Stochastic stabilisation of functional differential equations

Appleby, John A. D.; Mao, Xuerong

doi:10.1016/j.sysconle.2005.03.003

Cited by 86 publications

(61 citation statements)

References 6 publications

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“…In other words, we have showed that the assertion (3.8) holds as well when β < 2(α ∧ γ). 2 We can now prove Theorem 3.1. Proof.…”

Section: Lemma 32 Let α β and γ Be Three Real Numbers And Definementioning

confidence: 88%

“…The pioneering work in this area was due to Hasminskii [12, p.229], who stabilised an unstable system by using two white noise sources, and his work opened a new chapter in the study of stochastic stabilisation. There is an extensive literature concerned with the stabilisation by noise and we here mention [1,2,3,4,5,7,9,10,14,17,21,24,25]. It is now well known that noise can be used to stabilise a given unstable system or to make a system even more stable when it is already stable.…”

Section: Introductionmentioning

confidence: 99%

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Noise expresses exponential growth under regime switching

Liu

Mao

et al. 2009

Systems & Control Letters

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Consider a given system under regime switching whose solution grows at most polynomially, and suppose that the system is subject to environmental noise in some regimes. Can the regime switching and the environmental noise work together to make the system change significantly? The answer is yes. In this paper, we will show that the regime switching and the environmental noise will make the original system whose solution grows at most polynomially become a new system whose solution will grow exponentially. In other words, we reveal that the regime switching and the environmental noise will exppress the exponential growth.

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“…In other words, we have showed that the assertion (3.8) holds as well when β < 2(α ∧ γ). 2 We can now prove Theorem 3.1. Proof.…”

Section: Lemma 32 Let α β and γ Be Three Real Numbers And Definementioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Noise expresses exponential growth under regime switching

Liu

Mao

et al. 2009

Systems & Control Letters

Self Cite

View full text Add to dashboard Cite

show abstract

“…For this special case results on stability are obtained in e.g. [1]. It is shown there that noise may have a stabilizing effect, a result not obtained in the references mentioned above on stability of general stochastic evolutions.…”

Section: Introductionmentioning

confidence: 87%

Pathwise Stability of Degenerate Stochastic Evolutions

Bierkens

2010

Integr. Equ. Oper. Theory

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“…Thus, stability is not sensitive in this case to small delays. It is also worth mentioning that as a corollary of more general results for nonlinear stochastic systems, Appleby and Mao [1] showed that if r > 0 is such a small number that…”

Section: Dy(t) = αY(t)dt + βY(t − R )Dt + σ Y(t)dw(t)mentioning

confidence: 92%

Sensitivity to Small Delays of Pathwise Stability for Stochastic Retarded Evolution Equations

2017

J Theor Probab

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In this paper, we shall study the almost sure pathwise exponential stability property for a class of stochastic functional differential equations with delays, possibly, in the highest-order derivative terms driven by multiplicative noise. Instead of establishing a moment exponential stability as the first step and then proceeding to investigate the pathwise stability of the system under consideration, we shall develop a direct approach for this problem. As a consequence, we can show that some systems, which are not exponential momently stable, have the exponential stability not sensitive to small delays in the almost sure sense.

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Stochastic stabilisation of functional differential equations

Cited by 86 publications

References 6 publications

Noise expresses exponential growth under regime switching

Noise expresses exponential growth under regime switching

Pathwise Stability of Degenerate Stochastic Evolutions

Sensitivity to Small Delays of Pathwise Stability for Stochastic Retarded Evolution Equations

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