### Abstract

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

Language | English |
---|---|

Pages | 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

*SIAM Journal on Control and Optimization*,

*54*(4), 1919-1933. https://doi.org/10.1137/15M1019465

}

*SIAM Journal on Control and Optimization*, vol. 54, no. 4, pp. 1919-1933. https://doi.org/10.1137/15M1019465

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Almost sure exponential stability of stochastic differential delay equations

AU - Guo, Qian

AU - Mao, Xuerong

AU - Yue, Rongxian

PY - 2016/7/27

Y1 - 2016/7/27

N2 - 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

AB - 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

KW - almost sure exponential stability

KW - stochastic differential delay equations

KW - Ito formula

KW - Brownian motion

KW - stochastic stabilization

U2 - 10.1137/15M1019465

DO - 10.1137/15M1019465

M3 - Article

VL - 54

SP - 1919

EP - 1933

JO - SIAM Journal on Control and Optimization

T2 - SIAM Journal on Control and Optimization

JF - SIAM Journal on Control and Optimization

SN - 0363-0129

IS - 4

ER -