The Stability Of Stochastic Differential Systems Without Linear Growth Condition And Noise Suppress Explosive Solution Of Differential Systems

The theory of stochastic systems has been developed rapidly since the introduc-tion of stochastic integration by Ito in 1951. As an issue of general academic in-terest, the stability of stochastic differential systems has significant meaning in the theory and application area. Generally speaking, most of the classical stability results of stochastic systems require the coefficients to satisfy linear growth condition and Lipschitz condition(including global Lipschitz condition). The assumptions of linear growth condition are very strict in practice, which are impossible for many important stochastic models. When using a linear function to simulate the growth rate or rela-tive growth rate of some variables, the corresponding models can be simplized, and many instruments can be used for such models, but more errors will appear. When us-ing non-linear function, such as relatively simple polynomial function, to simulate the growth rate or relative growth rate of some variables, the corresponding models may approximate to the realistic situation more in some respects. However, the forms of those models may be more complex. Compared with the linear case, it becomes more difficult to find a suitable Lyapunov function and the systematic algebraic tools will not be suit for analyzing their properties. Whether applying Lyapunov direct methods or Laslle invariance principle, the construction of Lyapunov functions or Lyapunov functional is necessary in the study of the stability of stochastic systems. Moreover it is difficult to construct an appropriate Lyapunov functions or Lyapunov functional for the systems with coefficients not satisfying the linear growth condition. When the coefficients do not obey the linear growth condition and a suitable Lyapunov func-tion or Lyapunov is lack, we assume that the diffusion and drift coefficients can be controlled by corresponding polynomial growth conditions. There are academic and application significance in the research of the stability of such stochastic systems by applying some novel techniques to estimate and discuss the polynomial parameters.At the same time, more and more scholars are interested in the study of the in-fluence of dynamic behaviors that caused by the noise to a deterministic system solu-tions. It is well known that noise can be used to stabilise a given unstable system or to make a system even more stable when it is already stable. Most of classical results require the coefficients of the determinate systems to satisfy linear growth condition (including one side linear growth condition). In fact, many important models raised in engineering do not obey the linear growth condition, such as, van der Pol's system, Duffing's systems and Lotka-Volterra systems. When the coefficients do not obey the linear growth condition, the systems may explode in a finite time. It is of great aca-demic significance and practical value for the investigation of suppression explosive solutions by noise for nonlinear deterministic differential system with coefficients not satisfying linear growth conditions.The dissertation focuses on the investigation of stability for several stochastic systems and the problem of suppression explosive solutions by noise for nonlinear deterministic differential system. On the assumption that the local Lipschitz condi-tion is satisfied, for asymptotic stability and exponential stability of different kinds of stochastic differential systems, this dissertation further assumes that the growth of the drifting coefficients and diffusion coefficients are controlled by the corresponding polynomial condition. In virtue of the convergence theorem of nonnegative semi-martingales, Kolmogorov-Centson theorem, stochastic integral inequality and Bar-balat lemma, together with novel techniques, especially a positive definite criterion for a class of high order polynomial, some sufficient criteria are established which en-sure the asymptotic stability and the stability for such SDSs, SDSwMSs and SDDSs. By using the exponential martingale inequality and the Borel-Cantelli lemma, this dissertation also investigates the problem of suppression explosive solutions by noise for nonlinear deterministic differential system with coefficients satisfying a more gen-eral one-sided polynomial growth condition. The main work of this dissertation is as follows:Firstly, this dissertation aims at the stability of SDSs, SDSwMSs and SDDSs with coefficients not satisfying the linear growth condition. The polynomial growth con-ditions imposed on different kinds of stochastic systems can guarantee the existence and uniqueness of the global solution, the uniform boundedness and continuity of pth-order moment for corresponding systems. By applying some novel techniques to estimate and discuss the polynomial parameters, and combining with some stochastic analysis techniques, sufficient criteria on exponential stability and asymptotic stability are established. The criteria are expressed in the parameters of polynomial conditions, which can be easily verified. Based on these criteria, the range of the order p for moment asymptotic stability and moment exponential stability, and the Lyapunov ex-ponent are presented.Secondly, the problem of suppression explosive solutions by noise for nonlinear deterministic differential system is also investigated. Given a deterministic differential system with coefficients satisfying more general one-sided polynomial growth condi-tions, we introduce Brownian noise feedback and therefore stochastically perturb this system into the nonlinear stochastic differential system. We show that appropriate noise guarantee that this stochastic system exists as a unique global solution although the corresponding deterministic systems may explode in a finite time. Under some weaker conditions, we reveal that the single noise can also make almost every path of the solution of corresponding stochastically perturbed system grow at most polynomi-ally.Finally, the concluding remarks are summarized, and the future works which may be further invested are presented. Overall, the dissertation explores the expo-nential and asymptotic stabilities of several kinds of stochastic differential systems, discusses the influence on one kind of deterministic system caused by Brownian noise, and extends the research methods of stochastic differential system. Numerical examples illustrate the validity of the results and the effectiveness of the proposed methods.