Almost sure exponential stability of stochastic differential delay equations

Qian Guo, Xuerong Mao, Rongxian Yue

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

This paper is concerned with the almost sure exponential stability of the multidimensional nonlinear stochastic differential delay equation (SDDE) with variable delays of the form dx(t) = f(x(t−δ1(t)), t)dt+g(x(t−δ2(t)), t)dB(t), where δ1, δ2 : R+ → [0, τ ] stand for variable delays. We show that if the corresponding (nondelay) stochastic differential equation (SDE) dy(t) = f(y(t), t)dt + g(y(t), t)dB(t)
admits a Lyapunov function (which in particular implies the almost sure exponential stability of the SDE) then there exists a positive number τ ∗
such that the SDDE is also almost sure exponentially stable as long as the delay is bounded by τ ∗ . We provide an implicit lower bound for τ ∗ which can be computed numerically. Moreover, our new theory enables us to design stochastic delay feedback controls in order to stabilize unstable differential equations
LanguageEnglish
Pages1919-1933
Number of pages15
JournalSIAM Journal on Control and Optimization
Volume54
Issue number4
DOIs
Publication statusPublished - 27 Jul 2016

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Almost Sure Exponential Stability
Stochastic Differential Delay Equations
Asymptotic stability
Variable Delay
Differential equations
Differential equation
Stochastic Equations
Lyapunov functions
Lyapunov Function
Feedback Control
Feedback control
Unstable
Lower bound
Imply

Keywords

  • almost sure exponential stability
  • stochastic differential delay equations
  • Ito formula
  • Brownian motion
  • stochastic stabilization

Cite this

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Almost sure exponential stability of stochastic differential delay equations. / Guo, Qian; Mao, Xuerong; Yue, Rongxian.

In: SIAM Journal on Control and Optimization, Vol. 54, No. 4, 27.07.2016, p. 1919-1933.

Research output: Contribution to journalArticle

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