Numerical Method And Stability For Delay Systems With Impulsive And Stochastic Effects

This paper deals with the convergence and stability of numerical method for impul-sive delay differential equations and stochastic delay integro-differential equations and thestability for impulsive stochastic delay differential equations. Impulsive delay differentialequations have considerable applications to many fields such as biology, control scienceand physics and so on. It is difficult to obtain the explicit expressions of the solutionsfor the impulsive delay differential equations, hence developing the numerical methodsand discussing the properties of the numerical solutions possess great significance to thetheory and practice. In addition, if we take the noise of the environment into account, wemust deal with the impulsive stochastic delay differential equations.This paper first presents the background of some applications and surveys the devel-opment of impulsive delay differential equations and impulsive delay difference equationsand impulsive stochastic delay differential equations and their numerical methods, espe-cially focuses on the stability for these kinds of equations.The stability is considered for the trivial solutions of impulsive delay differenceequations by Lyapunov functions. The exponential stability is studied by Razumikhintechnique, the sufficient conditions are given to guarantee the exponential stability. Thecriterion is presented for the stabilization of delay difference systems by impulse. Somesufficient conditions are obtained for mean stability and p- moment exponential stabilityof the trivial solutions.The fixed stepsize Euler scheme is presented for a kind of linear impulsive delaydifferential equation. The convergence of Euler scheme is considered and it is provedthe Euler method is of order 1. Using the results we got for the stability of impulsivedelay difference equations, we deal with the stability of the Euler method and get somesufficient conditions for exponential stability of the Euler scheme.For a kind of linear stochastic delay integro-differential equation, we consider theconvergence and stability of the semi-Euler method. The convergence order for semi-Euler method is 0.5. Some sufficient conditions are obtained which guarantee the meansquare asymptotic stability of the numerical solutions. The p-moment exponential stability and almost surely exponential stability of triv-ial solutions are considered for the impulsive stochastic delay differential equations byLyapunov-Razumikhin method. The sufficient conditions are given for the p-momentexponential stability of trivial solution and the relationship is presented between thep-moment exponential stability and almost surely exponential stability. The stabiliza-tion is studied for stochastic delay differential equations by impulse, and some sufficientconditions are obtained to guarantee the p?moment exponential stability of trivial solu-tions.