Existence, continuation, and uniqueness problems of stochastic impulsive systems with time delay

Alwan, Mohamad S.; Liu, Xinzhi; Xie, Wei‐Chau

doi:10.1016/j.jfranklin.2010.06.005

Cited by 52 publications

(23 citation statements)

References 10 publications

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“…As a standing hypothesis, we assume that, for any

ξ \in P C_{{scriptF}_{t_{0}}}^{b} ([- τ, 0]; {double-struckR}^{n})

, there exists a unique solution process, denoted by x ( t ; t 0 , ξ ), to system (see ). For stability analysis, we further assume that f ( t ,0,⋯,0) ≡ 0, g ( t ,0,⋯,0) ≡ 0, I k ( t k ,0) ≡ 0 for all

t \geq t_{0}

and

k \in double-struckN

, then system admits a trivial solution x ( t ) ≡ 0.…”

Section: Preliminariesmentioning

confidence: 99%

“…is regarded as a PC([− , 0]; R n )-valued stochastic process; initial data , impulsive perturbation I k and impulse sequence {t k } k∈N are the same as in system (3). It is assumed that, for any ∈ PC b  t 0 ([− , 0]; R n ), there exists a unique solution process to system (19) (see [21]), which is denoted by x(t; t 0 , ). Theorem 2.…”

Section: Aii Conditions For Stability Of Sfdswiementioning

confidence: 99%

See 1 more Smart Citation

Exponential Stability Analysis for Stochastic Delayed Differential Systems with Impulsive Effects: Average Impulsive Interval Approach

Yao

Cao

Li³

et al. 2016

Asian Journal of Control

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This paper is concerned with the exponential stability analysis of stochastic delayed systems with impulsive effects. By using the average impulsive interval approach, and together with comparison lemma and Razumikhin techniques, sufficient conditions ensuring the moment exponential stability of the systems under consideration are established. A stability criterion for non‐delayed stochastic systems with impulses is also derived as a corollary. Compared with the existing stability results in the literature, which are usually based on the supremum or infimum of impulsive intervals, the results reported in this paper are less conservative. Two illustrative examples are provided to validate the effectiveness and advantages of our theoretical results.

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“…As a standing hypothesis, we assume that, for any

ξ \in P C_{{scriptF}_{t_{0}}}^{b} ([- τ, 0]; {double-struckR}^{n})

t \geq t_{0}

and

k \in double-struckN

, then system admits a trivial solution x ( t ) ≡ 0.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Aii Conditions For Stability Of Sfdswiementioning

confidence: 99%

Exponential Stability Analysis for Stochastic Delayed Differential Systems with Impulsive Effects: Average Impulsive Interval Approach

Yao

Cao

Li³

et al. 2016

Asian Journal of Control

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show abstract

“…One may refer to [21,22] for the results on the existence and uniqueness of solutions of impulsive stochastic systems. The solution x(t) = (x 1 (t), .…”

Section: Model and Preliminariesmentioning

confidence: 99%

Impulsive stabilization of stochastic differential equations with time delays

2013

Mathematical and Computer Modelling

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a b s t r a c tIn this paper, the impulsive stabilization of stochastic differential equations with time delays is investigated. By establishing a delay differential inequality and using the stochastic analysis technique, some sufficient conditions for mean square exponential stability are obtained. It is shown that an unstable stochastic delay system can be successfully stabilized by impulses. An example is also discussed to illustrate the efficiency of the obtained results.

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“…Many interesting results in dealing with impulsive stochastic delay differential systems can be found in previous works. () These results touch on many aspects such as stability, invariance, attractivity, controllability, existence, and uniqueness of the impulsive stochastic delay differential systems. However, the corresponding theory for the impulsive stochastic delay difference systems has not been fully developed, especially on the boundedness.…”

Section: Introductionmentioning

confidence: 99%

Exponential ultimate boundedness of impulsive stochastic delay difference systems

2017

Intl J Robust & Nonlinear

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Summary This paper is concerned with the exponential ultimate boundedness problems for the impulsive stochastic delay difference systems. Several sufficient conditions on the global pth moment exponential ultimate boundedness are presented by using the Lyapunov methods and the algebraic inequality techniques, and the estimated exponential convergence rate and the ultimate bound are provided as well. As an application, the boundedness criteria are applied to a class of discrete impulsive stochastic neural networks with delays. The obtained results show that the impulses not only can stabilize an unstable stochastic difference delay system but also can make an unbounded stochastic difference delay system into a bounded system. Examples and simulations are also provided to demonstrate the effectiveness of the derived theoretical results.

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Existence, continuation, and uniqueness problems of stochastic impulsive systems with time delay

Cited by 52 publications

References 10 publications

Exponential Stability Analysis for Stochastic Delayed Differential Systems with Impulsive Effects: Average Impulsive Interval Approach

Exponential Stability Analysis for Stochastic Delayed Differential Systems with Impulsive Effects: Average Impulsive Interval Approach

Impulsive stabilization of stochastic differential equations with time delays

Exponential ultimate boundedness of impulsive stochastic delay difference systems

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