Almost Sure Exponential Stability of Stochastic Differential Delay Equations

نویسندگان

  • Qian Guo
  • Xuerong Mao
  • Rong-Xian Yue
چکیده

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.

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عنوان ژورنال:
  • SIAM J. Control and Optimization

دوره 54  شماره 

صفحات  -

تاریخ انتشار 2016