Stability In Terms Of Two Measures For Impulsive Stochastic Functional Differential Systems With Applications

Stochastic perturbation is inevitable in any real system. In order to describe the real systems more exactly and thereby design better controllers, it is necessary to take stochas-tic perturbation fully into account during systems modeling. Besides, both impulses and delays are commonly encountered in practice, and they often coexist in certain system resulting in an impulsive system with delays. If considering the effects of stochastic perturbation on impulsive systems with delays, one should naturally resort to impul-sive stochastic differential equations with delays or, more general, impulsive stochastic functional differential equations. In this dissertation, based on the Lyapunov stability theory, the functional differential equations theory as well as the Ito stochastic calculus, the stability in terms of two measures for general impulsive stochastic functional differen-tial systems (ISFDSs) are investigated systematically by employing piecewise continuous Lyapunov functions, the comparison principles and the Razumikhin technique. Also, the results obtained are applied to impulsive stabilization of stochastic functional differential systems and stability analysis of impulsive stochastic neural networks with mixed delays (ISNNMDs).The main contributions of this dissertation are summarized as follows:1. First, the background, research significance and current situation of the selected topic are reviewed. Then, the basic idea and research progress of the two methods used in this dissertation, the comparison principle and the Razumikhin technique, are introduced briefly.2. The comparison principles are generalized to ISFDSs for the first time. Then, by using the comparison principles established and piecewise continuous Lyapunov functions, the stability, asymptotical stability and instability in terms of two measures for ISFDSs are investigated. Also, the results derived are applied to several special cases such as impulsive stochastic ordinary differential systems, ISFDSs with infinite delays (ISFDSs-I) and stochastic functional differential systems (SFDSs) without impulses.3. By using the comparison principles and piecewise continuous Lyapunov func-tions, the exponential stability, globally exponential stability, exponential instability and exponential divergence in terms of two measures for ISFDSs are investigated. The sta-bility results obtained can be applied to ISFDSs-I. Besides, weaker conditions ensuring the globally exponential stability of ISFDSs-F are given, which allow us to include more functions as Lyapunov functions. 4. Based on the comparison principles as well as the asymptotical and exponential stability of a scalar impulsive delayed system, sufficient conditions ensuring that SFDSs can be mean square asymptotically and exponentially stabilized by impulses are obtained. Moreover, the specific design methods of impulsive controllers are proposed.5. Utilizing the Lyapunov-Razumikhin technique, the stability and asymptotical stability in terms of two measures for ISFDSs are investigated. The results obtained can be applied to both ISFDSs-F and ISFDSs-I.6. Utilizing the Lyapunov-Razumikhin technique and the Gronwall-Bellman inequal-ity, the exponential stability in terms of two measures for ISFDSs are investigated, and the results obtained can also be applied to both ISFDSs-F and ISFDSs-I. As a special case, impulsive stochastic systems with mixed delays are discussed in detail.7. Applying some inequalities and the stability results derived before, the sufficient conditions ensuring the pth moment asymptotical stability, pth moment exponential sta-bility and pth moment globally exponential stability of ISNNMDs are established.Finally, the main results of the dissertation are concluded and some issues for future research are proposed.