The Stability Of Nonlinear Neutral Stochastic Differential Equations

A stochastic model is needed for an actual system in order to study its stability and dynamics better. However, despite the development of the stability theory of the stochastic differential equation since it was introduced by ?Ito, the theory is far from perfect, especially the insufficiency of the study in the stability theory of nonlinear neutral stochastic system with time-varying delay. In recent years, Luo Jiaowan studied the stability of the linear neutral stochastic system with time-varying delay by the contraction mapping principle, which made this method gradually catching people’s attention. But the usage of the inequalities of linear matrix is still in the booming development.This paper is based on the above two methods. On the one hand, this paper studied the stability of two kinds of nonlinear neutral stochastic system with time-varying delay by the contraction mapping principle. On the other hand, this paper studied the robust stability of one kind of nonlinear neutral stochastic system with time-varying delay by the inequalities of linear matrix.This paper consists of five parts. Firstly, we briefly introduce the background of this paper. Secondly, we elaborate the related knowledge of this paper. Thirdly,we studied the asympototic p stability and exponential p stability of one kind of nonlinear neutral stochastic system with time-varying delay. By constructing an appropriate contraction operator, we got the sufficient conditions for this type of stochastic system to be asympototic p stable and exponential p stable. Fourthly, also by using the contraction mapping principle, we studied the asympototic p stability and exponential p stability of one kind of nonlinear neutral stochastic system with mixed time delays and we got the sufficient conditions for this type of stochastic system to be asympototic p stable and exponential p stable. At last, we gave the appropriate examples to illustrate our result. Fifthly, we studied the robust stability of one kind of nonlinear neutral stochastic system with time-varying delay. By using the Lyapunov stability theory combining the techniques of the LMI, we got the stable conditions for this type of system.