1137/070712043
[8] Hong, Y.; Wang, J.; Cheng, D.: Adaptive finite-time control of nonlinear systems with parametric uncertainty, IEEE transactions on automatic control 51, 858-862 (2006)
[9] Huang, X.; Lin, W.; Yang, B.: Global finite-time stabilization of a class of uncertain nonlinear systems, Automatica 41, 881-888 (2005) · Zbl 1098.93032 · doi:10.1016/j.automatica.2004.11.036
[10] Jiang, Z. P.; Mareels, I.: A small-gain control method for nonlinear cascaded systems with dynamic uncertainties, IEEE transactions on automatic control 42, 292-308 (1997) · Zbl 0869.93004 · doi:10.1109/9.557574
[11] Krstić, M.; Deng, H.: Stabilization of uncertain nonlinear systems, (1998) · Zbl 0906.93001
[12] Lipster, R. S.; Shiryayev, A. N.: Theory of martingales, (1989)
[13] Liu, S. J.; Zhang, J. F.: Output-feedback control of a class of stochastic nonlinear systems with linearly bounded unmeasurable states, International journal of robust and nonlinear control 18, 665-687 (2008) · Zbl 1284.93241
[14] Liu, Y. G.; Zhang, J. F.: Practical output-feedback risk-sensitive control for stochastic nonlinear systems with stable zero-dynamics, SIAM journal on control and optimization 45, 885-926 (2006) · Zbl 1117.93067 · doi:10.1137/S0363012903439185
[15] Liu, S. J.; Zhang, J. F.; Jiang, Z. P.: Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems, Automatica 43, 238-251 (2007) · Zbl 1115.93076 · doi:10.1016/j.automatica.2006.08.028
[16] Mao, X.: Stochastic differential equations and applications, (2008) · Zbl 1157.60061
[17] Moulay, E.; Dambrine, M.; Yeganefar, N.; Perruquetti, W.: Finite-time stability and stabilization of time-delay systems, Systems control letters 57, 561-566 (2008) · Zbl 1140.93447 · doi:10.1016/j.sysconle.2007.12.002
[18] Moulay, E.; Perruquetti, W.: Finite-time stability and stabilization of a class of continuous systems, Journal of mathematical analysis and applications 323, 1430-1443 (2006) · Zbl 1131.93043 · doi:10.1016/j.jmaa.2005.11.046
[19] Moulay, E.; Perruquetti, W.: Finite time stability conditions for non-autonomous continuous systems, International journal of control 81, 797-803 (2008) · Zbl 1152.34353 · doi:10.1080/00207170701650303
[20] Pan, Z.; Başar, T.: Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion, SIAM journal on control and optimization 37, 957-995 (1999) · Zbl 0924.93046 · doi:10.1137/S0363012996307059
[21] Situ, R.: Theory of stochastic differential equations with jumps and applications: mathematical and analysis techniques with applications to engineering, (2005) · Zbl 1070.60002
[22] Wu, Y.; Yu, X.; Man, Z.: Terminal sliding mode control design for uncertain dynamic systems, Systems control letters 34, No. 5, 281-287 (1998) · Zbl 0909.93005 · doi:10.1016/S0167-6911(98)00036-X
[23] Xie, X. J.; Duan, N.: Output tracking of high-order stochastic nonlinear systems with application to benchmark mechanical system, IEEE transactions on automatic control 55, 1197-1202 (2010)
[24] Xie, X. J.; Tian, J.: State-feedback stabilization for high-order stochastic nonlinear systems with stochastic inverse dynamics, International journal of robust and nonlinear control 17, 1343-1362 (2007) · Zbl 1127.93354 · doi:10.1002/rnc.1177
[25] Xie, X. J.; Tian, J.: Adaptive state-feedback stabilization of high-order stochastic systems with nonlinear parameterization, Automatica 45, 126-133 (2009) · Zbl 1154.93427 · doi:10.1016/j.automatica.2008.10.006
[26] Yu, X.; Man, Z.: Fast terminal sliding-mode control design for nonlinear dynamical systems, IEEE transactions on circuits and systems I 49, No. 2, 261-264 (2002)
[27] Yu, X.; Xie, X. J.: Output feedback regulation of stochastic nonlinear systems with stochastic iiss inverse dynamics, IEEE transactions on automatic control 55, 304-320 (2010)
[28] Yu, S.; Yu, X.; Shirinzadeh, B.; Man, Z.: Continuous finite-time control for robotic manipulators with terminal sliding mode, Automatica 41, No. 11, 1957-2196 (2005) · Zbl 1125.93423 · doi:10.1016/j.automatica.2005.07.001