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Zbl 0996.60064
Mao, Xuerong
A note on the LaSalle-type theorems for stochastic differential delay equations. (English)
[J] J. Math. Anal. Appl. 268, No.1, 125-142 (2002). ISSN 0022-247X

The author presents an improvement of results obtained in an earlier paper [J. Math. Anal. Appl. 236, No. 2, 350-369 (1999; Zbl 0958.60057)]. In the article $n$-dimensional stochastic differential delay equations are considered, which are of the form $$dx(t)=f(x(t),x(t-\tau),t) dt + g(x(t),x(t-\tau),t) dB(t),\tag 1$$ where $B(t)$ denotes $m$-dimensional Brownian motion. The main theorem is a stochastic version of the LaSalle theorem, providing criteria for the determination of the almost sure asymptotic behaviour of the solution of (1). The improvement concerns the assumptions on the coefficient functions. The local Lipschitz and local linear growth conditions on $f$ and $g$ are relaxed to local boundedness in the first two arguments and uniform boundedness in the last argument of $f$ and $g$, in addition the existence and uniqueness of a solution of (1) is required. The results can thus be applied to a larger class of equations. The proof of the theorem, some corollaries and an extension to the multiple delay case are given. Several examples are presented, demonstrating the usefulness of the results.
[Evelyn Buckwar (Berlin)]

MSC 2000:: *60H10 Stochastic ordinary differential equations
34K50 Stochastic delay equations
93D05 Lyapunov and other classical stabilities of control systems

Keywords: stochastic delay differential equations; stochastic stability; Lyapunov stability; LaSalle theorem

Citations: Zbl 0958.60057

Cited in: Zbl 1082.60055

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