Qualitative Analysis Of Delay System With Impulsive And Stochastic Perturbations

This paper is concerned with the existence-uniqueness of solutions of stochastic functional differential equations and the asymptotic properties of solutions of impulsive differential equations with delays, impulsive stochastic functional differential equations, impulsive stochastic difference equations, respectively.In Chapter 1, some basic theories of stochastic functional differential equations of Ito-type are developed. By employing the local Lipschitz condition and Picard sequence, the local existence-uniqueness of solutions of stochastic functional differential equations of Ito-type is firstly obtained. Furthermore, a continuation theorem for stochastic functional differential equations of Ito-type is given by using stochastic analysis technique and the quasi-boundedness condition. Finally, by establishing some delay differential inequalities and using properties of H_m-functions, a stochastic version of Wintner theorem and the global existence-uniqueness of solutions of stochastic functional differential equations of Ito-type are given. The results generalize the earlier publications.In Chapter 2, the exponential stability of two kinds of impulsive differential equations with delays is considered. First, a singular impulsive differential inequality with delays is established. Then, by using this inequality and the properties of M-cone, transforming the n-dimensional impulsive neutral differential equation with delays to a 2n-dimensional singular impulsive differential equation with delays, some sufficient conditions ensuring the exponential stability of the impulsive neutral differential equation with delays are obtained. Finally, by developing an impulsive integro-differential inequality with delays, the exponential stability of impulsive Cohen-Grossberg systems with mixed delays is discussed.In Chapter 3, by establishing a L-operator inequality and using the properties of M-matrix and stochastic analysis techniques, some sufficient conditions ensuring the exponential dissipativity in mean square of impulsive stochastic functional differential equations of Ito-type are obtained. Furthermore, the results are effective for stochastic delayed differential equations.In Chapter 4, by establishing a impulsive difference inequality and using some analysis techniques, some sufficient conditions ensuring the exponential stability in mean square of impulsive stochastic difference equations are given. Moreover, a exponential convergence rate is estimated.