Almost sure exponential stability of stochastic differential delay equations

Qian Guo, Xuerong Mao, Rongxian Yue

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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
Original languageEnglish
Pages (from-to)1919-1933
Number of pages15
JournalSIAM Journal on Control and Optimization
Volume54
Issue number4
DOIs
Publication statusPublished - 27 Jul 2016

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Keywords

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

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