zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Lyapunov function for nonuniform in time global asymptotic stability in probability with application to feedback stabilization. (English) Zbl 1229.93159
The aim of this paper is to extend the well known Artstein-Sontag theorem to the concept of stochastic control Lyapunov function when nonuniform in time stochastic systems are considered. A stabilizer for a wider class of SDE is designed. The main tools used here are the stochastic Lyapunov theorem proved by Khasminsskii and La Salle’s invariance theorem.
MSC:
93E15Stochastic stability
60H10Stochastic ordinary differential equations
93C10Nonlinear control systems
93D05Lyapunov and other classical stabilities of control systems
93D15Stabilization of systems by feedback
93D21Adaptive or robust stabilization
References:
[1]Abedi, F., Abu Hassan, M., Suleiman, M.: Feedback stabilization and adaptive stabilization of stochastic nonlinear systems by the control Lyapunov function. Stochastics: Int. J. Prob. Stochast. Proc. 83(2), 179–201 (2011) · Zbl 1222.60039 · doi:10.1080/17442508.2011.552723
[2]Artstein, Z.: Stabilization with relaxed controls. Nonlinear Anal. 7, 1163–1173 (1983) · Zbl 0525.93053 · doi:10.1016/0362-546X(83)90049-4
[3]Deng, H., Krstic, M., Williams, J.R.: Stabilization of stochastic nonlinear systems driven by noise of unknown covariance. Trans. Autom. Control, 46(8), 1237–1253 (2001) · Zbl 1008.93068 · doi:10.1109/9.940927
[4]Florchinger, P.: A stochastic Jurdjevic-Quinn theorem. SIAM J. Control Optim. 41, 83–88 (2002) · Zbl 1014.60062 · doi:10.1137/S0363012900370788
[5]Florchinger, P.: Lyapunov-like techniques for stochastic stability. SIAM J. Control Optim. 33, 1151–1169 (1995) · doi:10.1137/S0363012993252309
[6]Florchinger, P.: A universal formula for the stabilization of control stochastic differential equations. Stoch. Anal. Appl., 11(2), 155–162 (1993) · Zbl 0770.60058 · doi:10.1080/07362999308809308
[7]Gao, Y.Z., Ahmed, U.N.: Feedback stabilizability of nonlinear stochastic systems with state dependent noise. Int. J. Control 45, 729–737 (1987) · Zbl 0618.93068 · doi:10.1080/00207178708933764
[8]Handel, V.R.: Almost global stochastic stability. SIAM J. Control Optim. 45, 1297–1313 (2006) · Zbl 1124.34033 · doi:10.1137/040618850
[9]Jurdjevic, V., Quinn, P.J.: Controllability and stability. J. Differ. Equ. 28, 381–389 (1978) · Zbl 0417.93012 · doi:10.1016/0022-0396(78)90135-3
[10]Khasminskii, F.R.: On the stability of the trajectories of Markov processes. J. Appl. Math. Mech. 26, 1554–1565 (1962) · Zbl 0137.35806 · doi:10.1016/0021-8928(62)90192-2
[11]Khasminskii, Z.R.: Stochastic Stability of Differential Equation. Sijthoff Noordhoff, Alphen aan den Rijn (1980)
[12]Krstic, M., Deng, H.: Stabilization of Uncertain Nonlinear Systems. Springer, New York (1998)
[13]Korzweil, J.: On the inversion of Lyapunov’s second theorem on stability of motion. In: American Mathematical Society Translations, Series 2, vol. 24, pp. 19–77 (1956)
[14]Kushner, J.H.: Converse theorems for stochastic Lyapunov functions. SIAM J. Control Optim. 5, 228–233 (1967) · Zbl 0183.19401 · doi:10.1137/0305015
[15]Kushner, J.H.: Stochastic stability, in stability of stochastic dynamical systems. In: Curtain, R. (ed.) Lecture Notes in Math., vol. 294, pp. 97–124. Springer, Berlin (1972)
[16]Kushner, J.H.: On the stability of stochastic dynamical systems. Proc. Natl. Acad. Sci. USA 53, 8–12 (1965) · doi:10.1073/pnas.53.1.8
[17]Rogers, G.C.L., Williams, D.: Diffusions, Markov Processes and Martingales, 2nd ed., vol. 1. Wiley, New York (1994)
[18]Sontag, D.E.: A universal construction of Artstein’s theorem on nonlinear stabilization. Syst. Control Lett. 13, 117–123 (1989) · Zbl 0684.93063 · doi:10.1016/0167-6911(89)90028-5
[19]Speyer, L.J., Chung, H.W.: Stochastic Process, Estimation and Control. SIAM, Philadelphia (2008)