### 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

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 language | English |
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Pages (from-to) | 1919-1933 |

Number of pages | 15 |

Journal | SIAM Journal on Control and Optimization |

Volume | 54 |

Issue number | 4 |

DOIs | |

Publication status | Published - 27 Jul 2016 |

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### Keywords

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

### Cite this

Guo, Q., Mao, X., & Yue, R. (2016). Almost sure exponential stability of stochastic differential delay equations.

*SIAM Journal on Control and Optimization*,*54*(4), 1919-1933. https://doi.org/10.1137/15M1019465