Retraction Note: Existence of weak solutions of stochastic delay differential systems with Schrödinger–Brownian motions

This paper is concerned with stabilization of impulsive stochastic delay differential systems. Based on the Razumikhin techniques and Lyapunov functions, several criteria onpth moment and almost sure exponential stability are established. Our results show that stochastic functional differential systems may be exponentially stabilized by impulses.

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Boundedness of Stochastic Delay Differential Systems with Impulsive Control and Impulsive Disturbance

Mathematical Problems in Engineering ◽

10.1155/2015/707069 ◽

2015 ◽

Vol 2015 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Liming Wang ◽

Baoqing Yang ◽

Xiaohua Ding ◽

Kai-Ning Wu

Keyword(s):

Stochastic Analysis ◽

Impulsive Control ◽

Sufficient Conditions ◽

Differential Systems ◽

Stochastic Delay ◽

Analysis Techniques ◽

Moment Boundedness ◽

Delay Differential System ◽

Delay Differential Systems ◽

Delay Differential

This paper considers thep-moment boundedness of nonlinear impulsive stochastic delay differential systems (ISDDSs). Using the Lyapunov-Razumikhin method and stochastic analysis techniques, we obtain sufficient conditions which guarantee thep-moment boundedness of ISDDSs. Two cases are considered, one is that the stochastic delay differential system (SDDS) may not be bounded, and how an impulsive strategy should be taken to make the SDDS be bounded. The other is that the SDDS is bounded, and an impulsive disturbance appears in this SDDS, then what restrictions on the impulsive disturbance should be adopted to maintain the boundedness of the SDDS. Our results provide sufficient criteria for these two cases. At last, two examples are given to illustrate the correctness of our results.

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Exponential Stability of Impulsive Stochastic Delay Differential Systems

Discrete Dynamics in Nature and Society ◽

10.1155/2012/296136 ◽

2012 ◽

Vol 2012 ◽

pp. 1-15 ◽

Cited By ~ 4

Author(s):

Xiaotai Wu ◽

Litan Yan ◽

Wenbing Zhang ◽

Liang Chen

Keyword(s):

Exponential Stability ◽

The Other ◽

Stability Criteria ◽

Differential Systems ◽

Stochastic Delay ◽

Almost Sure Exponential Stability ◽

Delay Differential Systems ◽

The Stability ◽

Delay Differential ◽

Average Impulsive Interval

This paper investigates the stability of stochastic delay differential systems with two kinds of impulses, that is, destabilizing impulses and stabilizing impulses. Both thepth moment and almost sure exponential stability criteria are established by using the average impulsive interval. When the impulses are regarded as disturbances, a lower bound of average impulsive interval is obtained; it means that the impulses should not happen too frequently. On the other hand, when the impulses are used to stabilize the system, an upper bound of average impulsive interval is derived; namely, enough impulses are needed to stabilize the system. The effectiveness of the proposed results is illustrated by two examples.

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