Exponential stability of non-linear stochastic delay differential system with generalized delay-dependent impulsive points

Highlights：

• To generalize the particular impulsive point Δx(δr)=kx(δr−) in the existing system, we proposed the nonlinear stochastic delay system with the more general impulsive point stated as Δx(δr)=Jr(t,x(δr−)), here Jr:[−δ,∞)×R→R is the continuous function such that Jr(t,x(δr−))≠−x(δr−) when x(δr−)≠0, Jr(t,0)=0.

• Comparing the results with existing literature, it is obtained that the relation between the solution of the equivalent model of SDDEs without impulses corresponding to the solution of ISDDEs. Then the conditions of the exponential stability of the proposed ISDDEs are obtained by deriving stability criteria of SDDEs without impulses.

• Numerically, the results are illustrated with the exponential stability of SDDEs by using the MRKM method and justified with corresponding ISDDEs. This shows that the proposed theory gives no need to construct a numerical scheme for ISDDEs.

摘要

•Based on the motivation of impulsive dynamical systems, the present work is focused on the exponential stability of the nonlinear Impulsive Stochastic Delay Differential Equations (ISDDEs) obtained by using the Modified Runge-Kutta-Maruyama (MRKM) method.•To generalize the particular impulsive point Δx(δr)=kx(δr−) in the existing system, we proposed the nonlinear stochastic delay system with the more general impulsive point stated as Δx(δr)=Jr(t,x(δr−)), here Jr:[−δ,∞)×R→R is the continuous function such that Jr(t,x(δr−))≠−x(δr−) when x(δr−)≠0, Jr(t,0)=0.•Comparing the results with existing literature, it is obtained that the relation between the solution of the equivalent model of SDDEs without impulses corresponding to the solution of ISDDEs. Then the conditions of the exponential stability of the proposed ISDDEs are obtained by deriving stability criteria of SDDEs without impulses.•Numerically, the results are illustrated with the exponential stability of SDDEs by using the MRKM method and justified with corresponding ISDDEs. This shows that the proposed theory gives no need to construct a numerical scheme for ISDDEs.