zbMATH — the first resource for mathematics

Stochastic versions of the LaSalle theorem. (English) Zbl 0921.34057
The famous LaSalle theorem [see J. R. LaSalle, J. Differ. Equations 4, 57-65 (1968; Zbl 0159.12002)] for locating limit sets of nonautonomous systems is generalized to the case of ordinary stochastic differential equations driven by \(m\)-dimensional Brownian motion under the hypothesis of local Lipschitz continuity and at most linear-polynomial growth of drift and diffusion parts.
The proofs are carried out by a lemma due to R. Sh. Liptser and A. N. Shiryaev [ Martingale theory. Vyp. 38. Moskva: ‘Nauka’. (1986; Zbl 0654.60035)] on asymptotics of semimartingales, the well-known Kolmogorov-Centsov continuity theorem [I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus. Graduate Texts in Mathematics, 113. New York etc.: Springer-Verlag. (1991; Zbl 0734.60060; 1988; Zbl 0638.60065)], Burkholder type estimates and the uniform continuity of stochastic \(L^p\)-integrable martingales, and of course, by the help of Ito’s formula.
Thus, attracting deterministic sets, an estimation of exponential and polynomial growth rates can be found for stochastic differential equations. A series of examples illustrates the immense power of stochastic versions of LaSalle’s theorem and striking importance of that author’s work.

34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G35 Signal detection and filtering (aspects of stochastic processes)
93E15 Stochastic stability in control theory
Full Text: DOI
[1] Arnold, L., Stochastic differential equations: theory and applications, (1972), Wiley New York
[2] Friedman, A., Stochastic differential equations and their applications, (1976), Academic Press San Diego
[3] Hale, J.K.; Lunel, S.M.V., Introduction to functional differential equations, (1993), Springer-Verlag Berlin/New York
[4] Has’minskii, R.Z., Stochastic stability of differential equations, (1981), Sijthoff and Noordhoff Rockville · Zbl 0276.60059
[5] Ikeda, N.; Watanabe, S., Stochastic differential equations and diffusion processes, (1981), North-Holland Amsterdam · Zbl 0495.60005
[6] Karatzas, I.; Shreve, S.E., Brownian motion and stochastic calculus, (1991), Springer-Verlag Berlin/New York · Zbl 0734.60060
[7] Kushner, H.J., Stochastic stability and control, (1967), Academic Press San Diego · Zbl 0178.20003
[8] Kolmanovskii, V.B.; Myshkis, A., Applied theory of functional differential equations, (1992), Kluwer Academic Dordrecht/Norwell
[9] Ladde, G.S.; Lakshmikantham, V., Random differential inequalities, (1980), Academic Press San Diego
[10] LaSalle, J.P., Stability theory of ordinary differential equations, J. differential equations, 4, 57-65, (1968) · Zbl 0159.12002
[11] Liptser, R.Sh.; Shiryayev, A.N., Theory of martingales, (1989), Kluwer Academic · Zbl 0728.60048
[12] Mao, X., Stability of stochastic differential equations with respect to semimartingales, (1991), Longman Harlow/New York · Zbl 0724.60059
[13] Mao, X., Exponential stability of stochastic differential equations, (1994), Dekker New York · Zbl 0851.93074
[14] Mao, X., Stochastic differential equations and applications, (1997), Horwood Chichester · Zbl 0874.60050
[15] Mohammed, S.-E.A., Stochastic functional differential equations, (1986), Longman Harlow/New York · Zbl 0584.60066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.