Dynamic Analysis Of Stochastic Systems With General Decay Rate

Stochastic phenomena are ubiquitous natural phenomena in our natural world. For some systems, the accuracy is not required and some stochastic factors have not influence on the systems. In this case, the stochastic factors might be ignored and the systems could be described by deterministic mathematical models. However, in social economy and prac-tical engineering, stochastic factors have importance influence on the dynamic properties of the systems. Along with the rapid development of science and technology, the accuracy requirement of the system in engineering and technology is getting higher and higher. In or-der to describe these complex systems accurately, people should take stochastic factors into account. By introducing stochastic factors to deterministic systems, we immediately obtain the stochastic systems. Recently, the investigation of stochastic systems have received con-sideration attentions. Particularly, in1951, stochastic integration was introduced by Ito, and his work has established the basis of stochastic theory.Stability of stochastic systems as one of the important content has also been studied extensively, and obtained fruitful results. However, in real wold systems, people more con-cerned with the convergence rate of the system besides the stability of the system. It is no doubt that the exponential convergence is ideal. Actually, there are many systems which don’t converge to a stable state with exponential convergence rate. According to actual de-mand, some researchers gave the definitions of moment stability and almost sure stability with general decay, witch includes exponential stability, polynomial stability and logarith-mic stability et al. Hence, it generalizes the definition of classical stability. In practice, it has become an important index to evaluate the performance of control systems. Theoretically speaking, it provides a new perspective for analyzing systems. Particularly, we may consider the other stability with different decay rate when some methods invalid. Although there are abundant results in this field, we still need to further develop. Meanwhile, some difficulties are encountered when applying the classical methods. Hence, the study of stability theory with general decay rate has important theory meaning and practice value.As concerned above, this thesis will investigate the dynamic behaviors of some kinds stochastic systems with general decay rate, including neutral stochastic systems, stochas-tically perturbed systems and stochastic neural networks. For a class of neutral stochastic systems, under both the local Lipschitz condition and the nonlinear growth condition, we show that the solution is global and unique. Then through constructing a Lyapunov func-tional, combining with the nonnegative semi-martingale convergence theorem, some suffi- cient conditions for stability with general decay rate and moment boundedness of neutral stochastic functional differential systems.For a class of neutral stochastic delay system with Markov jump parameter, by intro-ducing a kind of ψ-function. Some criteria on the attractor with general decay rate are obtained. Furthermore, stability and robustness of neutral stochastic systems with Marko-vian switching are considered.Given a deterministic non-autonomous differential system, we introduce two indepen-dent Brownian motions and perturb this system into a new stochastic differential system. By using Lyapunov analysis method and some stochastic technique, we show that a polynomial Brownian noise can suppress the explosion and guarantee the existence, moment bounded-ness of global solution of the perturbed system. Especially, it can make the global solution grow at most polynomially. Meanwhile, another linear Brownian noise may stabilize this system with general decay rate. Moreover, the assertions could be extended to a general case.Considering non-autonomous BAM neural networks with mixed delays and stochastic disturbances, first, we introduce two new lemmas. Applying the new lemmas we studied the pth moment stability of stochastic BAM neural networks with general decay rate. Then combining with some stochastic techniques such as Chebishev inequality, Borel-Cantelli lemma, we also examined the almost sure stability with general decay of the system.For a class of discrete stochastic recurrent neural networks, we firstly generalize the dis-crete Hanalay inequality. By the new inequalities, we are concerned with discrete stochastic recurrent neural networks. Some sufficient conditions of square stability with general decay are derived. In the systems, time-delays may be bounded or unbounded.Finally, the concluding remarks are summarized, and the future works which may be further investigated are presented. In a word, the study on dynamic analysis of some kinds of stochastic systems with general decay rate provides a novel perspective for the research of many application systems, and enriches and develops the stochastic stability theory with general decay rate. Numerical examples illustrate the validity of the results and the effectiveness of the proposed methods.