Stability For Stochastic Delays Differential Systems With Nonlinear Perturbation

Stochastic delay differential equations arise from many physical, mechanical, biological, economical, financial, chemical system. In these system. Often appears nonlinear disturbance, which tend to change the system’s dynamic properties. Based on this, it has important theoretical and practical significance value to study stochastic system with nonlinear perturbation and delays.In this paper, some sufficient conditions on stability of stochastic differential equations with nonlinear perturbation and delays were given by constructing Lyapunov-krasovskii func-tional and using Ito formula, integral inequality, linear matrix inequality and ticcati matrix equation, based on stability theory for stochastic differential systems with time delay.Firstly, the asymptotically mean-square stability for a kind of stochastic differential sys-tems with nonlinear perturbation and time-varying delay was concerned. By establishing a Lyapunov-Krasovskii functional, using the Ito formula and by virtue of Lyapunov stability theory, some sufficient conditions for the asymptotically mean-square stability of the system were obtained in terms of the matrix Riccati equation. Finally, the numerical example was provided to demonstrate the effectiveness of the result received.Secondly, relevanting theoretical knowledge of riccati matrix equation, the mean square asvmptotical stability for a stochastic differential svstem with nonlinear disturbances and with discrete and distributed delays. By constructing Lyapnov-krasovskii functional, using Ito formula, some sufficient conditions were presented to ensure the mean square asymptotical stability for the system. It is easier to constructed Lyapunov-Krasovskii functional by translate the system into neutral stochastic delay differential equations.Thirdly, the robust mean square stability for stochastic differential systems with multiple time-varying delays and nonlinear perturbation in memory state feedback controller was dis-cussed. By establishing a Lyapunov-Krasovskii functional, using the Ito formula, introducing appropriate free-weighting matrices, making use of integral inequality and analytical technique and based on the linear matrix inequality (LMI) and Schur complement theorem, the robust mean square asymptotically stability and the robust mean square exponentially stability for the system were obtained. In addition, the corresponding state feedback controllers were con-structed. The results are dependent on delays and stochastic perturbation, which has enriched the existing results.