Research On Stability And Stabilization Of Impulsive Stochastic Differential Systems

This doctoral dissertation is devoted to the stability analysis and stabilization of several im-pulsive stochastic systems, such as impulsive stochastic differential systems, impulsive stochas-tic delay differential systems, impulsive stochastic functional differential systems. Based on the theory of Ito's stochastic calculus and Lyapunov stability theory, this dissertation studies the stability of impulsive stochastic functional differential systems, stability and state-feedback stabilization of impulsive stochastic delay differential systems and noise stabilization of impul-sive differential systems by means of Razumikhin techniques, Lyapunov-Krasovskii functional method combing the free-weighting matrix approach and some stochastic analysis techniques. Some important theoretical and practical results are obtained.The main contributions are summarized as follows:1. A introduction to the background and significance of impulsive stochastic differen-tial systems is given. Also, the latest progress in the stability and stabilization of stochastic differential systems, impulsive differential systems as well as impulsive stochastic differential systems are presented.2. The asymptotic stability of impulsive stochastic functional differential systems is inves-tigated by extending the Razumikhin-type techniques and some criteria on the pth moment uni-formly stable, pth moment uniformly asymptotically stable and globally pth moment uniformly asymptotically stable of impulsive stochastic functional differential systems are obtained. Ap-plying the criteria obtained, sufficient conditions ensuring the globally pth moment uniformly asymptotically stable of impulsive stochastic delay differential systems are also proposed.3. The stability and instability of impulsive stochastic functional differential systems with delayed impulses are concerned. Based on the Razumikhin-type techniques, Lyapunov func-tion method and some stochastic analysis techniques, some pth moment exponential stability and instability criteria are established; and an almost exponential stability criterion is also ob-tained by using the Burkholder-Davis-Gundy inequality, Borel-Cantelli Lemma. Furthermore, these results are applied to impulsive stochastic functional differential systems and impulsive stochastic differential delay systems. Compared with some recent works, there are two main contributions in our works. On the one hand, the state variables on the impulses that we add are related to the present state variables as well as former state variables, which is more general and more realistic. On the other hand, both the cases that the impulses effect as perturbations and controllers are considered and the results show that the system will be stable if the impulses' frequency and amplitude are suitably related to the increasement or decreasement of the con-tinuous flows. Part of our results are also new if the system is specialized to the case that the state variables on the impulses are only related to the present state variables. 4. The problem of delay-dependent stability of impulsive stochastic systems with time-varying delay is investigated by applying the Lyapunov-Krasovskii functional method combing the free-weighting matrix approach, some delay-dependent mean square exponential stability criteria are derived in terms of linear matrix inequalities. At the same time, the exponential convergence rate is estimated, which depends on system parameters and impulsive effects. In our criteria, we don't assume that both the continuous systems and the discrete systems are stable at the same. Our results are also effective for the special case that the systems without stochastic effects, and some results are less conservative than the existing ones.5. The robust stability and delayed-state-feedback stabilization of uncertain impulsive stochastic systems with time-varying delay are considered by using the Razumikhin-type tech-niques. The parametric uncertainties are assumed to be time-varying and norm-bounded, and the state variables on the impulses are assumed related to the present state variables as well as former state variables. Three classes of impulsive stochastic systems with time-varying delay are considered:the systems with stable/stabilizable continuous dynamics and unsta-ble/unstabilizable discrete dynamics, the systems with unstable/unstabilizable continuous dy-namics and stable/stabilizable discrete dynamics, and the systems where both the continuous dynamics and the discrete-time dynamics are stable/stabilizable. For each class, we first estab-lish the robust mean square exponential stability criterion in terms of linear matrix inequalities. And then based on these stability criteria, the robust delayed-state-feedback controllers are derived.6. Stabilization of impulsive differential systems by noise is investigated. Firstly, some al-most surely exponential stability and instability criteria for general nonlinear impulsive stochas-tic systems are established based on Lyapunov function method, Borel-Cantelli and martingale exponential inequalities. Then several criteria on stabilization and destabilization of impulsive differential systems are derived under the assumption that the coefficient of impulsive differen-tial systems satisfies the one-side linear growth condition.Finally, after giving the summary of this dissertation, some problems are proposed for further research.