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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
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 

Pages (fromto)  19191933 
Number of pages  15 
Journal  SIAM Journal on Control and Optimization 
Volume  54 
Issue number  4 
DOIs  
Publication status  Published  27 Jul 2016 
Keywords
 almost sure exponential stability
 stochastic differential delay equations
 Ito formula
 Brownian motion
 stochastic stabilization
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Profiles
Projects

Numerical Analysis of Stochastic Differential Equations: New Challenges
1/10/15 → 30/09/17
Project: Research Fellowship

Epsrc Doctoral Training Grant
EPSRC (Engineering and Physical Sciences Research Council)
1/10/12 → 30/09/16
Project: Research  Studentship